Particle Kinetics of Gas-Solid Particle Mixtures Citation Haas, Roger Allison (1969) Particle Kinetics of Gas-Solid Particle Mixtures. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BT3R-BW68. https://resolver.caltech.edu/CaltechTHESIS:03272017-145900980 Abstract In this thesis the interaction of a normal gas dynamic shock wave with a gas containing a distribution of small solid spherical particles of two distinct radii, σ 1 and σ 2 , is studied (1) to demonstrate that the methods of kinetic theory can be extended to treat solid particle collision phenomena in multidimensional gas-particle flows; (2) to elucidate some of the essential physical characteristics associated with particle-particle collision processes; and (3) to give some indication regarding the importance of particle collisions in particle-laden gas flows. It is assumed that upstream of the shock wave particles σ 1 are uniformly distributed while particles σ 2 are non-uniformly distributed parallel to the shock face and in much smaller numbers than particles σ 1 . Under these conditions the gas-particle σ 1 flow downstream of the shock wave is very nearly one-dimensional and independent of the presence of particles σ 2 . The usual shock relaxation zone is established by the interaction of particles σ and the gas downstream of the shock wave. The collisional model pro- posed by Marble 3 is then extended and used with a modified form of the mean free path method of kinetic theory to calculate the macroscopic distribution and velocity of particles σ 2 as determined by the particle σ 1 - particle σ 2 and particle σ 2 -gas interactions. Within the condition that the random velocity imparted to a particle σ 2 by a collision is damped by its viscous interaction with the gas before it suffers another collision, the kinetic theory method established here may be extended to include more general particle-particle and particle-gas interaction laws than those used by Marble. However, the collisional model employed is particularly important because the criteria for its application are easy to establish and because it admits a wide class of physically interesting situations. Within the restrictions of this collision model, it is possible to analyze the macroscopic motion of particles σ 2 in three important limiting cases: (σ 2 /σ 1 ) 2 >> ⊥,(σ 2 /σ 1 ) 2 << ⊥ and (σ 2 /σ 1 ) 2 ~ ⊥. It is found that when (σ 2 /σ 1 ) 2 >> ⊥ there is essentially no redistribution of particles σ 2 normal to the gas flow. The only effect of particle σ 1 -particle σ 2 encounters is a drag force acting to slow down particles σ 2 . When (σ 2 /σ 1 ) 2 << ⊥ it is found that particles σ 2 . may have many collisions during their passage through the shock relaxation zone. As a consequence there may be a substantial redistribution of particles σ 2 downstream of the shock wave. The physical features of this process are studied in detail together with the range of validity of this diffusion model. The case (σ 2 /σ 1 ) 2 ~ ⊥ is analyzed under the condition particles σ 2 have at most one collision during their passage through the shock relaxation zone. It is found that when the gas or particle σ 1 density is low, the single collision effects may be important even when σ 2 /σ 1 differs significantly from unity and the particles are not very small. Under most conditions of practical significance, because there is invariably a distribution of particles sizes present in a dusty gas, the calculation of the particle distribution in the shock relaxation zone should account for the effects of particle-particle encounters. It is suggested that an experimental observation of particle size distribution in a shock relaxation zone can yield significant information on particle-particle and particle-gas interaction laws. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Engineering Science and Physics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Engineering Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Marble, Frank E. Thesis Committee: Unknown, Unknown Defense Date: 26 May 1969 Funders: Funding Agency Grant Number Caltech UNSPECIFIED Robert O. Law Foundation UNSPECIFIED Florence and Daniel Guggenheim Foundation UNSPECIFIED Alfred P. Sloan Foundation UNSPECIFIED Ford Foundation UNSPECIFIED NSF UNSPECIFIED Record Number: CaltechTHESIS:03272017-145900980 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:03272017-145900980 DOI: 10.7907/BT3R-BW68 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10109 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 28 Mar 2017 15:06 Last Modified: 29 Apr 2024 21:19 Thesis Files PDF - Final Version See Usage Policy. 56MB Repository Staff Only: item control page

PARTICLE KINETICS OF GAS - SOLID PARTICLE MIXTURES Thesis by Roger Allison Haas In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1969 (Submitted May 26, 1969) DEDICATED TO MY FATHER Mr. Allison Franklin Haas 11 ACKNOWLEDGMENTS It is a pleasure to express my deep gratitude to Professor Frank E. Marble for suggesting this problem and for his advice, guidance, and encouragement during the course of this study. I am also very grateful to Mrs. Virginia Conner and Mrs. Roberta Duffy for their excellent work in preparing this manuscript. I also wish to thank Mr. Frank T. Linton for helping prepare the figures. Financial support from the California Institute of Technology, the Robert O. Law Foundation, the Florence and Daniel Guggenheim Foundation, the Alfred P. Sloan Foundation, the Ford Foundation, and the National Science Foundation is gratefully acknowledged. Finally, I am particularly grateful to my wife, Donna, for her continuing understanding, cooperation, and encouragement. iii ABSTRACT In this thesis the interaction of a normal gas dynamic shock wave with a gas containing a distribution of small solid spherical particles of two distinct radii, <lj and oz, is studied (1) to demonstrate that the methods of kinetic theory can be extended to treat solid particle collision phenomena in multidimensional gas-particle flows; (2) to elucidate some of the essential physical characteristics associated with particle-particle collision processes; and (3) to give some indication regarding the importance of particle collisions in particleladen gas flows. It is assumed that upstream of the shock wave particles Oj are uniformly distributed while particles oz. are nonuniformly distributed parallel to the shock face and in much smaller numbers than particles Of. Oj Under these conditions the gas-particle flow downstream of the shock wave is very nearly one-dimensional and independent of the presence of particles <Ji. . The usual shock relaxation zone is established by the interaction of particles the gas downstream of the shock wave . posed by Marble 3 <JI and The collisional model pro- is then extended and used with a modified form of the mean free path method of kinetic theory to calculate the macroscopic distribution and velocity of particles <1l_ as determined by the particle o; - particle 02. and particle °2_-gas interactions. Within the condition that the random velocity imparted to a particle 02_ by a collision is damped by its viscous interaction with the gas before it suffers another collision, the kinetic theory method established here may be extended to include more general particle-particle arrl particlegas interaction laws than those used by Marble. However, the iv collisional model employed is particularly important because the criteria for its application are easy to establish and because it admits a wide class of physically interesting situations. Within the restrictions of this collision model, it is possible to analyze the macroscopic motion of particles o;_ in three important (~i./o;-) limiting cases: ( (J-"2/ CJ\ ) 2 rv l '2. >"> l (<12:/0j" f << l It is found that when and (CIV0\ 5J.. >> l there is essentially no redistribution of particles °2_ normal to the gas flow. The only effect of particleOj -particle a;_ encounters is a drag force 2 acting to slow down particles OZ:-. When (<1'.. /CJ(") l it is found << that particles <12. may have many collisions during their passage through the shock relaxation zone. As a consequence there may be a substantial redistribution of particles wave. CJ2. downstream of the shock The physical features of this process are studied in detail together with the range of validity of this diffusion model. (<Ji/<5j)'2 rv .L The case is analyzed under the condition particles 02 have at most one collision during their passage through the shock relaxation zone. It is found that when the gas or particle"j density is low, the single collision effects may be important even when <12/cr, differs significantly from unity and the particles are not very small. Under most conditions of practical significance, because there is invariably a distribution of particles sizes present in a dusty gas, the calculation of the particle distribution in the shock relaxation zone should account for the effects of particle-particle encounters. It is suggested that an experimental observation of particle size distribution v in a shock relaxation zone can yield significant information on particleparticle and particle-gas interaction laws. vi TABLE OF CONTENTS Part Title Acknowledgements ii Abstract iii Table of Contents vi I. INTRODUCTION l II. FUNDAMENTALS OF PARTICLE MOTION IN A GAS 5 III. l. Some Fundamental Considerations 5 2. Particle-Gas Interaction 14 3. Particle-Particle Interaction 25 4. Motion of a Te st Particle 29 NORMAL SHOCK WAVE IN A GAS-PARTICLE MIXTURE l. 2. IV . 39 Passage of a Normal Shock Wave Through a Uniform Distribution of Particles of a Single Size 41 Problem of a Shock Wave Passing Through a Gas Containing Particles of Two Distinct Sizes 49 PARTICLE-PARTICLE SCATTERING IN A SHOCK WAVE: Oi / Oj <.<. l , NON-UNIFORM DISTRIBUTION OF SMALL PARTICLES 52 l. Macroscopic Motion of Particles Oi 54 2. Motion of a Particle Impulsively Disturbed From Its Local Collisionless Motion 60 3. Calculation of the Scattering Flux 75 4. Physical Significance of Expression for 5. Solution for the Particle Density fpi. 108 6. Results of Numerical Calculations 122 '{)(\'"') T'. -pa. 93 vii TABLE OF CONTENTS (Continued) Part V. Title PARTICLE-PARTICLE SCATTERING IN A SHOCK WA VE Oi / <1j l'V i 1. Density Distribution of the Primary Beam 156 2. Density Distribution of the Secondary Beam 161 3. Uniform Distribution of Particles OZ" en ff2 ''1 ):c) VI. 153 =-ro 171 (I) (' 4. Fundamental Beam Solution ffa '''\'=m2.Nf'z.c:>C~) 175 5. General Beam Solution by Superposition 180 6. Results 188 SUMMARY AND CONCLUSIONS 199 References 205 Appendices A. Dynamics of the Collision Process 1. 2. 3. 4. B. 207 Collision of particles Cfj and 02: as viewed in the local collisionless velocity frame of particles 02_ 207 Statistics of encounters for particles relative to reference frame x\( i: 1 212 Collision of particles a; and <Ji as viewed in their center of mass reference frame 216 Statistics of encounters for particles as viewed in the center of mass frame 218 Cylindrical Symmetry; o;_ I "I << .1. 221 -1- I. INTRODUCTION If phase change and chemical reaction do not occur and ex- te:rnal forces are neglected, then the dynamics of solid particle clouds transported by gases is governed by the viscous forces exerted on the particles by the gas and by collisions between individual particles. The gas - particle and particle - particle interactions are not always distinguishable from one another, since the interaction of two or more particles is characterized not only by the collision of their surfaces but also by the coupling of their individual flow fields. If the number density of the particles is very high, the flow fields of many individual particles may be coupled together continuously as the mixture evolves, for example, as in a gas - fluidized bed. In this situation, the gas - particle mixture behaves almost as a fluid with modified properties since the particles and the gas are strongly locked to each other. On the other hand, if the average distance be- tween particles within the gas is much greater than the characteristic dimensions of their individual flow fields, the particles may move significant distances during which they interact only with the gas, and collisions involving more. than two particles will be rare. Under these circumstances, a collision between two particles, when viewed on a macroscopic time scale, is characterized by a rapid and very complicated transfer of momentum and energy between the particles. This investigation will deal exclusively with dilute particulate suspension. For a general discussion of methods applicable to the treatment of gas - particle flows in which the particle densities are high, the reader is referred to the books by Zenz and Othmer 1 and -22 Soo . 3 With the exception of Marble 1 s work , previous investiga4 tions -B into the dynamics of gas-particle mixtures, in which the particle densities were not large, have neglected the effects of particle-particle interactions. This approach is valid only if the particle number densities are quite small or if the particles are very nearly the same size so the particles have little tendency to collide. Since these circumstances do not generally occur in nature or in problems of practical significance, the effect of particleparticle interactions must usually be accounted for in describing the macroscopic motion of the gas-particle system. Fortunately, it seems that in many problems of practical significance the gasparticle and particle -particle interactions are sufficiently independent phenomena, with regard to the motion of a particle through the mixture, that the methods of kinetic theory may be used in principle to compute the macroscopic motion of the particles. The complexity and considerable lack of knowledge regarding the particle-particle and gas-particle interaction laws , however, prohibit a comprehensive treatment of the dynamics of dilute particulate suspensions. The re do exist circumstances, however , under which the gas-particle and particle-particle interactions are simplified and the viscous damping of the particle motion between collisions simplifies the treatment of the collision process so that a detailed solution to the problem is possible . These circumstances were first studied by Marble one-dimensional gas-particle flows . 3 for The present work proposes: first, to demonstrate, by generalizing Marble 1 s collision model, that -3- the methods of kinetic theory can indeed be extended to treat solid particle collision phenomena in multi-dimensional gas -particle flows; second, to elucidate some of the fundamental physical characteristics of the dispersion of particles in a gas-particle flow field due to particle-particle encounters; and finally, to assess the importance of particle collisions in particle-laden gas flows. To attain these objectives, this study is divided into two parts. Because of the gen- eral complexity of this subject, it is appropriate to first establish a good qualitative understanding of the physical aspects of gas-particle flows. The foundation is particularly important here, because the kinetic theory method is considerably simplified if use is made of the underlying symmetry principles that govern particle-gas motion and interaction. Therefore, the first part of this study presents a qualitative discussion of the essential physical features of gasparticle flows . Important dimensionless parameters are introduced and their general physical significance is indicated. The analysis is broken into four parts. First, a reasonably general model for the gas -particle system to be used throughout this thesis is outlined. Then the fundamental characteristics of single particle motion in gas flows commensurate with this model are described. Third, a particularly important particle-particle inter- action model and conditions required for its validity are presented. Some experimental results obtained by McLaughlin 9 which tend to support this model are also described. These general concepts of single particle motion and particle-particle interaction are then combined to treat qualitatively the motion of a test particle in a gas- -4particle flow including collisions. In part two, the interaction of a normal gasdynamic shock wave with a dusty gas is studied with appropriate mathematical rigor by using the powerful methods of kinetic theory together with ideas generated in part one. Throughout the calculation, an effort is made to ascertain the validity of the computational model in order to establish the physical significance of the results. This approach also serves to point out ways of increasing the quantitative accuracy of the calculation. It should also become evident as we proceed that the necessary conditions for the application of kinetic theory procedures to the study of gas -particle flows are sufficiently weak that, in addition to the shock wave problem, they admit a wide class of physically interesting problems. Finally, it should be noted that the study of shock waves passing through gas-particle mixtures appears to have fundamental as well as practical implications. For instance, this investigation re- veals that the study of shock waves in gas -particle suspensions may be particularly suited as a means for investigating particle-particle interactions and other non-equilibrium phenomena in solid particle gas flows. This is an important result, since at the present time there is little experimental evidence regarding gas - particle and particle - particle interactions and their effect on the dynamics of gas -particle systems. -5II. FUNDAMENTALS OF PAR TIC LE MOTION IN A GAS 1. Some Fundamental Considerations Consider a perfect gas containing a dilute distribution of small, solid, spherical particles of two distinct radii, 0( and OZ:- • In principle, it is straightforward to extend the following considerations to treat particle distributions that consist of more than two distinct particle sizes. Suppose that the state of the gas and the composition of the particles are such that the following assumptions are valid. Individual particles do not vaporize, condense, agglomerate, or chemically react with the gas during their motion through it. · Then the mechanical properties of the particles are dynamical invariants . The particles have a large thermal conductivity, so their internal temperature is uniform. Furthermore, heat exchange occurs only by thermal trans- fer between particles and gas; radiative heat transfer is neglected. The distribution of particles is dilute in the sense that the average distance between particles, within any region in the gas, is much greater than the characteristic dimensions of their individual flow fields. In general, the flow fields of individual particles do not over- lap continuously during their motion through the gas. In fact, the particles may travel significant distances during which they interact only with the gas. A collisional encounter between particles occurs when they approach sufficiently close that the force exerted by the gas upon each particle is significantly altered. In essence, a collision occurs when -6there is a substantial coupling of the flow fields of the individual particles involved. Consequently, if the particle number densities are not too large, collision events involving more than two particles will be rare. To be more specific, suppose Y\.pi. ml , m2 are the masses, are the local number densities, and are the local mass densities of particles of radius "'l and 02: , respectively. particles of radius The local average distance between , and is then approximately 07 J the local average distance between particles of radius cles of radius 02" ~2 ~ is -Y3 ( nr 1 + Ylp2.) Now if ~ and parti- Li is the characteristic radial dimension of the flow field of an isolated particle of radius , the distribution is called dilute if the following inequalities are satisfied throughout the flow; SI ~ Ylp;~ ') > 2 L:, (2. 1) r~l "' -y~ - nf1 > > 2 "2:2 (2. 2) (2. 3) The length 2:.z.. depends upon the local particle Reynolds number Rei -: : . f CJ;, l ~ - 1Jfd /rand Knudsen number kpt. ~ , Mach number M~i=I~ -ytd/Ov Ac/ OZ , where and Q are the local gas density, viscosity, velocity, and sonic speed, respectively; ~c is the molecular mean free path, and the local average velocity of particles of radius ~r~ Since the is -7Knudsen number can generally be expressed as a function of Mach number, Zi,-== Li CRec,)Mri.). If corresponds to l<rc. << l, the gas behaves as a continuum with regard to its interaction with the particles. When corresponding to k r~ > > 1. ' the gas- particle interaction may be described by the methods of rarefied gas dynamics. In this case, if the particle velocity and mass are suffi- ciently small, Brownian motion of the particles may be important. Cases where the effects of slip flow, transition flow, and free molecule flow are significant will not be discussed. It will be as- and that the motion of the gas and the sumed that particle can be computed on the basis of continuum mechanics. When the velocity of individual particles relative to the gas is much less than the sound speed, the gas behaves as an incompressible medium in response to the particle motion, and Li..-:::: L~(R.ei,). The disturbance created in the gas by the motion of a particle is usually spatially anisotropic, emphatically so when a wake is formed. The significance of L {., is illustrated by the data of Taneda lO presented in Figure 1. for .LO < Re.:.< '300 wake. For the steady motion of a sphere , the particle motion is accompanied by a R.ei..>24, that grows in Re(., increases. For Re. >l~O This wake consists of a vortex ring, size and decreases in stability as (, it oscillates and gives rise to time dependent forces on the particle. Generally, we will assume L·L to be the maximum radial characteristic dimension ofthe particle disturbance unless, as in the case of a large wake, L"t, has no useful significance. -8- 1.2 1.0 ~s 2<r. -~ " .£'.i·.6 zaz .h .4 Zoz .2.. .l.O 2.0 100 2.00 400 .1.000 Re l· Figure 1. Sphere wake dimensions as a function of particle 10 Reynolds number {Taneda). Note; .:[i "'-' q -9Using the previous results, we can rewrite (2. 3) in the form (2. 4) f5 , then If all particles are composed of material of density Wl I .::: 111 \S o-1 '3 and m 2 :::: 4 ll fs02.a . 0 k~ ==:fr;,/ f Then, defining the ratio of the mass density of particles of radius CJZ density of the gas, and supposing for convenience 0( > <{ to the mass , we can rewrite (2. 4): or In a similar fashion, relations (2. 2) and (2. 3) become: (2. 7) and (2 . 8) The quantity kl is a measure of the local total interaction force per unit mass between the particles of radius <J. l. and the gas. -10, the presence of particles of radius If negligible effect on the dynamics of the gas. has If the parti- cle density may be very high, a situation with which we are not presently concerned. radius <1i. When, however, Kc.~ 1 , the gas and particles of are coupled together and local gas properties are modified by the momentum, energy, and heat transfer between the particles and the gas. The situation Ki. "£ .L occurs, for instance, in the passage of shock waves through dusty atmospheres 5-7 and in the ex- haust plume of metallized solid propellant rocket motors. 8 If local f , is less than or on the same order as the standard gas density, atmospheric density, and the particles are composed of solid material, ~ fs/p ~ J..O . Usually, L::",~°t , ~'V D.z:', and there would appear to be no difficulty in satisfying (2. 6 ), (2. 7), and (2. 8) if too large. For particles in liquids, since IS/j CJ('/<{ is not is not large, condi- tions (2. 6), (2. 7), and (2. 8) are much more restrictive. These con- side rations suggest that there exist physically interesting gas - solid particle flows in which the particles modify local gas properties but may travel substantial distances between interactions. The system is defined to be in its equilibrium state when the density, temperature, and velocity of the gas are uniform and the particles move with the gas and have the same temperature as the gas. The particles may be non-uniformly distributed throughout the gas. The dynamical evolution of a non-equilibrium state of such a heterogeneous system depends on the following time scales or their c or responding characteristic lengths: -11- -Ca , duration of a coll is ion between two molecules within (a) the gas. For gases at standard conditions, -(2. lb~ .Lo sec. -Cc. , the average time between successive molecular (b) collisions within the gas. -Cf -Cc~ lo For gases at standard conditions, sec. (c) -C . • , the time of a collision between particles of radius OLj and particles of radius J , which is characterized by the time during which there is a significant interaction between the flow field CJi: of particle and the flow field created by one particle to another is explicitly neglected. for particles with <J:[/\J l.O f standard conditions '1::;° (d) -Cui radius 0 •. "'\.> J "Bon ding" of Note: -C 0 i.j and relative velocity N .la. -5 -h 10 -10 = Toji at sec. , the velocity equilibration time for particles of Oi, , a measure of the time required for the motion of a partiFor particles with -2. in gases at standard conditions, -Cu-1\J _lo sec. cle to respond to changes in local gas flow. 10 fl radius " (e) -C-Ti. , the thermal equilibration time for particles of 0. z, , a measure of the time required for the temperature of a particle to respond to changes in the local gas temperature. (f) -cc .. of radius 0£. l.J , the average time between encounters for particles with particles of radius the same magnitude as (g) -C appreciably. For '-C lJ". ' t a-; J Generally, -Cc1 is -c lJ'J• • , the time over which the entire flow system changes Generally, -C > LO-4 sec. -12- Associated with these relaxation times are the following length scales: (a} ~o"' ~<L potential, where is approximately the range of the intermolecular ()., is the local sonic speed and is nearly the gaseous thermal speed. (b} \_'V-CcCL 8 is the molecular mean free path within the gas. \ \.J ~ "°Go.-~ l\.fD. -q,j· \ - l (.. -, particle interaction~ where cm. is the range of the particle - \rfi istic velocities of particles -s Ac""' 10 In gases at standard conditions, (c} ~~ ~a""' 10 cm. In gases at standard conditions, and and Vyj cJ, J are the local character- \ -3 respectively. A<Jt'\110 J cm. , the velocity equilibration length, is a measure of the distance covered by a particle during its response to changes in the local gas velocity; provided at standard conditions, (e} Ji-rL ~ A0- ~ LO L -c1t, l qf~ \ , the l V'fi..1/\/ l CL g in gases cm. thermal equilibration length, is a measure of the distance covered by a particle during its response to changes in the local gas temperature. (f} A,..·. l\J T vtj ~ G~ ( ~f i. l particles of radius of radius (g> 0-: 2i Generally, is the average distance traveled by between successive collisions with particles CG" J lb l ~ ~ l~ I is the characteristic geometric dimen- sion of the entire gas particle flow system and teristic velocity of the system. Generally, I~ \ is the charac- l l I~ lOl cm. If the number density of the particles becomes too large and (2. 1} - (2. 3} are no longer satisfied, the particle relaxation times -13and lengths will in general be well-defined no longer. When the flow fields of many individual particles are coupled together, the time of a particle-particle collision and the average time between successive collisions lose their meaning. We shall assume that characteristic times and lengths are well defined and that conditions (2. 1 ), (2. 2), and (2. 3} are satisfied. For sufficiently dilute gas-particle mixtures, the characteristic parameters are related in the following manner: <.. ~ 'Cc < < -C · · "1:" J < <: -r:._. ~ J Cl.J) -Coi' <:.< Ac, < <. ~oD <:: < lu~ ) Acij > L } (2. 9) In relations (2. 9} it is assumed that the gas behaves as a continuum in its interaction with particles. Because fs/ f is large and the particles are small, it is reasonable to suppose that, when kc,~ l , the volume occupied by the particles is negligible compared to that of the gas. When treating the average macroscopic motion of the gas, it is assumed that, on the scale of particles, disturbances caused by the particle motions may be neglected. This implies that the momentum defect on the particle scale introduced into the gas is immediately diffused to a neighborhood of the particle spacing scale. The same assumption applies to the energy dissipated in the gas by the particle motion through it. In microscopic detail, of course, the energy first ap- pears in the particle wake as a velocity disturbance which is, in turn, dissipated to thermal energy. In our approximation, the energy is -14- dissipated immediately and uniformly throughout a local volume of particle spacing dimensions. Provided no large gradients in gas properties are present, such as gasdynamic shocks, we assume that the gas is inviscid except for the drag it exerts on the particles. The specific heats of the gas and particles are assumed constant. As a consequence of these assumptions, the particles move through a gas with "smoothed" properties. Since t:0 z.j < < Lcij corresponding to Aotj C < ~Cij , the dynamics of particles can be computed using the averaged gas properties. Specifically, the dynamics of an encounter between two particles within the gas may be studied approximately by assuming 11 smoothed 11 local properties of the gas. Large gradients may, how- ever, be created within the gas during an encounter between two particles. Therefore, on the microscopic scale of particle-particle interactions, viscous effects cannot be ignored. 2. Particle - Gas Interaction The motion of a single small, solid, spherical particle in a non-uniform gas flow is governed by the viscous forces exerted upon the particle by the relative gas motion and by heat transfer between the particle and the surrounding gas. This problem has been dis- cussed in some detail by Torobin and Gauvin 10 8 11 , Hoglund , Fuks , 4 2 Marble , and Soo. If the temperature of the particle is different from the tem- perature of the gas through which it is moving, there will be a variation in fluid viscosity around the particle because of heat transfer -15and the corresponding temperature distribution. Variations in vis- cosity alter the velocity field of the gas and, hence, the viscous drag on the particle. Fortunately, the slow variation of viscosity with temperature exhibited by gases makes this effect fairly minor if the temperature of the particle does not differ greatly from the temperature of the gas. We shall assume that the particle resistance may be computed utilizing the local viscosity coefficient, neglecting local thermal effects of the particle. If we neglect non-steady effects upon the particle resistance, the equations for particle motion are ~ -i£l!f(~) :. - ~cl) CR.~M; f(~raJ)lrcr'l(t!O~p(:hJ)-Mpc-tJ)\ 1A-~vet)J( 2~r;~f d~r Ct) :::. ~t\;) d..-t4(.t) and !.lf(_1::) where cle at time -i; . (2. 11) are the position and velocity of the parti- The drag coefficient, particle Reynolds number C.I> , is a function of the R.e :::= l(~))O- l!:!Cx,E.t-))-~(t) l/f<, and Mach number M. =. \ ~ ('ft(;b) ) - ~(.t) l / Q.(~p(t)) , where \ ~ , CL , and respectively. are the gas density, sonic speed, and viscosity, For low values of M , the medium may be consid- ered incompressible and the drag coefficient is a function of only the particle Reynolds number. For simplicity, we shall assume that the individual particle motion obeys the classical Stokes law and that the heat transfer between the particle and the gas takes place with a Nusselt number of unity based on the particle radius. Although these drag and heat -16transfer laws hold strictly for a single sphere in steady motion through a uniform medium, the errors involved in applying them to a dilute suspension are minor. Moreover, the significant physical features of the collective gas-particle motion do not depend strongly on the details of the drag law. Consider the motion of a particle of mass Wt in a steady, one-dimensional gas flow of velocity position, x~ (t~ ' velocity, 1!f(-b) and radius 1.l(XJ ~)( ' and temperature, . 0- The Tr Eb) ' of the particle satisfy, using Stokes Drag Law and Nusselt Number of unity, cr CLL ex I (-t)) ..ex - llr (:{:;') ( 2. 12) G, w k C> I- ( 2. 13) and ( 2. 14) In equations (2. 12) - (2. 14), specific heat of the solid, and gas. j-'l~ is the local gas viscosity, Cs the the thermal conductivity of the Define the velocity equilibration time "'Cu ::: and the thermal equilibration time ( 2. 15) -17- ( 2. 16) Cp where is the specific heat of the gas at constant pressure. Since the Prandtl number, Pr :::. Cr/{k, for most gases is approxi- mately two-thirds, the velocity and thermal equilibration times are about equal. Rewriting (2. 12) and (2. 13) using the definitions (2. 15) and (2. 16), --!:.<-v ( LL ex - -r ~) ( 2. 17) (T-T) ( ~)J_ s- ~ r In most situations for solid particles (2. 17) we see that <:lJ (2. 18) Cs /\.J Cr . From the form of is approximately the time required for the velocity of the particle to respond to changes in the local gas velocity. If the gas is uniform, it becomes the time required for the particle slip velocity Us value. =Ll e'X _ Llf A similar interpretation of variation of particle temperature. to decay to e-l of its initial ""Gr is useful to describe the The spatial relaxation of the par- ticle velocity and temperature is characterized by the velocity and thermal equilibration lengths, respectively. They are usually de- fined as: ( 2. 19) ( 2. 20) -18where u. 0 ters Av radius () is a velocity characterizing the gas flow. The parame- ~T are measures of the distance that a particle of and will be transported before its velocity and temperature equilibrate reasonably with those of the gas stream. If f is the mass density of the gas, of the particle, and -z> the mass density the kinematic viscosity of the gas, then the equilibration parameters may be written (2.21) · (2. 22) For metallic solids in gases at standard atmospheric conditions, is on the order of lcS ts// -:--4 For a typical particle of radius CY'VlO cm the velocity equilibrationtime at standard atmospheric conditions in air is -Cu /'\J radius. -+ 10 sec, and increases as the square of the particle When the viscosity of the gas, the gas flow, '"Lu ft , is nearly constant over is also nearly constant. Under this condition, it is apparent from equation (2. 17) that the motion of the particle normal to the gas flow direction is independent of its motion parallel to the gas flow. This transverse motion is damped in a time of order -Cu • Suppose at time t =-o and position X : 0 a particle is injected into the gas flow with a velocity (2. 23) different from the local gas velocity. The velocity components of the -19particle are governed by ®i(-t) := dt (2. 24) (2. 25) (2. 26) (2. 27) Equations (2. 25) and (2. 27) are readily solved, and the motion of the particle perpendicular to the direction of gas flow is given by: ~.1.. (t) =- ~.L(o) -t/-cu e (2. 28) (2. 29) The velocity of the particle normal to the gas flow is damped exponentially in a time of order -Cu so that it reaches a limiting trans- verse position (2. 30) The length '><p.L (co) is called the transverse range of the particle; it is the maximum distance the particle can move across the gas flow field if its initial transverse velocity component is value of ~t .L(oo) Ufl. (o) . The depends on the particle's initial transverse veloc- -20ity and is independent of variations in parallel velocity components. The concept of a transverse range may be extended to circumstances where -Cu varies with "lC. The motion of a particle parallel to the gas flow is generally complicated and is most conveniently treated within the framework of a given problem. The dynamics of the particle depend strongly on the magnitude of ( Ao/L) /\..J (-c\)/-c) where L teristic geometric length of the gas flow field and is the charac- T ""'u0 l proximately the time it takes a particle to travel a distance When l:'0 > > -C corresponding to is ap- L ,\v Y> L , Figure 2, the particle motion is relatively unaffected by local changes in the gas flow field. The force acting between the particle and the gas is small compared with particle inertial forces and may be treated as a perturbation on the motion the particle would have in the absence of the gas. The particle motion through a region of length L within the gas is determined principally by its motion at the time it enters the region and, in the absence of external forces, the particle trajectory will be nearly linear. On the other hand, if --C-0 <<. -C , corresponding to \,<.<. L, the particle adjusts rapidly to the local gas velocity. Referring to Figure 3, the particle motion is strongly coupled to the gas motion because its relaxation time is much less than the time over which significant changes in the flow field occur. Because the initial mo- ti on of the particle across the gas flow is damped out in a time of order ""t"0 , its motion is nearly parallel to the direction of the one- dimensional gas flow for times larger than "t"v Under these con- l~_tP)\ t I" l~rj.1 /'Ji:., I- "' i::--1 ~ "Cu > > ""C Ufo' Up ~ -t l~r..f°)/ ~ l\!ei._Co>\:\J I l~r.J -·-. x r L Au:>> L l~J.I lip(()) tlf S pa..cio..I the gas when -Cu/'\:. -= Av /L Y> l ~~ j- ,-...,L-1 Particle relaxation effects; motion of a particle through a region of ;____ _ _ _ /"-' AC) I ~fTC\. I ... - - - - - - - - - - - - - --------- Figure 2. j.- rv'"t',,____,j IJ.\t.1ol I Tempora..I ,., I I I-' N x l~f°)lt .,"'°SI I\ "U.p I l~~~I l~p.1.l Irv,\.,I IV'"C U.(t) rv ~ L .1. I I j~(O)I. I I IU,..£0 )-cv{ I1!:p.L I >I l~rJ ~(0) \v<<L I""~) I I\ 1Llr Spa.c;a.I x U.(X) /"VL J when -c., /-c. = .\..,/L << .l. Particle relaxation effects; motion of a particle through a region of the gas "t:v <..<.. "t t \Jf01 Figure 3. ;-sl liJ:p.ll Te m.pcra..I I N N I )( -23ditions, Marble 4 has shown that after its initial rapid relaxation for , the particle adjusts itself to the one- dimensional gas motion by taking on a velocity relative to the gas, the slip velocity, that provides the force to accelerate or decelerate the particle at nearly the local rate of acceleration or deceleration of the gas. Then, for t )) -C., corresponding to state of motion of the particle is approximately given by ( 2. 31) (2. 3 2) and (2. 33) The particle has essentially lost its memory of its state of motion at time t ...::0 ; its motion is relatively independent of its previous history. When -Coj-c /\J 1'/L is neither large nor small, the motion of the particle depends on the entire history of its motion through regions within the gas of dimension L Apparently, Figure 4, no great simplifications may be made in treating the particle motion parallel to the flow. The transverse displacement, however, has al- most reached its limiting value. A similar discussion holds for the relationship between the local particle and gas temperatures. For a more detailed discussion of the gas-particle interaction, the reader is referred to the work by 4'.t. t .., t 1--- I!:¥zr·~ I -11 ~ .1..I I 0 I A --v L I< L llp ,,...., A.v \ "'i:, rv L x .NL----- - - - - - - - - - - - - - - - - ::.-=-:::---= - -- - - - l~p.i,. l '"'I l~r~I 11!f'l-I Soa.c ia. \ Particle relaxation effects; motion of a particle through a region of the gas when -r:: v/--c = >..u / L "'-1 1. · l~r.fo) [L;, /'-.Ji:: I--- AJTu I \;urv -C 11co: Figure 4. 1~~ I l'l!f1.' tu. ~,,..t UY' Te m. f oret.l ~ I N I x. -254 Marble , which also includes some illuminating examples of the physical significance of ~ 3. j-c . Particle-Particle Interaction The collision of two spherical particles in a viscous fluid is complicated by the fact that, during collision, the distance between particle surfaces in relative motion becomes so small that viscous forces dominate. The problem is further complicated when the particle Reynolds numbers are so large that wakes are formed. Then the interaction mechanism loses its symmetry; this is probably of importance when particles collide while moving parallel to each other through the fluid. In spite of these complexities, it is reasonable to expect tonditions under which the particle-particle collision is nearly elastic and is characterized primarily by the particles 1 properties. When two particles approach each other, their individual flow fields interact and the force exerted by the gas upon each particle is altered. If this viscous force had a negligible effect upon the collision, the momentum and energy exchange between the particles then depends on the nature of their contact. Marble 3 has suggested that a necessary condition for this behavior is that the time required for each particle to traverse the flow field of the other must be small in comparison with the velocity equilibration time. This condition is reasonable since, if it were not satisfied, the particle would have sufficient time during the encounter to respond to the flow field of the other particle. To make this criterion quantitative, let L l. be the charac- teristic radial dimension of the flow field of the particle whose radius -26and is ~f(, be the local collisionless velocity. during which particles of radii /t L"~ ~~ -~fj \ ~ and J The time interact is of the order Consequently, in order that the collision be- tween particles of radius °l and oZ be nearly independent of the structure of their flow fields, it is necessary that l:ul >> 1 (2. 34) >>1 ( 2. 3 5) Utilizing the definition of the velocity equilibration time <='tf particle of radius er-:: J the same material so of a J and assuming the particles are composed of fs,;::::. fs~ \"S, this may be written f) >> ~ (f) (f Ol l~, -!Jr~\) ( 1 (2. 36) ( 2. 3 7) The second bracketed term is a Reynolds number based upon the relative velocity of the two particles and will generally be of order unity. For our purposes here, it will suffice to take and conditions (2. 36) and (2. 37) become 2:' rv Oj 1 and ~"V ~· (2. 38) (2. 39) For solid particles in gases, the density ratio fyl is large and, for particles having Reynolds numbers of relative motion of '3 order unity, expressions (2. 38) and (2. 39) are of magnitude LO Further information is required to determine just how large expressions (2. 38) and (2. 39) must be to assure that a collision is essentially elastic . Toward this end, a detailed experimental investigation has been made by McLaughlin 9 into the momentum and energy losses re- sulting from the collision of spherical particles with an infinite wall in the presence of a viscous fluid. This is a geometry which tends to maximize the energy and momentum losses of the sphere. The ex- periments were performed using steel spheres in glycerin-water mixtures. His results, most conveniently given in terms of fSU.(r/ f'- For ~U.<3/ ~ less than showed two definite interaction regimes. about 30, the kinetic energy of the particle was dissipated in ap'2. proaching the wall and no rebound occurred. Above f~IV 10, over two-thirds of the energy dissipation of the sphere ~curred within a tenth of a sphere radius from the wall. At higher fsl,\<J/(<- , the sphere rebounded from the wall, -28and as was increased, the loss became a smaller frac- tion of the kinetic energy of the sphere. At rsu.~':Yo 5 x io1 ' the loss had been reduced to approximately eleven percent. Further- more, as the Reynolds number was increased, the length scale over which the momentum loss occurred was observed to decrease. I shows the variation of momentum loss with the fsLLcr/j"\- cated by selected examples of McLaughlin's experiment. fsua-/r-~ to+ Table as indiAt the collision is reasonably elastic. TABLE I Radius, cm Viscosity poise Velocity cm/sec ~'.iu.() ft, Momentum Loss, o/o o. 318 9. 25 12. 7 5 8. 7 100 o. 476 9. 25 26. 9 28.6 100 0.476 2.40 55. 3 214 69 0.476 o. 59 95. 3 1475 39 o. 238 0.096 103 40300 18 The experimental results point to a thin-film energy loss mechanism over a wide range of To confirm this mechanism, McLaughlin also performed experiments to measure the loss of a sphere rebounding from a wall covered by only a thin film of the liquid. These results showed that a critical film thickness ex- isted, above which the energy loss did not significantly increase with increasing film thickness. case was The critical thickness for a particular , considerably smaller than the particle radius. The experimental results of McLaughlin suggest that flows -29generated by colliding particles exhibit high pressure gradients perpendicular to the line of approach, supported by high shear stresses generated by squeezing liquid from between the surfaces. For values , elastic deformation of the solid occurs to such an extent that the collision may be considered elastic. 4. Motion of a Test Particle Let us now consider some qualitative aspects of the effects of particle collisions on the motion of particles within a gas. a perfect gas containing number densities cles having radii CJ{ and Ylp \ 02_ , respectively. and Consider of parti- The dynamic behavior of this complete system is intimately related to the relative magnitudes of the several characteristic time scales or their associated lengths. Assume that the gas behaves as a continuum and that the particle number densities are low enough that only binary collisions between particles are important. Finally, assume that the average time of an encounter is much less than the velocity equilibration time of either particle, the average times between encounters and the time over which the flow field changes significantly. Under these conditions, an encounter between two particles appears as an almost discontinuous transfer of momentum and energy between the particles. only with the gas. previously, Between encounters, the particles interact In terms of characteristic time scales described -30The dynamics of this system may be described in terms of (i) the momentum and energy transfer in particle-particle encounters, and (ii) the motion of the particles between encounters. Suppose that the particle-particle interaction law is known and we wish to consider the motion of the particles between encounters. The particles respond to local changes in the gas flow field through the viscous force exerted upon them. There is a transfer of momentum between the particles and the gas. The magnitude of this momentum transfer and its overall effect on the motion of the particles depends on the time between encounters . Consequently, the relative imp or- tance of these momentum and energy transfer processes, and therefore the motion of the particles due to collisions, is dependent upon /-ccll"V Aol /'>..en the ratios Lu\ ""t"o2 /""Ccz.i '\.J Avz..) ~C2.\ , and These ratios will depend strongly on the particle radius ratio and the number densities of the two particle species. The velocity equilibration time of a particle depends on the particle radius 2 because it is a factor in the drag law; in the Stokes regime, T~f'" J. The average time, -Cc.i,j , between successive encounters depends on the sum of the radial dimensions of their flow fields, their relative velocity, of particles. \lf p(. -- !.lf j ( Lt,+ L J • , , and the number density In practical problems a wide distribution of particle sizes occurs, and one should expect a corresponding variation of velocity equilibration and collision parameters. Suppose that Y\.f I ) > Y\ rl. so that ; the effect of collisions on the motion of particles -31of radius <:Ji can be neglected. Particles of radius CJf may, how- ever, have a significant effect on the character of the gas flow field. Consequently, the gas-particle flow field will remain one-dimensional provided the mass density of particles of radius less than the mass density of the gas. motions of 02. ~ is also much Under these circumstances, the particles, caused by gas-particle and particle- particle interactions, may be studied without consideration of their effect on the gas or particles of radius CJt Ol >oz: ; parti- For the purposes of discussion, suppose that cles of radius locities LLr 1 and and 02, will have different collisionless ve- Llpt. and different slip velocities The particle velocities L\.f 1 indicate local collisionles s velocity of the particles. Usi=· U.-U.f 1 and and Urt. will The local col- lisionless velocity is the local velocity that particles would have if there were no collisions. Now if the gaseous acceleration is negative in the direction of the positive x axis lly 1)Uf2. The relative velocity of these two particle sizes will lead to collisions between particles of radius oZ and particles of radius CJ"t • If the velocity equilibration time of particles of radius 02._ is short in comparison with the time between collisions, ""t:U2.. < ""[ cz.l , a particle of radius .r2 that is disturbed from its local slip velocity by a collision will return to its local slip velocity before it encounters another particle . The collision frequency 1{ computed in the classical manner is 'l (2. 52) and the mean free path between collisions -32(2. 53) Collisions may be neglected if -Cc::; ) ' ) 1: 21 , corresponding to \~l)) L . The approximate time -Cci.i to be expected between encounters experienced by a particle <Ji.. is (2. 54) and it is of interest to compare this time with the effective velocity I equilibration time -C0 2.. , for a particle 02._. Using (2. 55) where local particle Reynolds number and Mach number are given by and M'2.. :::. 1 LL - u.rz. 1 / Q respectively we have as the ratio of the characteristic times; The magnitude of each of the bracketed terms may be estimated. If the local Reynolds number and Mach number are sufficiently small, , L:1N ~ equation 2. 57 becomes , Z:~ l\J ~ and -33- t (f.C1+~~1t;.-"'d)(t)(c"'~t'l)2.ss1 ~l The quantity l ("iT<li:) Iur,- (lf>-1 Ij'v is the Reynolds number of particles based on their relative motion and in the Stokes law regime is less than or of order unity. than unity. The quantity fri /f is generally less The last term involves only the two particle radii and <li/<Jj becomes small as becomes small. Therefore it is not restric- tive to assume that I Lo:z. T~l < l /V (2. 59) where relative particle velocities are not large. Outside the Stokes regime for particle Reynolds nurrbers greater than unity and Mach numbers less than or on the same order as unity, the drag coefficient of a sphere is of order unity. Except for unusual situations such as particle shock waves or extended tur. l ev1.d ence lO suggests t h at b u l ent wa k es, exper1menta Consequently even outside the Stokes regime . there seems to be no difficulty in satisfying condition (2. sq) if the density of particles of radius O[ is not too large and <Ji/a; is sufficiently small. Consider the motion of a test particle of radius -C"h/ -Cc~, < l ct;_ , . If the particle were subject only to interaction with the gas, it would, after an initial transient, slip relative to the gas at a rate such that the viscous drag force accelerates the particle at very nearly the local gas acceleration rate. When the particle is disturbed from this state, it will return to the appropriate local state -34of slip within a time of order l:"v2. . Now the collisions experienced by the test particle disturb the particle from the local equilibrium slip condition. The scattering velocities that result from collisions will have components normal and tangential to the directions of average gas velocity. Consider fir st the scattering velocity component normal to the average motion a s indicated in Figure 5 . Following a collision the normal velocity component decays toward zero with a time constant Consequently if the time interval T'_ between -i:::;zl successiv e collisions is larger than its velocity equilibration time -c.,2. , the particle enters into each collision with negligible velocity normal to the direction of gas flow. Furthermore, the particle has moved across the flow field, its transverse range based on the scattering velocity Ypi.CP) normal to the gas flow. regime, is approximately The transverse range , in the Stokes ~l(o) ~i The scattering velocity parallel to the gas motion, see Figure 5 , is a bit more complicated. Prior to a collision the test particle is moving with its local collisionless slip velocity. After a collision, the parallel components of the particle velocity decays toward its collisionless slip velocity with time constant --COZ- . restriction insures that particles of radii Consequently the °T and OZ. are moving at very nearly their local collisionless velocities prior to each collision. a;_ particles of radius We conclude that when -Cvz.. /-cc ;G .L 21 will "diffuse" across the gas flow. There is an interesting analogy with the manner in which a low density , weakly ionized plasma diffuses across a strong magnetic field, Figure 6 . The guiding center Figure 5. ~"'~~1-i "ii ~ '\JAu1 I;~ ~~,--J 1"'~,I ' j.--.-vA111 "1 The positions of successive collisions for test particle<l2, are marked along the X axis UF --+--..,.~' ~"'r~ I x. I \J1 w x x, X:1 r- 1'.~i~ "r,?'11"'-"2 ry ~ x.. -- 0 o;_ x x: I I O' w + +' 11 ri°'2. l t:;;2. t'\c(~) + ~B ~ I'""~v1 I Figure 6. If a particle of radius O"'.a. undergoes many collisions it will "diffuse" across the gas flow much like an ion diffuses across a weakly ionized low density plasma that is confined by a strong magnetic field. ~ ~ - 3 7of a typical ion in the plasma can move only a finite distance across the magnetic field due to collisions. fc 'V\·(l\\} -Cc where "'Ge, is the cyclotron by its cyclotron radius period. This distance is characterized The requirement that l p>) \p-2. so that collisions had pi;) a negligible effect on the dynamics of particles of radius matter of convenience only. O!, was a It assured that the initially one-dimen- sional gas flow would remain one-dimensional; if f r1 fr'2. "\J I'\) f the gas flow will usually deviate from one-dimensionality. The following physical picture of the gas-particle flow process will hold: If fpt ~ \1-z. and <;G o Cf) Lvz. <: -Cc-21 L"I < -Cell J -C ) "t"cz..z. cl2. <::-c <:'-c then (1) the collisionless slip velocities of the particles are deter- mined by the local acceleration of the equilibrium gas-particle flow field; (2) the random particle velocities resulting from inter-particle collisions have essentially decayed to zero before the next collision takes place. (3) all collisions take place with a relative velocity equal to the difference between the local collisionless velocities of the two particle sizes and with a direction parallel to the local gas acceleration. Consequently collisions between like particles are unimportant. (4) collisions between particles result in a transfer of momen- tum, due to viscous interaction, from the particles to the gas in the -38direction normal to the local gas acceleration. initially one-dimensional if \pil\J f fl."-.> f A gas flow that is will most likely deviate from one-dimensionality as a consequence of particle -particle inter actions. -39llI. NORMAL SHOCK WAVE IN A GAS-PARTICLE MIXTURE The considerations of Chapter II suggest that there exist a fairly wide class of physically interesting gas-particle flows for which conditions << -Cc << -c o:J.. << -Co·) -Cc·· u 'tj ) -C Ao << Ac << ).cjl-..J << Aoi) Acij) L To hold. Collisions between particles within the gas are well-defined events so that particle-gas and particle-particle interactions, which govern the evolution of the system, are readily distinguished. Be- tween collisions, the particles interact primarily with the gas. In addition, the experiments suggest that when -Coi:j << Tui,>'"'C'v j' the collisions between particles are nearly elastic and the effect of the gas is relatively negligible. Throughout the remainder of the analysis we shall consider the encounter to be elastic, recognizing that even with its weakness, this assumption is the most reasonable within the present state of our knowledge. Indeed, it will turn out that many of our results, if com- pared with a suitable experiment, should provide insight into the true character of the particle -particle interaction law in a dilute gas particle system. With the picture of the physical phenomena which we have formed, let us consider the passage of a normal shock wave through a gas containing a distribution of small, solid, spherical particles. This example is intended: -40( 1) to demonstrate that the principles of kinetic theory may be extended to treat quantitatively the dynamics of dilute gas-particle flows in which particle-particle interactions are important; (2) to determine novel physical features introduced by particle -particle collisions; and (3) to ascertain the general importance of collisions in the dynamics of dilute gas -particle suspensions. We shall suppose the dust is composed of particles of two distinct sizes with radii <1(" and "i To insure simplicity at a later stage of the calculation, we shall let particles of radius tributed uniformly upstream of the shock wave. particles with radius <12. U,- be dis- On the other hand, are non-uniformly distributed in a direc- tion parallel to the face of the normal shock wave with a density much crr less than that of particles radius 6j eJi Under these conditions, particles of have little effect on the motion of the gas and particles The motions of the gas and particles with radius ever, are strongly coupled. G( , how- These circumstances permit us to ex- amine the effect of collisions on the motion of the cloud of particles a;_ downstream of the normal shock wave. We shall compute first the dynamics of the gas-particle mixture due to the presence of the shock wave, neglecting the presence of particles cr; Then, using these results, we shall analyze in detail the motions of particles <12 in the known gas -particle flow field. Finally, primarily to simplify the analysis, we shall make several restrictive assumptions regarding the gas-particle and -41- particle-particle interactions. Stokes drag law will be used to de- scribe the gas-particle interaction. The collisions between particles will be approximated by collisions between rigid elastic spheres, considering the gas to have negligible effect on the particle-particle interaction. Throughout the discussion, every effort will be made to as- certain the validity of these assumptions and their general effect on the solution. It will be shown, quite clearly, that their primary ef- feet on the results is quantitative. Qualitatively, they do not alter the essential physical features of the gas-particle flow downstream of the gasdynamic shock wave. 1. Passage of a Normal Shock Wave through a Uniform Distribution of Particles of a Single Size. Consider a normal shock wave in a mixture of perfect gas and a collection of small, solid, spherical particles of essentially uniform °T . So long as the solid particles are large with respect to the molecular mean free path of the gas, °l > > Ac , the thickness of radius a gasdynamic shock is negligible in comparison with the momentum and thermal ranges of the particles. Thus, the structure of a normal shock wave in a dilute particle-gas mixture may be thought of as a conventional gasdynamic shock which produces temperature and velocity conditions of the gas different from those of the particles. Follow- ing this essentially discontinuous variation of the gas properties is a relaxation zone in which equilibrium between the particles Oj and the gas is gradually re-established through the mechanisms of particle drag and heat transfer . If the particles represent a significant frac- -42ti on of the mass of the gas-particle CJj mixture, their response to changes in the gas, in turn, modifies the state of the gas. The statement and analysis equations of this problem were . . b y c arrier . 5. apparent 1y f irst given Various aspects of the relaxation process were also studied by other investigators, for instance, 4 6 7 Marble , Rudinger , and Kriebel. In the present presentation we will follow the analysis by Marble 4 since his approximations are con- sistent with ours. The number density of particles 61 is sufficiently low that pairs or groups of particles may be considered non-interacting. The volume occupied by the particulate matter is assumed negligible. At any point in space not occupied by a particle, the state of the gas is defined by its local pressure, density, temperature, and three components of velocity. The properties of the gas are assumed to be smoothed; only the average effects of particle motions are accounted for. The energy and momentum present in the particle wakes are as- sumed to be dissipated immediately and uniformly through a volume with dimension of the order of particle spacing. The gas is treated as inviscid except for its interaction with the particles. The state of any particle is defined by its velocity components and its temperature, both generally differing from those of the gas. As a conse- quence of these assumptions, it is admissible to employ the known behavior of isolated particles in uniform gas fields to calculate the interaction between the two phases. The shock wave and shock structure are stationary in the reference frame employed. Under these restrictions, the method presented by Marble 4 -43may be used to establish the equations of motion for this gas-particle Denote the gas velocity and density by U. and mixture. f , re- spectively, and designate the average velocity and density of particles <Ji by llp1 and phase is conserved, so , respectively. The mass flow of each pu. = . 'ft'\ (3. 1) ff1uf,::::. k1 '°"where M {3. 2) is the gas mass flow rate per unit area and mass flow rate of the solid particles 0( k, ~ per unit area. is the Likewise , the momentum equations for the two phases are (3. 3) - -~fl (3 . 4) p is the local gas pressure and is the force exerted upon a unit volume of the gas by the particles. The partial pressure where of the particle or cloud is negligible. of non-interacting particles Stokes drag law, l=r 1 For a number density Y\.pl C1[ , assuming that the particles obey i s given by {3. 5) Since the particles are non-interacting, may be interpreted as the local collisionless velocity of particles with radius Furthermore, ~ul:: '{\o./ k>1rf-c:Jj cJj . is defined as the velocity -44equilibration length of particles cf and a. is the local gaseous sound speed. It will be assumed that the local gas viscosity that the ratio r;Q is a constant, and consequently f<.l\J ( Y2., so A.0 1 is ais o constant. The first law of thermodynamics for the gas is (3. 6) where Q~! i s the heat transferred per unit volume per second from the particles to the gas; the term (u_r 1-u..)Fp 1 represents the dissipative work done on the gas by the particles passing through the gas . The specific heat, sumed constant. Cf , at constant pressure , is as- To the same approximation as Stokes law, we as - sumed the heat transfer between the particles and the gas takes place at a Nusselt number (based on particle radius) of unity, and Qfl may be written as (3. 7) where is the thermal equilibration length for particles d'j value of the Prandtl number, a constant and is essentially equal to 2/3 , so that Usually, the , is assumed to be and are considered equal. The first law of thermodynamics for the solid phase is \...,.. I\ l I -45- (3. 8) where Cs is the constant specific heat of the solid material. To- gether with the equation of state for a perfect gas, (3. 9) the above equations provide a complete description of the nonequilibrium gas-particle 0( flow downstream of the gasdynamic shock wave. Equations (3. 3 ), (3. 4 ), (3. 6 ), and (3. 8) together with the equations of continuity (3. 1) and (3. 3) may be combined and integrated directly to give the conservation laws (3. 10) c2.. (3. 11) where ct and Cz. are constants. The conservation laws (3. 1 ), (3. 2), (3. 10), and (3. 11) are sometimes referred to as the mass, momentum, and energy integrals of the flow. They are valid even across the shock wave . We now denote, as indicated in Figure 7, the conditions upstream of the shock, where the gas and particles Oj are in mechan- ical and thermal equilibrium, by 1 ; the conditions immediately downstream of the gasdynamic shock by 2; and by ro, the condition far u. Figure 7 . 0.Cl) • 5 .l..O .l.S X./A.vi 2 .o '2,5 3.0 ~.5 (co) shock wave M, = 3 .2 , "6°= l.4 , kl= .2.5 , C-:s/cp = Au 1/\1j = l. Spatial non- equilibriw:n of a gas-particleO\ flow downstream of a gas dynamic (".n lk 2..) ~05 ..1..0 l.O Go..s Ve\oc'1t7 T I "' I ~ 2.01Z1) 2.5 J... .5 Po.rt1de '1 Veloclt7 °i TempeY"'Q-\-ure 3.0 .1..5 2,.0 2.S ~.o Go.s Tem.pe~o..+ut"e -47downstream where velocity and thermal equilibrium has been reestablished between the solid particles and the gas. Then, since the particles do not affect the structure of the gasdynamic shock, the change in state of the gas from 1 to 2 is given by the conventional gasdynamic shock relations - UCZ.) --UC1) (3. 12) P\= U..Ct) /W:..t) is the Mach number of the shock relative to \(2) L<L) where the gas and Y:::: Cp /cv is the specific heat ratio of the gas. The equilibrium conditions far downstream of the shock wave are altered by the particles and may be related to the conditions upstream of the shock wave by applying the mass, momentum, and energy integrals of the gas-particle points. llCcn) l1.<t) For simplicity, if we set °I mixture between these two c:r/<s :::=_L , then =(l+k I ( i)fl +(~ M'2.} u Ceo) (2i ))\'({+l ~ (::=fl ==fr\(L) (3.14) 1 =-fS-0 d' t-J. M .~ lit>,(1) p C~) \ r' p(oo) By compatison of these results with (3. 12), we have a relation between the gas velocity immediately downstream of the shock wave and the gas velocity far downstream : -48- (3. 15) This relation indicates the general effect of the particles on the gas; k 1 the significance of For situations of interest in this thesis, consequently, for atomic gas, U(oofu(2) 'f"' l .4 corresponding to a di- '5S l/2.. . The detailed variations of the properties of the gas and particles 0/ are determined by numerical evaluation of equations (3 . 1), (3. 2), (3. 4), (3. 8), (3. 10), and (3 . 11), utilizing evaluated upstream of the shock wave. tions at )( =C X~o The appropriate initial condi- for the evaluation of the particle equations of mo- ti on are ur1(0);:::: u.. cl) and (3. 11) for CI and Cz T and When solving (3.10) and U... , care must be taken to as sure that as from downstream of the shock wave, U.-7 U(2.)and T --3'T(2.). A typical solution, which illustrates the general character of the relaxation zone downstream of the gasdynamic shock wave, is presented in Figure 7. Downstream of the shock wave the particles and the gas are out of equilibrium. Because of their greater velocity, the particles do work on and transfer momentum to the gas; as a result, they are decelerated. The force applied to the gas tends to es- tablish a positive pressure gradient in the gas as well as to increase its momentum flux. The dissipation and heat transfer associated with the particles decrease and increase its density, respectively. Gener- ally, the properties of the gas and particles are such that the gas velocity is reduced downstream of the shock wave. -49Rudinger 6 has investigated the variations in structure of the relaxation zone when different assumptions are made for the drag and heat transfer laws which govern the gas -particle "I interaction. His results show that quantitatively both the particle velocity and gas pressure are significantly affected by the assumption for the drag coefficient. As should be expected, the Stokes law produces a relative- ly large zone. In contrast, the assumptions for the heat transfer law have only a small effect. In general, however, the essential physical features of the relaxation zone are qualitatively correct in all cases using the Stokes drag law and heat transfer law with Nusselt number unity . 2. Problem of a Shock Wave Passing through a Gas Containing Particles of Two Distinct Sizes. When particles of a different size, gas under the condition that f~ << l) ffl CZ. , are present in the , the shock relaxation zone is established solely by the interaction of particles gas. The motion of particles CJ2 °l and the is then governed by their interac- tion with the gas and collisions with particles of radius Oj' This situation may be studied by considering the magnitude of the particle size ratio, <12. / 'f" When at.jdj >> ..L , the length of the shock relaxation zone, established by the interaction of particles Oj" small compared to the velocity equilibration length, particles zone, particles and the gas, is A°'- , of the Consequently, throughout most of the relaxation 02_ are moving through a uniform distribution - 50composed of much smaller particles. This phenomenon is similar to the motion of a steel ball through still air. exerted on the particle of radius Cf CJ2. There will be a drag force due to encounters with particles in addition to the viscous drag associated with its motion relative to the gas. On the other hand, there will be essentially no redistribu- tion of particles Cf.2. in the directions parallel to the shock face. We shall not consider this case in further detail. 02./a-; <.<: l When , the length of the relaxation zone of the shock wave is large compared to the velocity equilibration length OZ of particles particles ticles 02 Oj Under these conditions, the redistribution of is collision dominated. When the distribution of par- (J2_ - is dilute, binary collisions 0\ are important, CJ2. dur- and because of the large momentum transferred to particles ing collision there may be a substantial spreading of the cloud of particles OZ downstream of the shock wave . This situation is studied in detail in Chapter IV. When <:Ji/°I /\J .L , the particles have very nearly the same velocity,and consequently have at most one collision during their passage through the shock relaxation zone. The particles collide while moving at their local collisionless velocities. The particles are of nearly the same size, so their velocity equilibration lengths, and Ao2.. , are approximately equal. ticles move a distance of order Au2. relative to the characteristic length zone. Following a collision 1 ~ par- normal to the gas flow field, ~111 of the shock relaxation Therefore, a substantial redistribution of particles single collisions is possible. A(] °2 due to This case will be considered in detail -51in Chapter V. -52 IV. PARTICLE-PARTICLE SCATTERING IN A SHOCK WAVE; Yo;<<l. NON-UNIFORM DISTRIBUTION OF SMALL PARTICLES When <Ji(~ << l, a particle of radius ~ collisions with particles of radius q- -co << -cc <<::: -c02,l << -c..,2 may have many satisfying the conditions <T(;;-z.\. <<.. -C"l ( 4. 1) \,<<:::: \: <<:. Ao2 l << Au2. < >. C2.\ <::C:::::. .kv, during its motion through the shock relaxation zone of thickness of order Xo1 . It will become evident later that since conditions (4. 1) can be met, provided the particles are sufficiently small, namely O[ ~ i.O- 100/{- . The large difference between the collisionless velocities of the two types of particles is indicated in Figure 8 . Under conditions (4. 1 ), only binary encounters are im- portant and, between successive collisions with particles particles <J;_ relax to their local collisionless velocities. 0\ There- fore, the macroscopic motion of the non-uniform distribution of particle s of radius a;: , downstream of the gasdynamic shock, is expected to be diffusive or collision dominated. Consequently, there may be a substantial redistribution of the cloud of particles of radius ~ in directions normal to the gas flow. Let us now analyze the process in detail, supposing that conditions (4. 1) are satisfied throughout the relaxation zone and that the particle-gas interaction is governed by Stokes law. v U(2.) Figure 8. 'r u.f2.(X) 'V.\ul -, ...J Collisionless velocity relaxation conditions when r-~I ~~oy -===-~(~) _--_ tJ.fL«;p)_, - - U.(I Cl2. /a;- << l x I u.i I (J1 -54- 1. C!z. Macroscopic Motion of Particles The dynamics of particle cloud suming ffi ) f > )> \ p2 <J2 will be computed by as - so that the presence of particles CJ2: has essentially no effect on the motion of the gas and the cloud of particles 0( downstream of the shock wave. Upstream of the shock wave, all particles are in mechanical and thermal equilibrium with the gas, but particles of radius a;_ are non-uniformly distributed in a direction parallel to the shock face, while particles of radius °I are uniformly distributed. If we denote by 1 conditions upstream of the shock wave; by 2, conditions immediately downstream of the shock; and by ro, conditions far downstream where mechanical and thermal equilibrium between the particles and the gas has been re-established, then upstream of the gasdynamic shock, url(l) tp,(I) - \lf2 (1) U.(l) °1?z. Cl) \(I) __, \fl (I) - Kl \Cl) • (4. 2) (1) ' fe t) == fr c1, -c ) 2 ( 2 Throughout the calculation the right-handed Cartesian coordinate systern ~L\~ is oriented so and -:Z:. face, Figure Ci . X measures distances normal to the shock measure distances parallel to the shock face, The origin of X l\ -:C is located on the downstream side of the mathematically infinitely thin shock wave. Because of their size and mass, the particles are unaffected by passing through the gasdynamic shock, while the gas undergoes a compressive change of state. The change in the state of the gas is -55computed from the usual gasdynamic jump conditions. The macro- scopic properties of the particles are continuous across the shock wave. Consequently, immediately downstream of the shock wave, at X := 0 , the states of the gas and particles are related to their conditions upstream of the shock by _ U( l) _ 2 ~ l. _ ~) - 0'-rl)M~ L ~ -c~f~ a.Cl)) \(\) iy~2) ::: ip le 2.) T<L) Uf"(-i,):::.. uft(2.) = U..Ct) (4. 3) • 1 f r~ ('2..) By changing the state of the gas , the gasdynamic shock has established a non-equilibrium situation between the particles and the gas. The relaxation of this gas -particle mixture to a new equilibrium state , where the particles and the gas are again in mechanical and thermal equilibrium, is governed by the following considerations. Because the number density of particles of radius small, the particles o;_ or the particles of radius °2 is very have no effect on the dynamics of the gas °I The gas -particle 'f flow re- mains one-dimensional downstream of the shock and is governed by -56the exchange of momentum and energy between the gas and particles of radius <J{ This gas -particle detail in Chapter III. fr ~1 1 and , °I problem has been treated in , T, Henceforth, we will assume that lf 1 are known functions of X de rived by numerical solution of the problem formulated in Chapter III. Within this approximation, the equilibrium state achieved by the gas and particles of radius radius °i is independent of the presence of particles of CJ;_ 02_ On the other hand, the motion of particles downstream of the gasdynamic shock is governed by viscous interaction with the gas and by collisions with particles of particles of radius cJi °i The dynamics of the cloud downstream of the shock can be viewed as the spreading of a non-uniform stream or beam of particles moving perpendicularly away from the shock face. The spreading or redis- tribution of this beam is due to collisions between particles of radius OZ and the particles of radius OJ" which are unaffected by the beam. Consequently, the final distribution in the beam of particles °2. will depend on their complete interaction with the gas -particle 0\ mixture. The macroscopic states of particles of radius 02_ are con- tinuous across the gasdynamic shock wave and are therefore known at x: :::: 0 Although the distribution of particles CJ2_ is dilute, the number density is sufficiently high that fluctuations in the macroscopic properties of the particle cloud may be neglected and the methods of continuum mechanics are applicable. This assumption implies that within an infinitesimal volume element there are suffi- -57cient particles Y\.r that particle density eJi. scopic properties of the particle cloud are well defined. The equation of continuity for particles in the usual manner. and other macro- 2 o;_ is established A. surface is of arbitrary shape, but closed, with area of particles of radius V whose Consider a fixed element of volume <l2, The mass within this volume is S Fr~ JV (4. 4) v is the mass density of particles where the integration is over the volume V is d~ of the surface bounding ~ is the local mass flux .!fz where density of particles of radius and V . The mass of particles CJ2: flowing in unit time through an element the volume OZ. 02. . Its direction is that of the ave rage motion of the particles, while its magnitude equals the mass of particles of radius 02_ flowing in unit time through a unit area perpendic- ular to the average velocity, The magnitude of the vector "\JP._ , of the particles; it"-=' ff'-Y~ · J. A is equal to the area of the surface element and its direction is along the local normal. we take J~ positive along the outward normal. By convention Then positive if the particles are flowing out of the volume tive if they are flowing into the volume. of particles 02_ ~2. ,JB is Y , and nega- Consequently, the total mass flowing out of the volume V in unit time is I ~r · otA (4. 5) A where the integration is over the whole closed surface the volume V. A surrounding -58Since the mechanical properties of particles (J2_ are invariants of their motion through the gas and during encounters with particles Oj , the decrease per unit time in the mass of particles lJi. volume in the V ( 4. 6} must equal the total mass of particles V in unit time. <;:. flowing out of the volume Therefore , equating (4. 5} and (4. 6), (4. 7} we establish the integral form of the conservation of mass for parti- 02. . The surface integral can be transformed by cles of radius Green's Theorem to a volume integral; then equation (4. 7) becomes ( 4. 8} Since the particular choice of V is completely arbitrary, the inte- grand in equation (4. 8} must vanish everywhere in the shock relaxation zone . Therefore , in differential form, the equation of continuity of particles CJ2 is ~r~ + ~ a-t where ii'>? -: : . P.r- -f2 ti: -l- radius 02_ • if ~a i. (4. 9} is the local mass flux density of particles of \ Since there are no oscillations or other time-dependent phenomena present in the shock relaxation zone, equation (4. 9} becomes ~ fyz. ::::Q and -59- 7. i -r-i : : 0 1d.r-:::: \J-f:z..C..X) e,x; Define o;: of particles ( 4. 10) as the local collisionless velocity CJ;:_ is the velocity particles would have if they passed through the shock relaxation zone without encountering a\ . Then, since we have assumed that the any particles of radius particles interact with the gas according to the first order Stokes law, ~l. is given by Uy~~rz ( 4. 11) dx ~u 2 -: rn.._a./ b1ffw ~ 0-2.. 11:z.(o)::::. Ll( I) X:::O, where immediately behind the shock wave at is the velocity equilibration length for particles and is a constant since we assumed o/j"t , the ratio of the gaseous sound speed to the gas viscosity,is a constant. Without loss of generality, the local mass flux density of particles of radius ~ ~j2 , may be written ( 4. 12) This decomposition simplifies the treatment of the motion of particles CS: \..[) (r) between collisions and allows 1f'2. to be computed by kinetic theory methods. '-P(r) 1f'2.. Now is the local mass flux density of particles ~ at the origin of a non-inertial reference frame which is moving at the local collisionless velocity of particles collisions, ...CJ of particles "!y a;: 2 .::::: fr Ufl. ~'>< 02 due to collision of particles 2 ; consequently, In the absence of '° (r-) _:tF is the flux o;: with particles 61 · '°::!:.y In general, -60(I") depends on 2 fr 1 , ~\ p , U.. • ff'- • Llp"2... • , and their spatial derivatives, namely, ~ r) _~Ct")( p LL., p ll p u._) Cr) l fl' r') \) >1 f2) l'2. When is known in this form, equation (4. 10) leads to an equation -rz.. i?y'l. , the mass density of particles -1f2- <12: \Fl, The values of , and U.. are known from the solution of the gas-particle 01 problem, and Llpz. is obtained from ( 4. 11 ). satis- Therefore, fies together with appropriate boundary conditions. '° .!f'2..(1"') can be found, with good approx- In the present problem, imation, by a straightforward extension of the 11 mean free path 11 12 method of kinetic theory. The particle flux particle - OZ ~ (<"') -r- depends fundamentally on the °'I particle interaction law and the characteristics of the motion of particles <J;_ between collisions . It will prove convenient to view this motion as an observer at rest in a reference frame moving at the local collisionless velocity. 2. Motion of a Particle Impulsively Disturbed from Its Local Collisionles s Motion Before proceeding with the detailed calculation of sider the motion of a particle of radius ~Cr), con-rz. OZ" , following a collision with a particle CT/ , as viewed from a reference frame moving at the local collisionles s velocity of particles of radius the particle £l2" Oi . Since -Cu 2. < ~Z.\ , is nearly at rest in this reference frame if the col- lision takes place near the origin. For simplicity, we will assume - 61that the origin of this reference frame is coincident with the center of ffi. the particle prior to the encounter. are taken parallel to the Xi~ L.\ ~ ~o X .:::. 0 >\:[c~c: axes which define the rest frame of 9. the gasdynamic shock, Figure velocity of reference frame The Cartesian axes At time Xc~c -:t:-c ~ ='lo t:: 0 , the position and relative to the shock wave are ' and lAyi.(c) ::= U ( \) • The cal- Ci' may be culation of the local collisionless velocity of particles carried out using either Eulerian or Lagrangian coordinates. Both are useful for our purposes. In Eulerian coordinates, the local collisionless velocity, Ufz.. ( X) , of particles of radius 02. , since the flow is steady, is determined by the solution of together with the condition immediately downstream of the gasdynamic shock wave; at '>(..:: O , fines a velocity field ~(o).:::: U( l) . The solution of (4. 14) de- 1.lyzS X.) • In Lagrangian coordinates, the local collisionless velocity, (C) , is given by the solution of the pair of coupled differential ~2. CS) equations (<:) M'2- dur2.'!i ) . . . . ( 4. 15) J:s and (c:) ::::. ~2. C$) ( 4. 16) where S is a dummy time parameter. At time 'S =O , the position / / ,, , l~o - - -i:;c: - - -- - - - - - I I - _y I / / ~c; / , ~ / ,. Xra (c") U (X <"< 1 ) f2. f-1 ><c. ')<... Figure 9. Definition of reference frames. The shock wav·e is stationary in >''i~ and is located at .x-:::o. The cartesian reference frame X\.\~ together with clocks is an inertial reference frame. The reference frame x'"\''i/ , moving at the local collisionless velocity, u.r~CX) , of particles 02_ , is decelerating relative to X'-\.t . ,, ,, / / I~ I I O"' N -63(c) )( ffo) of the particle is Cc.> =0 Uffo) = Ll(\) and its velocity is Equations (4. 15) and (4. 16) are the characteristic equations of (4. 14). S :::!O They describe the trajectory of a particle that at ately downstream of the gasdynamic shock. is immedi- Since the gas flow is steady, the subsequent motion of the particle is such that at time ( <:.) X. == X'.' P~(s, > the particle will be at position (c.) ity will be and its veloc- (c:) llf~ -= U.fz.(S1) Conse- u..f' ( XFL (~) ) quently, the link between the two descriptions is simply ~) lAp2 (XP2 ) = Cc) u~ ( (<) scxt1.) ) (4. 17) or -- ( 4. 18) The position of the origin of the reference frame for t >0 S placing may be computed from (4. 15) and (4. 16) by simply re- t by Furthermore, the velocity of this reference frame is related to the collisionless velocity field, a;_ particles by ( 4. 1 7) and ( 4. 18 ). Let us now calculate the motion of particle rest at the origin of CJ! Uf2.(X), of Xe ~c ~c. a;: ; initially at following a collision with a particle The encotmter between the two particles occurs in the re- -h =:. t laxation zone of the shock wave at time xc~c ~ when the origin of is located at (c:.) ur2..(X..:,).:::::. 0 At -up. ( +o) ~ ~c> :::- (x".) L\ o }::.o) t .::::t , when its position was 0 , the particle received an impulse from its encounter with particle ~ , giving it an instantaneous increment -64in velocity without changing its physical position. During this en- counter, the effect of the gas on the interaction of the two particles is negligible. The details of such an encounter are classical and are given in Appendix A-1. t >t Following the impulse, 0 , the motion of the particle may be described in Lagrangian coordinates relative to the inertial rest frame of the shock wave M,_ dgr2C-b) -==bit" jtCX'ti--frJ) Ci ( U.(~-C>) ~ - ~i.(-b)) {4. 19) clt J~p:z.Ct) Vf2-(-t) { 4. 20) di where ..6.fz..lt) cle relative to the cosity Yrz-C::h) and X~ :C are the position and velocity of the parti- frame. In many cases of interest, the vis- )-<.. of the gas may be treated as constant. If the velocity equilibration time length Au2. := Cl \:02. -Cu'l. :::::: M'2.. J~ t\ f<-O:Z. and are introduced, equation {4. 19) may be re- written =- Q( X?2.(t>) (u..(Xyz.(-b>) €" - lI'~(tJ) I · - " - I { 4. 21 ) A<JL Immediately after the impulse, the particle at Kf2-:::.. ( X01 ~0}Jwill have a velocity (c.) '\fr-Ct) = Uy,__(~) e>< where }( = Xo {4. 22) is given by {Al. 5 - Al. 8), evaluated at , and is the velocity increment due to the collision, All -65values of the impact parameter for the collision are considered equally likely. is complicated The subsequent motion of the particle by variation in the gas flow field in the region of the shock relaxation zone through which the particle moves. oZ motion of the particle Fortunately, the approximate is particularly simple when viewed in the reference frame. is a non-inertial ref- Since erence frame, the apparent acceleration effects on the particle CJ2 must be accounted for in determining the motion of the particle This is simply done by decomposing the ve- relative to this frame. locity and position of the particle into the sums, (G) (r) Yyz.C-h) ::::: lip~Ct) €x + ~, (t) ( 4. 23} (4. 24} (c.) where Xr2. (1;) are the velocity and position of the and reference frame at time origin of the t they are the velocity and position which the particle had at time i; Physically, <J2: t=i if the re had been no collision at time (c;) Consequently, and ~ra.Ct) t frame at time t >t 0 "2. 0 are appropriate solutions of ::.0 equations (4. 15} and (4. 16} with, at Cc) (r) ~2..Co):::::. LL(\) . Evidently Xri.Ct) and and velocity of particle would have (c) ' :'ri.G>):::: (o J ~o) ~) and C'r) Uyz. Ct) are the position relative to the origin of the following the collision. Xe ~c-=tc. The geometry of the conditions (4. 23} and (4. 24) is presented in Figure lO. Substituting (4. 23) and (4. 24) into (4. 20) and (4. 21), one obtains -66- '' ''' ' '' '' ' '' ' '-- - '' '' -67- d c~) (c) (r-) c~> +-¥rz. ctt c-t) ===QC_ xr2.Ct) + -xezc-c ))fucx l'" r2.c-c) -+ xf2c-t)) (c) .Au2 (4. 2s CrJ - Urfl)ex - ~f2Et) > ) (Y") + ~ (t) 02 Now if the particle does not move "far" from the origin · of the decelerating reference frame Q we may expand and U, (c) (.:) (r) Q(xf2.C-t) Xe L\ c 'i-c following the encounter, in a Taylor series about the collision- xf2. C-tJ ' the origin of less position ( 4. 26) (c.) Xc::.'-tc. -t:-c: ( ..... ) J L at time \; • (c.) + xfi-Ct) )~ o.cxy:z.Ct)) + xf'" Ct)~ (xf2.C-t)) + · · · (4. 21> dvx Cr> I (c.) , .._ OV)I.. I~ + x~ (t) ~ex.'" (-b)) +··· (4. 28) Since Aul is the characteristic length over which Cl and ll change, (r) X'.f 2 (t) , the component of the distance moved by the particle X. relative to the origin of the \ x~)C-t) I<< A11, ~ ~c. =t--..-. frame, is not "far" if • In this case, the first terms in (4. 27) and (4. 28) are really adequate. Substituting (4. 27) and (4. 28) into equa- tions (4. 25) and (4. 26), we have ~x I Cr) (c) (s::.) (c) + a~ Ci)-=- Q(xp,Ct>) ( u.(x..f_-\:)) e - U..,Ct.) e ·) :&;' 2 J' -X ,z. -?< A<.11. (4. 29) -68(_c) (c::) Ur/t) and Since Xfz.Ct) are solutions of equations ( 4. 15) (-\:;) , and (4. 16) together with appropriate initial conditions as discussed ~~=) earlier, we find the velocity, particle er;_ relative to and position, X'f~rb;) , of iXcttc.:t:;c:: due to the collision impulse at t-=-i 0 are determined by (r-) (c> ) (\"') c:l.Y. cri. Ct ) :::: - Q( x f2 C-t]_ ufl. Ct) nr + \02. ( 4. 30) (r) d _xri.C-t) ( 4. 31) ol-t (r)(-l:-) = x~ a 0 with the initial conditions at time er.) Ct) L12 (to} ::: ~2( ~) jA.., ~ 0.. 0( In many cases of interest, constant in the shock relaxation zone, as pointed out in previous calculations. that the local variation of Q. and f" and hence Assuming may be neglected, the solution of (4. 30) and (4. 31) is .r1·) VT' ~pi. CXo) -0. - >.1.1 (t-t) () e z. + ( 4. 32) and (4. 33) Introducing the velocity equilibration time, rewrite (4. 32) and (4. 33) as ""C"v'l..-= \,./o... , we can + (4. 34) Furthermore, there is no difficulty in showing that for gas flow in the relaxation zone of a gasdynamic shock, xr- /t\u, /V C.r) I 1 0 ~(t-°t) <<~ , that \! 2 Au, <.< l. . Consequently, relative to the col- lisionless motion, the gas appears locally uniform to particles the increment in velocity of a CJ2: o;_ and particle due to a collision is damped exponentially. From (4. 35) we reach the important conclusion that, following the collision, the trajectory of particle linear to order that for a;: as seen in X'G '1<=- :C-c is Au-z-/>..., <<j_ . From (4. 35) it is also apparent 1 a< (-t --ho) << -c\), the particle az. moves only a finite l c. i-c: before it comes to rest. distance relative to the origin of Xe: Furthermore, since this limiting position is reached in a time that the particle will have a collision before it has reached this limiting position. The fact that \:""2.. < Tcl-z. << -C.U 1 makes it unneces(r) sary to include higher order terms in the computation of and Cr) Ur2. ( {-) Xr2 (-C) than those contained in (4. 34) and (4. 35). The particle will have another collision before ( 4. 34) and (4. 35) are significantly invalidated. distance moved by the particle ~ Under these conditions, the finite based on (4. 34) and (4. 35) is of particular significance, and we shall refer to it as the range, -70- ~ (X 0 ) , 02_ of the particle We might note that equations (4. 34) and (4. 35) are exact for particle cf motion normal to the gas flow,and the approximation enters only in the motion parallel to the direction of gas flow. The range for particle 02, is obtained and is given by ( 4. 36) We note that the range of particle o;:: RfG. (Xo) , depends only and the initial velocity increment received from the colli2. sion with particle Since ~=;Cf)(_;) , large, dense particles move the farthest for a given velocity increment. In general, it is expected that AU-i_ from (4. 36) and the definition Consequently, the condition 0 < ( t -t<l") << -Cul <fl = \ Uyz. (~) { ~ 0... -C-02. • Q Then \_Rp2 (Xo)l ~ ,\~ l~)(t) \<<Aul is well satisfied when • Using equations (Al. 5), (Al. 6), and (Al. 8), the components of the °i' particle range are ( ( R. J -= _!'\1-CU2 \ur, ex.)- U.r1(1'c,) l( l + cos 2.cf ) \'2- Xe. f.11 +Y"l.z.. ( 4. 3 7) {4. 38) (4. 39) -71Specifically, these are the components of the range of particle ffL due 0\ to a collision with a particle of radius whose impact parameter 1 I • I \ .. was \o;::: (Oj+<Ji) S\Y\'\" and azimuthal angle The geometry of the collision is depicted in Appendix A, Figure Al. Following the crz. can only move in the positive Xe:. direction. collision, particle This is a direct consequence of the manner in which momentum was transferred in the collision, since Up 1CX) > U,[><) throughout the shock relaxation zone. In analogy with the distribution of particle ties over a sphere in the velocity space of the °1,. recoil veloci- Xc:l{c. ~c reference frame, we also have a distribution of particle ranges over a similar sphere in the Xc.l\c=tc reference frame, Figure U. Now suppose that a collision between a particle of radius and a particle of radius or occurs at time t =to different from the plane in the the collision, the trajectory of particle at a position 'Xe_ L\,c'i=c system. a;: ~ Following is, by simple modifi- cation of previous arguments, linear to first order in ~ot../A.ul essentially finite length on the time scales of interest. This conclu- sion is accurate so long as Xe where position of the encounter relative to the origin of the and of is the ><c ~ c. "tc. C'f) reference frame. and range However, the tips of the velocity Rr2. Cx:, +Xe) vectors for particle urz. CXo+ Xe) <£" are no longer distributed over a perfectly spherical surface because of the variation of the relative velocity of particles along the direction of the gas flow. The surface is, however, nearly spherical because of the slow variation of the relative velocity of the particles compared to the ap- -72- Figure 11. Range sphere for particles with particles Oj. oz. due to collisions Reference frame Xe:: jc 'l:c. moves at the local collisionless velocity of particles <1z • -73proximate radius of the range sphere. Upstream of the origin of the range sphere is slightly larger and elongated, and downstream it is slightly smaller and compressed in the Xe direction. Note, however, that the scattering is still entirely in the for- ward or positive Xe. direction. The situation is sketched in Fig- ure i2. It should also be noted that the previous results can be ex- tended to three-dimensional flows and different drag laws. The con- cept of particle range, which we will use often in .the following analysis, is independent of the form of the drag law. The linear Stokes law has been used because of its computational simplicity, and serves well in elucidating the fundamental physical features of the collisional process. Summarizing, we can make the following general statement , is essentially independent of the particle -particle and gas -particle interaction laws. As viewed in a reference frame moving at the local collisionless velocity of particles radius cli , between successive collisions with particles of °l , a particle <1z: maximum length, the range moves along a linear trajectory whose R~ (X) on the momentum transferred to particle its velocity equilibration time. of particles o;:_ cJ2 , depends during collision and Furthermore, it should be clear that the previous results hold for encounters between particles Cfj and throughout the shock relaxation zone as viewed from local reference frames moving at the collisionless velocity of particles of radius Llf 2 (X) eJt:. a; I "' -' - "' :Z.c sr'nerica..l 1 - - __ 1,,"' "' / 2. // " Xe Xe ~(, <:lphe.r-i.ca.{ no+ 'ifheY"-l ca.\ I Figure 12 . Variation in the range of particles a;_ near the origin of -"c:~c re. assuming collisions producing the different surfaces occur at -C = t 0 with the origin of Xc.Y.c:"'rc. at .x position X 0 relati v e to the shock front. ~_, - - I ,,,,/ -xe, ~ ~'n.G<"ica.{ Sy>hertca..\ l~c I ~ I -J -75With these results, by appropriate superposition of the motions of particles , due to collisions with particles ~(1"') ()2.. '"°(!') calculate -f -f1. -t, -rz. The quantity Oi' particles of radius <r, is the mass flux density of at the origin of a non-inertial reference frame which is moving at the local collisionless velocity, Uf2- C X) o;_ of particles 3. , we can Calculation of the Scattering Flux. X '1 .:Z., moving at the local collisionless In the reference frame particles of radius (12_ between particles (J( particles (r) j f2. Ci velocity of particles is the mass flux density of at the origin of x~ -t and cJi. due to collisions in the vicinity of the origin. R.~ move a finite distance, , in )(. ~ ~ Since following an encounter, only encounters within a small region touching the origin of X. ~t can contribute to ~(~) ~ Consider a small volume element x~i of Assume particles ~ ..-1 olV , erect a Cartesian ref- -'-'_, X. l\ =l , with origin centered within iv C\ ,_, x- ~ frame is fixed with respect to the '1 located near the origin are at rest before a collision. In order to describe events relative to erence frame, d'/ xq-:c • The reference frame and its axes are parallel to the corresponding axes of the >< l\ ::t frame as described in Figure 13. From Appendix A, equation (AZ. 3 ), the number per second, ~~:tft. ( Xv) 0\ , o f particles 0::2 collisions within the element tive to the shock face, is J 1,•. by scattered into a so 1i d angle OUl. olV , located at position ~v rela- -76- Figure 13. -77- The element of solid angle is In the absence of collisions and neglecting, for the moment, transit time effects, the number per second of particles <l2, passing I J<s 1 , subtended by solid angle dSL,, on through an element of area the surface of a sphere of radius DI centered on I\. dv linearity of the particle trajectories , is equal to (4. 40}. ly assumed here that scattered out of JV R. due to the It is explicit- / is less than the range of particles OZ I J.Q.. • The element of sur- into solid angle face cut out by the solid angle is eR.1 ,JS where E:?R,t 1 :::::. RclsL. 2 JS':::: is a unit vector in the 01 (4.41) _,_ ,_, l'l. tors for a polar coordinate system in 1 direction. X ~ i. The basis vec(~ J ~f) are e\ 1 ) Using (4. 41) we may rewrite (4. 40) as ( 4. 42) where cl ~~Cl\v • ::: ny, ex,> rtro-<~ v J ( ur, cx, 1-u,..c.1<v J) ("it-q} ccslJ/d v [2 is the flux of particles of radius tive to ' ~' ""- lf ~ have assumed di:. , due to collisions in J ~2. I 11_/ (R:>'t'~ at the point l,1 d V • Since is constant over J..S In/ CA-.) r ~' ( rela- is small, we 1 • <4. 4 3 1 We note that -:!f~x,,J -782 has the units of particles/cm sec and is directly radially outward JV since particle trajectories are linear. The volume element cJ.\) may be viewed as a source of particles. However, there is no component of d_ff 2 ( ~ v ) in the negative -)( / direction out of dv relative to d Y since the scattering of particles from _1 occurs only in the forward 'I( direction. Let us now discuss the transit time effect. _,_(_/ X ~ -:C frames and ----:C. Now the reference , while stationary with respect to each XL\ other, are in motion parallel to the direction of gas flow in the shock relaxation zone. Consequently, the position Xv in (4. 40) and (4. 43) is changing with time, and there is a time dependence associ- ated with and Since the time required for X ~1- to move a significant distance through the shock relaxation zone is , then the characteristic time over which ~~ , and Ylr' vary significantly, as viewed by an , is also observer fixed in particles is non-zero only over distances of order rection downstream of JV , of Now the flux, Ao1 in the di- because of their finite range. Gener- ally, the time required for these particles to trace out their range is of order T 02 of particles Consequently, since olJpl..('~_) at a distance R -r:::",/ "Lu, <(< l. from dV varies slowly ~ to transverse the compared to the time required for particles distance R. 1 from correction of order the flux dV . Then, as an approximation which has a Tu2 /-c vi << J_ , we may neglect the motion of reference frame in the shock relaxation zone to simplify the calculation of Note that the transit time effect re- -79quires a correction that is of the same order as the correction required for the particle trajectory curvature effect. We have neglected '"C"i. is sufficiently less than -Cc21 collisions under the assumption Almost all the particles trace out their range between successive collisions. From this point on, the calculation is relatively straightforward and parallels to some extent the mean free path method of kinetic theory. Consider an element of volume ture located at 14). r::J.V R, fr , and <=\> with respect to i< ""\ =t.. {see Figure The flux of particles of radius ~ transit time effects, is the component of ~ d~15~) , which is given by x in the positive otrc ~)parallel to the -rz -v <{rf2.)((Xv) = -clr(x) f:i. -v co~{)-- where Lj"' 2. negative <~ <S. >t . There is no component of (4. 44) d~C~} in the X. - direction due to scattering of particles ume elements in the region ~ X -direction ct V , neglecting collisions and at the origin due to collisions in axis , of the gas-particle mix- ~ from vol- > O, since the scattering of particles out of a given volume element located in X ~ .:f is always for- ward. Using equation (4. 43) for ~ J~~Xv~ :::::-~ffXv) npt~v)(Ur 1 (X1 > -urfXv).X~~lco~clv'eo~tf (4. 45) ~ Expressing ~ dV in spherical polar coordinates with respect to -80- Ix IX . ~ ....... I,- -81( 4. 46) We rewrite (4. 45) as ~ J ~~l>v l "' -nr,vtr,_ cUr u I.) ("i +<>; 5'co:l\> cos{} SI\'\~ J.-&dr <IR (4. 47) t- ~.::::'}t"-~. where we understand that The total flux in the positive volume elements located in ~ where we note that x<O \( -direction due to collisions in is then in general, as depends on noted earlier, because of the variation of particle properties across the flow. As should be expected, from the considerations of Chapter V2, the integral summation is over a nearly spherical region upstream , see Figure .l.4. of the origin of X ~-!" simply because (Up ~~) 1 The region is not spherical varies locally,and consequently the parti- cle ranges are not independent of the position of the scattering event (M) is a measure of the maximum in rn (4. 28), --- Rp-i.ct,1) l\>, <y in direction lf, f distance Rl'2-(4, 1') particle range along the direction Only particles within a (M.) have sufficient momentum following a collision to overcome the viscous forces exerted upon them by the gas and reach the origin from 0-- to ¢ X.~ i' . Changing , we obtain from ( 4. 48) variables -82- ( 4. 49) Since there is no contribution from scattering events in the region ..-- X> 0 , ~ 12.X ~ () Then the net flux of particles of radius ""2.. _.. in the positive n- X -direction at the origin of ir;2 Xi, -:C is ~'IJ'l'J 111" r-J eos~<f s Ir{\> J<jJ.:li'. Jn,,vv ttp,-ur~) JI\. Co;+oz_ 0 0 ( 4. 50) 0 will be determined subsequently. where The flux of particles of radius the origin of , of particles the origin of X q~ the component of c!If 02_ in the negative i -axis at The flux, q -direction at due to collisions in a volume element R, ~ , and cy component of parallel to the is calculated in a similar manner. ctr~ cated at Gz_ JV lo- in the half space ~>o , is given by ~£ ~V) . parallel to the lj-axis. Taking this 2.. we have t J.~~ 1 "'° d~o~11 ) Sin.&-cos) and using ( 4. 43) and ( 4. 46) we obtain ( 4 . 51) -83The total downward flux is found by summing over all elements whose particles have ranges long enough so that they reach the origin of , see Figure 15 (i). The region is found to be nearly he mi- spherical, the deviation being due to the variation of the flow. (Ufl"LlrJ across Application of these considerations to ( 4. 52) yields Ctrt2 Kf2..C4_,~) 1Ii'2 1f ~~ ==Ji.~ =- 0;+aifjet>slf si ~S-JSjcos~)nr;nrl'il'lt•xlR( s31 4. l\ )O -172 o/2 The upward flux of particles parallel to in the region ~<o ~ 0 due to scattering events is obtained by taking the component of I in this direction: Jrc~v) -r,_ ( 4. 55) Using ( 4. 43) and ( 4. 46) one obtains The total upward flux of particles of radius °2. at the origin of --l >( is then found by integrating (4. 56) over the nearly hemispherical region sketched in Figure l5(ii). Again, this is a consequence of the i; -84IX l:r- lr\-1 . l() l'>< -85- The new upward flux of particles of radius 02_ at the origin of >l~t due to collisions in the neighborhood of this origin is given simply by ( 4. 59) The region of integration is the almost spherical volume described in Figure 14. In a similar manner, the net flux of particles of radius the ~ -direction is given by 02 in It is appropriate to note that the expressions (4. 50), (4. 60), and ( 4. 61) for the fluxes of the particles a;: at the origin of due to collisions, are accurate to zeroth order of "2.. since .L where -r:v,_/-c11 -::::. Av~';.. 0 (<ti/q-) << erence frame is located at .?'.( X ~ i:, l The I X~~ ref- with respect to the shock wave and moving at the local collisionless velocity of particles £T;_ , Ufz. Cx) . Let us now evaluate the expressions (4. 50), (4. 60}, and (4. 61) for ~~~Ji l , f\'z.ij()O , and ~'~~ l , re spe ctivel y, taking ace ount of the slow variation of the particle properties in the region of integration. The procedure to be employed necessitates approximations that involve the transit time and the curvature of particle trajectories. It will be shown that this approximation is of the same order as those made previously. R(VY\) The upper limit, , on the radial integral dR is a measure of the maximum particle 5J~ Y)rt_ cur,- ufd 0 range for particles the direction 'l2: scattered toward the origin of x;i along cp, ~ . This limit may be estimated by expanding about the origin. , of a particle <1l_ Using the components of velocity, that has had a collision at 2!'v as given by (A 1. 5 ), (A 1. 6 ), and (A 1. 8) together with the definition of the particle range ( 4. 62) then ( 4. 63) or IRri.l ==- ~ Cl-lr,- \,\f'>2. """"Cu~ c.os ~ (4. 64) Wt,+wi2. and short particle ranges, as indicated in Chapter V-2, the maximum range is nearly the same as the range in a uniform system. (m) proximation to R?l..(lf) er) We suppose that a suitable ap- is a Taylor expansion of (4. 64) about the range for a uniform system. This is the range .!Syz.. evaluated at Therefore, we approximate where x_ refers to the position of X~ "t:: wave and is independent of ~ or with respect to the shock <p . Then using equations (Al. 5), (Al. 6), and {Al. 8) together with (4. 62), we obtain ( R~ 2\.~ J~ :::.- rY\~l. 'M1+1'¥)2. ( ~,cx) -d·-lr,(X)) ( l + co"S 2.tP) ( 4 . 66) -88- ( 4. 6 7) (Rf2.(~J)~ == M;c,i. ( ur,0() - Uy:i..00) ~ ln.~4 S\n l (4. 68) YY\+M2 and "O Ry~~) Ox ~Rr2-( X) :::=. \N\Tu z. ~ W1 1+M2. 0-x c0y 1(X)- Llrl-(x)) cos4' ( 4. 69) ~r,C~) 0 ~ 0\ ( 4. 70) The radial integral may be written by defining F(~>RJ\f>Cf) =nr,c)(v)Y1r}~..,)(uffx~1 -Uy~Xv)) as ( 4. 71) ('M) !Zr2-( 4', 'Y) IR.Cl'i tP,1') = j Fe~, R, t\>><\') clR (4. 7 2) 0 Because of the slow variation of particle properties across the gas flow field, to a good approximation, we may expand series about the origin of )( f in a Taylor L\ ~: fo:()R,J~1) = fc~) xJ q>=t.) ~ Fe~) %q (~)1 + ~~ ( x:) z + .. . + ~<XJ x ex (4. 7 3) -89Expressing )(, ~ , and ~ R. in spherical polar coordinates 4 , and er ' namely, X = R. cos-\J- ::::. - R_co'S~ ( 4. 7 4) ~::::. Rs--~n-3' cosy - Rs\h~ cost ( 4. 7 5) ~ = R.-s(nif-s\n~ ( 4. 7 6) R~·\Y\~ cos\ and using them, we substitute (4. 73) together with (4. 65) into (4. 72) to obtain ~ R (X) ~- J[ FCXJ C) Since the gradients are of second order, we neglect products of them and (4. 77) becomes 4'_~ Fe.-{ R11-c~) + ~x.Ryp~.) (~p~_) h } Rrf~) +Jt~oo(-R.eo,.1\-) +~~i(R.s'1hlfcos'l) ~ c + ~C't.J ( Rs'1n~ sin<d di( Carrying out the integral in (4. 78) over ~ , and \ are independent of 4 78 (. > I\, R , recognizing that and using the definition of given in (4. 64), it is straightforward to show that X, -90- lR ~Fe~) R,.<~) + fCJD [ ~v!!) Cfy.\z + !r.Oo ) ~ex> l~J, + .~..Fe~) C~)- +£.fo:>C~ l..( < ~ Lax '2. j 4. 19 l '1 'CS d:t: * which may also be rewritten 4<.~ Fe!!. i ~i <1 i + ~ L f Fuo q].rl!I'. l + UR.r~ ~i Fo'->) ?ClriJx ox o-x 5 + ±1 ;~ RrI20 Fcdj (E,l"-)~ +~ where f~('Krft.J Fo~~J C~,_h IR= Ii<C~}h r) since ( 4. 80) Rrz. = £-rf~,t\>, cy) · Using the definition (4. 72), the fluxes (4. 50), (4. 60), and ( 4. 61) may be written as T/2 21t" ~,~Y l =(a + oa.),_ )Jt J<l.'f l sin.4 co?-+ T ~a', l[> 'f) z 1 " and (j () L 1 J (4. s 1 J -91- 11/z. 2.1\ - - (<Jj+ <>;_ f 1 j d<(> ~J'\' cost\> ~in'l.4 si,..~ IR.C~,<l>J't+ 4• 0 83 ) 0 Substituting (4. 80) into (4. 81), (4. 82), and (4. 83) and then carrying out the somewhat tedious integrations, one obtains Then the random flux, at the point X J;2.( X) , of particles of radius relative to the collisionless flux of particles uf '-cx) fr,c~J = a; CJ2 is given by + pC~) e f2~ -1 The character of the components of .fr2(~) as given by (4. 84), (4. 85), and (4. 86) may be accounted for by the following considera- -92tions. Because T 02 < T<z.i , the particles collide, on the average, only when moving at their local collisionless velocities. in the relaxation zone of the shock wave, X for all are characterized by particles ting them from behind. Consequently, collisions CJ\ overtaking particles 02_ Particles and this increases their flux in the a;: and hit- are always scattered forward, X -direction and accounts for the first term on the right hand side of ( 4. 84). However, because of the variation of the particle properties o:;: which are scat- across the gas-particle flow field when particles tered out of different regions in the flow come together, they are out of phase. A net flux of particles thus arises. There is a certain randomness introduced into the system by the encounters and the variation of the particle properties. This is the essential origin of the second term in (4. 84), The variation in the particle~ 0/ density normal to the gas-particle for the net flux of particles in this direction. lar in effect to the second term of (4. 84). rather brief. fl ow accounts These terms are simi- These statements are Later, we will take a much closer look at the interpre- tation of the character of the particle fluxes • .....Pc ... ) The random mass flux density j:f2. can be derived from (4. 84), (4. 85), and (4. 86) by multiplying by the mass of a particle o;:. -93'°(,.) Physical Significance of Expression for +. 4. ~~~_;__~~~~~~~~~~~~'--~~~~~--'"~~ In terms of the mass densities of the two particle clouds, f'r~ and = Y\'"'1_ Ylf 1 , the components of the mass flux density of particles of radius er.:2. relative to their local collisionless mass flux ( 4. 87) Cr) Cf!"-)'1 =: - : ("I+ <r, )\·n, l:V~ ~ ?r' fP< ( ur,- ul" ~ ) 2 ( c~,+m2.) ~ ( 4. 88) and ( 4. 89) Derivation of these relations involved assumptions, some of them concerning the kinetic theory method, others the particle-gas and particle-particle interaction laws. Because of the large numbers of particles involved, the restrictions introduced by the kinetic theory method are minor. Therefore, any inaccuracies stem from the validi- ty of the particular gas-particle and particle-particle interaction laws and the assumptions introduced to facilitate their application. Consider two important approximations: curvature of particle <Ji (i) to neglect the trajectories in reference frame X ~~ -94caused by viscous interaction of particles with the gas, and (ii) transit time effects in X q-=t; resulting from the finite velocity of ref- in the shock relaxation zone. <Ji: The trajectory of particle have a curvature correction of order tion. in the X (Av, j Au 1) q~ frame will to the linear mo- The transit time effects are expected to be of order Because ( <:r;./<Jj J'2.. <:: <:: .L same order. ~2. /-cur , these effects are small and of the Since the expression (4. 87) is the sum of two terms, the first of which is larger than the second, it is important to compare their relative magnitudes. Let us define, for purposes of examination, ( 4. 90) where ( 4. 91) and (4. 92) The characteristic length over which the relaxation zone is of order \pt ~ lll and (~,-~,_) vary in It is also to be expected that, because of collisional coupling of particles 62 and CJf sity of particles a;: teristic length. Consequently, it is to be expected that a. ( fpi fr a~ , the den- will vary significantly over the same charac- ~ t. ( ~f ,- uf:L) ) /\/ lf'- er~ (\LJ -Uy~)3 1 l\V~ -95so that ;S 0 (l) Since we conclude that (4. 94} ~l :g Therefore, relative to rection of order the flux term represents a cor- (o;:/61 )'2- , the same order as that to be expected from particle trajectory curvature and transit time effects. ~ to in contains only part of the term of order 'f1 ( At2 /Aul) Hence, relative Therefore, to make this calculation correct to first order ( 1" 2 />.u 1 ) , further restrictions are required. The relative magnitudes of (~(i-)\ -r· )~ (r) with respect to (~f2. ) 1 and become important in the equation of continuity, and there- fore it is appropriate to compare Suppose that i-l. A'1 'd'f~ with "Jx affr )~ 2 a L-\ d( f r.1) : ) -o:i- are the characteristic lengths of the and density distribution of particles of radius <I2_ in the y and z directions, "2 '2. respectively. Therefore ~p~ "\.J ~ d ~' 1~ and ~f'Z N d<t2. _fu A~ since -96only ff2- depends on y and z in (4. 87-4. 89). Using these approxima- tions, 'df'.l. ox d~2~ 01 ( 4. 95) 'CJ\ similarly, 'OX 'd(f(r)) -r- ~ IV 0 (C~J) ( 4. 96) d:C From (4. 95) and (4. 96) we can conclude that, provided ~ )2, <:~ Tt ( Av 1 (4. 97) 1 ().."YA.~) can be properly neglected. The detailed evaluation of ~I and A:z: depends on the upstream density corrections to of order fr2. (l~-=1:.) , of particles az. and the collisional spreading of (1) profile, this profile downstream of the gas dynamic shock wave. the rigorous justification for neglecting after calculating ff2... Therefore, 'f.2- must usually be established -97We shall restrict our selves to circumstances such that, when (~/Au 1 ) << l particles £l2: and transverse gradients in density f F of 2 are not small, we may write the components of mass and Then the total mass flux density of particles cs;_ at a given point in the relaxation zone is given by ; (4. 101) or in component form (4. 102) -98and The symbol -o e ?;I -1 denotes the transverse gradient operator. ':?r .l... 2 The representation of in other coordinate systems, for instance cylindrical polars, is readily obtained by expressing v.L in this coordinate system. (See Appendix B for treatment for the representation in a cylindrical coordinate system.) Equations (4 . 102) and (4. 103) express the mass flux density of particles of radius oZ as a function of several known quantities and the mass density lf2. of particles cr;: Consequently, by substi- tuting (4. 102) and (4. 103) into (4. 10) we obtain an equation for \y:z. which, together with appropriate boundary conditions, determines ffz. throughout the shock relaxation zone. Before proceeding with this calculation, let us consider our results in terms of more familiar kinetic theory ideas. quantities Introduce the ~ and,) where 'N-x) is a measure of the maximum distance moved by a particle of radius <Ji, following a collision, in excess of the distance it would have moved without collisions. Acx) = ~a. ( ur,(X) - u.r,_(x.)) ty\li-M-z. (4. 104) -99The quantity -V(X) is approximately the local frequency of collisions -l b e tween particles of radius cJ2: and <Ji" 1..e. -c-=1J<x) is the local average <::21 time between successive collisions. This may be written 1 f ~I Ufl - ~&.) -V(x) ~ lt(<ll +"l. f~ ( Uf,- U.r:z. J ~ 1( ( "( T"di ( (4. 105) V\'11 Physically, the length ~(X) is the maximum distance that a particle 02. can move normal to the gas flow following a collision at x. fact, it is the transverse range of particle 02. In Using the definitions of A and -V, Eqs . (4. 104) and (4. 105), the components of the mass flux density of particles ~ , Eq. (4. 102) and Eq. (4. 103 ), may be written in the form; C~rJx = ff'i ~· i- J \-v ~)( (4.106) and i-f'2 - -...L~ 't-u v..L \r:z. (4. 1 07) .1.. The motion of the particle cloud convection with velocity di is characterized by an average 1Jf1 == Uyz. + A-V parallel to the gas flow accompanied by a diffusive spreading of the particles .~ across the gas flow. Evidently the average convective velocity of particles ~ from E q . (4 . 106), is a sum consisting of the local collisionless velocity, -uy 2 , of particles OZ:- and a local diffusion velocity, \v , due to the collisions between particles of radius o- and particles of '2. radius 6j" . Marble 3 has pointed out that these collisional effects may be interpreted as a force per unit volume existing between the particle , -100- clouds. In other words there exists a "dynamical friction" between the cloud of particles 0( and the cloud of particles CJ2. . The effect of this interaction is to increase the average velocity of particles Ur1 , the local collisionles s velocity of particles ~ . toward '\/'f 2 degree to which magnitudes of -C ":z. The approaches Ufl depends upon the relative and '"C"..-~ • - ~1 The circumstance such that the average time, """Cc21 , a particle a;: spends between successive collisions with particles 61 exceeds its velocity equilibration time, -Cllz. , is of special interest. Before a collision particles OZ: are moving at their local collisionless velocity, 1.lp2 , which may differ significantly from Ur, , the local collisionless velocity of particles <Jj . Following O'i a collision the momentum and energy received by the particle during the collision are damped by the viscous forces exerted on the particle oZ by the gas. The particle returns to its local collisionless velocity ~J. before it has another encounter. cous force exerted on the particles Consequently the vis - a;_ during their motion between successive collisions tends to force particles of radius ~ to take on their local collisionles s velocity which is near the velocity of the gas, since Av•/)vJI is small. Therefore, the dynamical friction force competes with the viscous drag force. It is the interaction of these two forces which determines the average convective velocity of particles Oz , "U'r1..~U.fz. +AV , parallel to the gas -particle O)" flow. Let us compute the dynamical friction force , f Pf , acting on particle cloud Cf2. and compare it with the Stokes drag force. by Denote .f0 F the force per unit volume exerted on particle cloud oz due to collisions between particles cJiand <1j. f~F' depends on the rate -101- of transfer of momentum from particles <J{ to particles ~ resulting from collisions; its jth component may be written; (fc~)).:::: \ti( ~~0 2.(~ra_)). o\Npi.C~>.n.) DF where j J 4JC 6, (rn2.. Q~~~)"S2.) )j is the jth component of momentum change of a particle CZ. that scatters it into a solid angle ~r+ c~.,.Q.) d;tc).V (4.108) J d:tdV I is the number of particles ..J2 - _Q +dSL. a;: per second, per unit volume, at _2l that experience a collision and are scattered into a solid angle J2-JL +cliQ. Now since to a velocity difference and ,6(rnz.~~Q.))j is proportional ~V(Q) is independent of the reference frame from which the collisions are viewed , the reference frame in which the integral (4. 108) is evaluated is irrelevant. local collisionless velocity frame for particles o;_ since they are at rest in this reference frame prior to a collision. (Al. 5), We choose the Therefore from (Al. 6), (Al. 8), and (A2. 5) of Appendix A we have; ~ (1"('2_\:f~K/l) );( =- ml h'\;z. (Uy~><) - UrfJO)( .L + c.os 2~) (4. 1 09) (O\rM4 ) 6(m2.~C~)S2.)) == ~(ur 1 Cx) ~ u.rix)) Sin1.~~cs<? l (Mi+~) \ b, (r,vJr2 (X>-9-)):c= 3 M-i. (llf,<x) - ~~))'Si 112~ 3tY1l (M 1+m2.') and (4.110) (4.111) -102Then by substituting (4. 109) and (4. 112) into (4. 108) the force per unit or (_ft>f~J)/~ f}f><) ~~~) 1C(Of+o;.)c u.1,cx>-~Cx>) ILlf CxJ-Uy,_uo\ 1 (m 1+M2.) This result is valid for all values of (4.114) CZ:j°i when '"C'ui_ < Tczl The following discussion, however, is restricted to the present case where 02/<1 << 1 . An equal and opposite force, -(EI>~~)) X is exerted per unit volume on particle cloud <Jt • However, since this force is negligible compared to the viscous drag force acting on particle s OJ . In a similar manner, using ( 4. 110) and ( 4. 111 ), one finds, by carrying out the integral in ( 4. 108) (.foFoo)~ 0 This result is not surprising since in the calculation of (4. 115) ~r.~ d.tO.V all impact parameters for collisions within the volume element dV were taken equally probable. Under this condition there can be no net transfer of momentum normal to the X axis which is parallel to the direction of relative motion of the particles prior to an encounter. -103Consequently the dynamical force, upon particle cloud ex F (<S_).::::: Ft>F(x) - , exerted -t>F c;_ , acts only in the positive x direction and tends to increase the convective velocity of particles a;_ the relaxation zone. as they pass through This result is consistent with expressions 4. 106, 4. 107 for the mass flux density, ~p<. , of particles ~. We are now in a position to investigate the relative magnitudes of collisional interaction force and viscous forces acting on particles a;: . Under the assumption of Stokes law, the viscous drag force per unit volume acting on particles ()2_ is (4. 116) Then using (4. 114) and (4. 116); (4. 117) w her e -l T<:.il =. L> for particles is the average time between successive collisions 01_ Consequently since °I'/" CZ and \'Yl ")> h1 2 1 the collisional to viscous force ratio depends essentially on the product .j of the ratio, l::u1. L:"ci..l , of the velocity equilibration time to collisional time and the ratio, (~L"Ur2.) / (u.-21>2.), of the difference in the local collisionless velocities of the two particle sizes to the slip velocity of particles CJ2_ • For the simplified collision model employed here, -Cu2.. / Le-z1 is less than or equal to unity. evident that, when C!2/'1j << l From Figure 8 it is , the ratio (uf'2--Uy 1 1/(u.- L)>:z..) becomes large downstream of the shock wave. The solution for LAri, 1..Ay2.. and \A. , Figure S., show that, immediately downstream of the shock, Up(VUr 2 and U-~ is large; therefore viscous forces -104X ~ ~Oz. , however, the local collisionless velocity dominate. For of particles oz has decreased nearly to its small slip value; U- U pz l'\J Av2. -U. JLL __, Q.. (4.118) ~ Cue\Llrz.) may be large and consequently l\J (UCO - LA.('2.)) In this region the collision forces dominate. We may conclude, therefore, that there exist conditions, when -CU2../Lc:.21 < 0 ( 1) , under which collisions dominate viscous lJ2', forces in determining the macroscopic motion of the particle cloud under most circumstances when <12/"j << l . It is interesting to rewrite (4. 114) using expressions (4. 104) and (4. 105) for ~ and -V r~<~) (4.119) This suggests that the rate of increase of the diffusion velocity of particles OZ is limited by the relaxation time,~ , of particles ~ In addition, dividing (4. 114) by the Stokes force (4. 116), we obtain, \v (4. 120) (u-u.f'l) may be significantly greater than unity, the effective diffusion velocity, Al>, exceeds the slip velocity of particles CJ;::. Expression (4. 107), which gives the mass flux of particles Di normal to the direction of gas flow, indicates their motion to be diffusive but not in the sense that the collisions are completely random. The reason for this is the fact that collisions take place only when particles <L are overtaken by particles cr, and always recoil in '2. . 1 -105the direction downstreain of the collision site. Within this restriction, all impact parameters are equally probable and consequently the normal scattering introduces some random character into the transverse motion of particles 02:. The numerical effect of the anisotropy in the scattering may be seen by rewriting (4. 107) :j'.f?.1 == - D D :::: t ~ -V v,L frz (4.121) 2 where is the effective diffusion constant for the transverse dispersion of particles Oi_. (4. 122) If there were complete randomness we would expect a numerical factor of 1/3 instead of 1/6. It seems appropriate to ,interpret D as a diffusion constant, recognizing that the analogy is not complete. When a diffusion process is purely random the diffusion con. g1. ven by ,· 12 ' 1 3 st an t is (4. 123) where Lis the characteristic step size of the diffusing particle and ~ is the number of collisions per second. The motion of the particle appears as a random walk due to collisions. For in stance the diffusion of gas molecules of a given species, i, through a much more dense uniform distribution of gas molecules of another species, j, is described by the diffusion constant; (4. 124) -106where Ai.j is the average distance moved by a molecule of species i between successive collisions with molecules of type j and -Vi.j the local collision frequency for collision. in Fig. is This process is sketched 16a. A contrasting example exhibiting anisotropy is the diffusion of a weakly ionized gas which is drifting parallel to a strong uniform magnetic field B. The charged particles are nonuniformly distributed. Because the gas is weakly ionized, only charged particle-neutral collisions are important. In a reference frame drifting with the gas, the magnetic field causes the diffusion process to be anisotropic. Because the magnetic field is strong, the cyclotron radius fc of the ions, is much less than the m~an free path for ion-neutral collisions, Under these conditions the diffusion of ions across the magnetic field is determined by (4. 125) where -: : : . -u: fc. --Ve. ; -Ve :::l ~· • ~is the average velocity of ions I, perpendicular to the magnetic field, e/m~ the charge to mass ratio of the ion, and 1.-? 1.Y\ the collision frequency for ion-neutral collisions. The magnetic field does not affect the diffusion of ions parallel to it and consequently '2. Dil\l'V \iri. vi)'), The diffusion of a test ion is depicted in Figure 16-b. (4. 126) Since the motion of the ion across the magnetic field is constrained . The ion moves normal to the field only by colliding with a Figure 16. tion zone, L. ~ (C) "'-' )_C.z1 · ~-- B ~- )>j 1J.. == -U...<x) ~x I 0 --I ...... diffusion of an ion in a magnetic field B. Characteristic stepsize for motion across the flow is fc> The cyclotron radius, parallel to the flow in a drifting reference frame it is Ain the ion- neutral collision mean free path. <b) - Diffusion of a solid particle o-2. across a decelerating gas-particle oi flow in shock relaxa, characteristic stepsize across flow is A the transverse range. cr2. /a; << .l ....... r ,.,_, A<x> j.-,.,,). --,.( diffusion of a molecular test particle of species i. Characteristic step size of particle diffusion is mean fre e path Aij (0.) B 0 -108neutral molecule. The characteristic step size for the displacement is the cyclotron radius \c (4. 127) In light of this result, the correspondence of (4. 122) and (4. 123) suggest the following analogy. The motion of particles of radius 02, aero ss the gas flow, due to encounters with particles Oj', is a diffusion pro- cess in the sense that A(x) may be interpreted as the characteristic ~2..L A./ Ao2/ Aez.t < l In this sense\ corresponds to ~c for ion diffusion step size of the <lisper sion when (See Figure l6c). in a magnetic field and-Ccorresponds to Tc . Oz. a particle (f) 1L r2· .l. (,f.) Tu r'2..L 2. oz. moves a distance A=t.l.. Following a collision, across the gas flow where is the component of its scattering velocity normal to the direction of the gas flow. The gas, through the action of viscous forces, constrains the perpendicular motion of particles o;_ analogous to the magnetic field-ion interaction. 5. Solution for the Particle Density pp • 2 The equation of continuity (4.10 ) and the expressions (4.1P2J and (4.io?>) for the mass flux density, ~ -f2. , of particles of radius may be written; ( 4. 128) and (4. 129) -109with; (4. 130) (4. 131) In Equation 4. 131 "\i. is the gradient operator of the coordinate system used to describe the distribution in a plane parallel to the shock face, fr2. is the average density of particles of radius <Ji", ~the local collisionless velocity of particles of radius <Ii_, AandV are given by (4. 104) and (4. 105). \ (x. ") :=. ~To.z. ( Up, 0') - U fz..(x) ) VV\ 1*Yn..z. (4. 132) and 2 -V(X):::: TC (Oj +~) _ff'(X} ( lAyi ('() ~ ~(X)) (4. 133) vn1 Substituting (4. 131) and (4. 130) into (4. 128) using (4. 129) we obtain; (4. 134) where only f\.>:z.-;::. \f (.X) is a function of transverse coordinates. Sine e 2 'lfti.(X) is independent of these coordinates, (4. 134) may be rewritten; 2. - where ~2- kO<) =:. D(x) /u'rfX) K(X) \7.l (\ft.(~) \Jf2-(X)) -=: 0 (4. 13 5 ) . Now, as remarked earlier, fyi , Up,, and T~ are known functions of X. obtained by solution of Eqs. ( 3. I)- (3. 9) and Eq. (4. 11) together with conditions (4. 3) . determines Equation (4. 135) fr-== ff 2C~) when supplied with appropriate boundary and -11 OU sing 4. 130, the mass flux density ~ initial conditions. fi.x satisfies a diffusion equation; 2 d ~ (X)' - - f1- 0)( x with diffusion coefficient X)o, Eq. (4. 136) for and \c(x) \JJ.. ~C~J ::::: f2x (4. 136) }<(x'):::::.. t)(x:") }r-~). is parabolic. o Since KO<) > 0 The mathematical problem ~fzx is properly posed when 'f~(~) is given at X-=-0, for all ~(OJ :0,.L) r2x is bounded for for --oo<l ~.d<oo. 14 X 1 ~ , These conditions are met by our physical knowledge of the problem. To establish '*-rio )~J...) we observe that particles cYj" and <Ji pass unaffected through the gas dynamic shock. are continuous across the shock . of particles particles ~ Then if ~f All particle properties is the mass flux density 2)( OZ. in the X direction and ~..Lis the mass flux density of in the directions perpendicular to X, (4. 137) -==o Here, l denotes conditions immediately upstream of the shock wave and 2 denotes conditions immediately downstream. (4. 3), expressions (4. 106) and (4. 107) for and taking the limit of Using conditions ~fa.><. and ~2.j_' respectively, 'fp.12>(, ~p as x~ 0 from downstream of the -,2..l. shock wave, it follows that conditions (4. 137) are satisfied. the density, In addition ff'.2. , of particles ~ is continuous through the shock wave and this situation is consistent with continuity of particle fluxes as examined earlier . -111- The mass flux density of particles 02' is determined by (4. 138) together with; (4. 139) and the values of {1~~) downstream of the gas dynamic shock wave are frz.(~) =~cl~) "'"Uy.f><) where Uf._Cx) =f 0 (4. 140) 1-Ar~(X) + A(x)i.{x) for all X. "2:0. Consider the variation of ~ rz..'/(. with respect to the xyz carte sian system oriented such that the plane X-=O contains the shock face. Then (4. 138) and (4. 139) become; (4. 141) and; (4. 142) Transform (4. 141) to canonical form by writing x ~(>() = ~ k(><:'l dx:' 0 . ' x~o (4. 143) (4. 144) Because ~::::o when X=O the boundary condition (4. 142) holds when ~=O. -112- (4. 145) corresponds to a delta function distribution of particles 02, upstream of the gas-dynamic shock wave (4. 146) where the delta function has the properties Sccx-~ > = ~ e _(0(-o<. '2. 0 ) /4f ~~o~ J Seo(- f ro -(X} Consequently; Ol0 ) JD< - .l. (4. 147) 00 ~J f~Cl) =l:;) ~dt- Ny2 and we interpret Nfz. . -00 as the number of particles of radius <l2_ in a cross section of unit thickness in the X direction of the distribution of particles~ of the gas-dynamic shock wave. upstream We call this the fundamental beam solution. Because (4. 144) is linear, we may write the general beam solution of (4. 144) as a superposition of fundamental beams -~ (l-~I) '2..+(~-7//-J/4S6<) 00 ::f.p}"Jt,"') "" '>< 1 41tS(X) II' t coHi) e J.tt' u J J -oo 1 X. Using (4 . 140) and (4. 145) the density of particles function distribution of particles <72:' upstream, is; ~, (4. 148) for a delta (4. 149) (\) \tz. (~) =t) so that, for a general density distribution, r c~4()2.+(~ t:.)2.lf4 f l 5 sex) \.froo 4"T(~~ JJ ffi.ci):t:>e ~~1J~ (4.150) p ( x L\ =t:) - CU.Cl) ) lf'l- > ') - 00 I - \ (i") I I I - -00 2. X. utilizing prev~us results, :) 60 , we have; =J Do() dx' = ..LS A<~') (Aci<,-v>o/i )d.x" ~ix) 0 (o 0 (4. 151) lFr-fX/) (4. 152) A.ex.) = ~l:"oz.. ( Uyi (x ) - LLf'" ex )) (4. 153) m1+m.2 and -r>c~);:::? xcoi--+az ) fr ,oo cur, ex) - LArz oo) 2 (4. 154) YYl.1 As X becomes large ~I, value U(ao). }<. llpz. and 'U. all reach the same limiting Therefore because of the functional dependence of 1rf2. and on 11>:z. and Up! . ~6<)~ constant as x becomes large. Consequently the density of particles a;_ always reaches a limiting distribution far downstream of the shock wave . There is a "freezing" of the density distribution of particles of radius 0:- • 2. Later we will show that this final distribution is essentially reached when)(> 0(\1.1 ) . 1 equilibrium state of this gas-particle mixture is therefore The final -114ll(co) -:::. uf 1Coo) T Coo) :::. T p1 Coo) - and or CX) f'("">[i"°)= 1:lfl) I u.Coo) 4~CO'.)) \ } j -03 co where; S <;cro) :=:. k<K, dx' () In this state the particles and the gas are in velocity and thermal equilibrium but particles Q2_ are nonuniformly distributed normal to the X axis. For the present, restrict the analysis to a two-dimensional problem, that is one where particle a;_ distributions are independent of~· The appropriate fundamental and general beam solutions for this case may be obtained from (4. 144) or by integration in (4. 149); (4. 155) and pf'2Cx/p == (urn~ I ----, I f~li') e o\~I 1.ff (x)) 41tSCX) j l (4. 156) I i- _a) 2. In the pres e nt case 2 -~-~{) /4-:;oo cc -Nrz.-= 1vi SO;)pCl)(~')JL\ / ..::. ()O \ r:z. I is the number of particles in a cross section of unit thickness in the x and z direction upstream of the shock wave. Analytical evaluation of <;;(X) is not possible because ..\, -V , and '\f1'2 are obtained from numerical solutions of (3. 1) to (3. 9) and (4. 11 ). Consequently further analysis must also be done numerically. Before presenting results of such calculations, let us review the conditions for the validity of our solutions. Introduce the dimensionless variables; l'\J x.:::. Au l x ~ l ~ )wl i /\..) f =rel) f \r1 = \(1) fr1 Al "\.I U.=G\.CI) U rv T Tc1) T u.r, ;:=. Q(I) ur, ir, ~ Tct) 1f, /\.) '\) (I) fFL= \f 2.co fri. 1\1 (I) fr? f~l) ft?- /'\.J '\.) llyz.-;:: o..c,) u.r2.. Q :::. Q(\) a. (4. 157) ~ lpt. .=. TCl) lf.1. and (4 . 158) (\) where fr£. O is the characteristic magnitude of l~'l.(L-\) . The charac- teristic transverse range, A('l.), and collision frequency, 1>(2.), are -116· given by; A(2) ::::: ~ -c..,z ( U(O - ll,(2)) (4. 159) mi't-mz. -VC;2.) .:::X(crl+<r )~) (LLCl)-u.(2-))== 1t(<T +cr2..}~ Cu..c\)-t,L(2)) 4 m 1 m1 1 (4. 160) Substituting (4. 157) and (4. 158) into (4. 155) and (4. 156) we obtain; IV 11, ) cf)"' "'-' ff2 ( X) L-\ ') '\.1'21 (l\J -urf - ~ e ~ 4D~CX) rv rI x) \14 K D:S<.X'J rv ( 4. 161) I and Ac~~;=-~ M1 \~_I_ ___, \fl- ) \~~x)J.l4iro~cx~ \J C"f) where we have taken P. l -co f - '\I(.+) - -::::;.~'2.. /\Jf ( )~p (o)A , since ma.'N/ ~u 1 r'2. Av ri. I rz. \f :2. r dimensions of a density, and has l (4. 163) fo , addition rewrite (4. 151) and (4. 152) using (4. 157)-(4. 160); "V Av x. r A.J x )( (-.J "-'/ "" jy I ~c~)= ~/>!,o-.=: J kG)o\?C:::: j ~c:; d.1( = J"<.X) ~x,V~x) (4: 164) l\J C> ".J A,J \ I\) (\I 0 VfjX) AJ \ \ ,"1 I 0 I VfJX') and (4. 165) where; -117- V(2) ~(2.)-V(2) (4. 166) Q(I") The dimensionless quantities D and V characterize the relative dis persion of the density of particles °2. downstream of the shock wave. Using Figure 8, the definitions (4. 151), (4. 153), (4. 154), (4. 159) and (4. 160), Figure 17 indicates the approximate variation of the v/-V(2.), and (u.fl Upz.) V>..cz..) with x. Ur Urz.• LL , 1 The dependence of vM-2.)and Y.x.(z.) on <-ct 21 stems directly from the collisions model valid whe n ~ They reach a maximum at Xl'V ~11 2 and since 'AvL/>.._.., 1 <:< _L maxima occur close to the shock wave. , the Because of the relatively slow variation of the properties of the gas and particles ~ in this region, (-v(x)) Ma.,c_ /'\.J -V (2) and (\°x) )WYl>./ l\J A(2..) . -/' \(z.) are defined by (4. 159) and (4. 160). where --V(2) and Consequently'L/(2)is approxi- mately the maximum collision frequency for particle ())- particle ~ encounters within the shock relaxation zone. maximum value of Similarly A(2-)is the A , the transverse range of particles ll2_ , through- out the shock relaxation zone. The physical conditions underlying the validity of the collision model and its consequences, Equations (4. 149) and (4. 150) are as follows: (a) ly, The region over which the gas flow field varies significant- Au1 , is much larger than the velocity equilibration length of par- ticles~ (4. 167) -118- U.(1) "'>-.iJ wo--------"' ~ul _ _ _ _ _ _ _ ____,... x (i) .L.O -V/-J(2) x loGP--------- ""'f..ul-------~ Figure 17. Approximate variation of particle collisional parameters downstream of the shock face -119(b) The particle Reynolds nllITlber and Mach number is such that the Stokes drag law is valid. Re2. -= f Oi l ll - Uf z. \ < .L l\J Ill- lA.e2. I <<l tL Re =- \ o; \ U-Llr I < l 1 l (c) (4. 168) 0.. ft f'V \ U-Lle 1 [ a. <<:::::.l (4. 169) The time between successive collisions for a given particle must be as large as or larger than the velocity equilibration time of the particles (4.170) (d) The velocity equilibration time must be large in comparison with the time required for one particle to pass through the flow field of the other. Analytically, this may be written; ~ (~ )(F 02 Ii- ur~l)C-f) >>1. ~ (~)(E"l l~,-Ur~l) ( (e) The mass density the mass densities f) >> J_ (4. 171) (4.172) l\'2- of particles <:Ji is much less than ff\ of particles C\ and f of the gas; (4.173) (f) "large" The transverse gradients of the density of particles o-2. are -120- (4.174) When Stokes law is violated in the shock relaxation zone, the mag nitude of the gas-particle interaction is underestimated. particle o;:_ rang e , used to compute the mass flux density based on Stokes law is therefore too large. The '£\2. , Consequently, we may expect our results to be an upper bound on the actual dispersion of particles 02_. 0( /\.J I 0 Jl For strong shock waves the maximum particle Gj" M1'V ~ , and particles with Reynolds number varies between 10 and several thousand and the maximum particle Mach number is less than or of order unity. On the other hand, values of elastic collision parameters, (4. 171) and (4 . 172), generally exceed several thousand so that in this re g ime the effect of the presence of t he gas on the collision is negligible. The elasticity of the collision depends on the composition of the particles. The only effects which may be significant are: ( 1) the effect of the compressibility of the gas since the Mach number of particle O"j is of order unity prior to the collision, and (2) the effect of the wake of particle 0( on particle oz:- following the collision. These effects are not well understood at the present time . Let us now consider condition (4. 170) in more detail by writing; -c\,2. (4. 175) ~2.1 The local collision frequency -V is given by (4. 154). a maximum value V(2) ly; Since-V attains (4. 175) may be expressed more conservative- -121(4.176) Tu~'2.)== (olf/<(z.)~ 1 which may where -z.)(2.) is defined by (4. 160) and be rewritten; __!;, (f(2.) 02_ ((, Since the particle (llCl) ~ (2) cr;_;<Jj << .L ~ll(z_) ))(~ f )(.l + _§)-Z. q (2.) < l (4.177) this condition represents a restriction on a; Reynolds number, Rez_(2) =: f Cz)CJ2 (U(l) -ucz.)) jtC2 ) immediately downstream of the shock wave, the strength of the shock wave, and the density of particles 02,. < Q.(2.YAoz. Alternatively we may rewrite condition (4. 176) as -VC2) and obtain a restriction on D and VQ.)the collisional parameters appearing in (4. 161) and (4. 162) . Using the definitions (4. 163) and (4. 166) we find D < ~ (ACzJ)(~) (o..c2 Avt .A~ 1) (4.178) QCl) (4. 179) Consequently there is a maximum allowable diffusion velocity and effective diffusion constant. In other words as a consequence of the requirement that particles may only collide while moving at nearly their local slip velocities , there is a limit on the magnitude of the mass flux density of particles C£ arising from collisions. In numerical cal- culations we have found (4. l 77), (4. 178) and (4. 179) useful as a guide to guarantee the satisfaction of -c"uz.-0 < l . The limitations imposed by this condition and the variation of-C-"i..L) in the shock relation zone -122are best understood by computing some examples. variation of -i)~ In general the is such that conditions (4. 177), (4. 178) and (4. 179) 2. may be violated without significantly invalidating our results. The violation of (4. 1 74) results in a correction of order '2.. (02/<r;) << l which, for our purposes, is small. It will become evident from the calculation, however, that this condition is usually satisfied. 6. Results of Numerical Calculations. By introducing several numerical examples we shall ( 1) inves- tigate the macroscopic motion of particle cloud '2 downstream from the shock wave on the basis of our specific collisional processes, and (2) examine the significance of particle-particle interactions when the present collisional model is valid. Consider the collisional dispersion of particles'12:" downstream from the shock wave when their densities upstream are given by; ( 4. 18 0) and (4. 181) 0 The density distribution of particles a;_ for x > 0, when their upstream density is (4. 180), may be determined from (4. 155) or (4. 161), the fundamental beam solution. This example represents the limiting case of very thin upstream density distributions whose widths are much less than Au 1 , the characteristic dimension of the shock relaxation zone. When the upstream density of particles OZ" is given by (4. 181) their -123 density downstream of the shock wave may be determined by integration in (4.156) or (4.162) and is (4. 182) or introducing dimensionless variables; f where the dimensionless quantities are defined in (4. 157)-(4. 166), frl.= frl..(eo) fr1 and error function of~· This distribution is representative of a distri- 2 f :t: -)1'2 JTC" e I JI er ( i:) is the bution whose width is on the same order as the characteristic length of the shock relaxation zone. In the following calculations we will take \b .=::. #2 \0l. (a) General Variation of variation of ~ 2 ) ('~)· As an example of the /\.ff2..(X) (f) /\J "\I l\J """ "\J L\ ) , and\( x.) ~ ) throughout the shock relaxa- tion zone consider Figures l8 and 1c1, respectively. ing variations of l.J.. '> The correspond- Uyi ) U.t>'1- > A,-V, K J and AV , which determine the collisional redistribution of particles <J2 in the shock relaxation zone, are described in Figure 2.0. The physical state of the gas- particle mixture upstream of the shock wave is defined by the values of the physical parameters listed in the Figures . , k 1:::::.25 C.s/ cf>= \uj\Tl =-1.. "t = ...L .4 ' I .1 I .2 :fA ~ .3 -.-- A :::::--==---+-- I I .5 oz. fundamental beam density distribution downstream :-:---..- , fs/fc11-== \d , <Ji/CJ( = l/4 , C1! = 8.0 fl , /\-= ~u 1= SJ ctvi) -3 0 Upstream conditions; Gas-Air, ec17~l.3x1o'l/cm~, Tco-=2.ac)M 1-=-3;z., ' = X//\ Variation of particle I of the shock front. Figure 18. 0 2 4 w)IO fri 12 14 ....... I *"' N .8 Variation of particle .I fA 3.94 .3 X//\ fs/ fCtl :::::: 10 3 , <:r-z-/ °t =- l/4- .4 .5 a;: uniform b e am density distribution downstream of the shock front. .2 I I I I I I I I I I , o-1 == 8.0 jl. , /\=)..." 1= 31 cm , k 1 ::: .25' , Cs. /cf' = ;\u 1 j A1j-= l -3 Upstr e am conditions; Gas-Air, pco==l.3x 10 9/cm~TC0=20°c) M1 =~.'2., Y-:::l..4, Figure 19. 0 .2 A .6 lf~ro) (fz. P, I l.Or----- - - - - - - - - --j I I N U1 ..... -126- 3.0 v 2.5 0..(1) 2.0 1.5 8 2 0 4 10 8 3 >.v f(b IJ.pz. ~6 )( ,,..... lii1) 2 3 40 ~4 :-i '-' 2 X//\ Figure 20. 2 Variation of collisional properties of particles 02. downstream of the shock front for density distributions described in Figures 18 and 19. -127- a.t 50. 30 - X//\ 10 0 Figure 21. .I .2 Variation of particle CJi fundamental beam density distri- bution downstream of the shock front. Upstream conditions; Gas-Air, fC•1= 1.3x10-~9/cm~,\(1)=2o~c)CS'==t.4 :> Ml:::=.L.b, k 1 =.2S- , a-7../0j = 1/4 , eJt:::: 8.o~ , A=Ao -:::.'3lcm, 1 Cs/ cf' = Au1/A.T = j_ • 1 ps: lfCO == 10 3 , ▪ -128- C) O (1 Cl) .. t-H 0 „ • cd • 11 11 „.-• 66 cn O • 0 z ti —• • (Y) rir) 0 •rl r 4-, ) „••••<7 • t .1) 0 11 ▪ N < VI i • \-) L. • tos n -0 • .0 8-, °°1) -4 )( (11. II 0 C; <+-1 ..a4 u cd •r4 • o o , O \-/ II rs `el... u • 0— c e 4-3 N* C`l • • 0 rag 4-1 14 II -129(l) }[ 1.4 <lCI) 1.2 .6 4 .5 3 .4 ~o -l .3 2 ><. r-. ~ iJ ~.2 ,( '-./ .I 0 0 2.0 :4 (iii) .3 1.5 Av ~o v.r2- -l x ,-...1.0 .2 ~ ')J. v .5 Figure 23. .I Variation of collisional properties of particles Oi downstream of the shock front for density distributions described in Figures 21 and 22. -130- Upstream of the shock wave the particles 62 are distributed in the y = 0 plane. They pass unaffected through the shock wave so that this distribution is retained immediately downstream of the shock wave. Downstream of the shock wave the gas and particles are out of equilibrium, and collisions occur due to the unequal velocity equilibration lengths -V('2.. )-C\12 (2) ~~~< Au 1 :=. /\ Since, from Figure 2D(ii), < ..L, most of the particles have experienced a collision after moving a distance X."V .2S Au range of the particles 1 • Now the maximum transverse <Ii. scattered in the region is, from Figure .2.0(ii), about A('2..) "1 .O'lS O ~ X ~ _2c;, Au 1 \", ; From previous discussions A(2) is the maximum distance a particle OZ:- can move across the gas flow following a collision with a particle <I\ at position X X.""\./. 2S ~\1 1 Therefore one would expect that at a distance downstream from the shock wave most of the particles 02:. would be t..\""'VA('l.)""' .01Slv1 of the '1 axis. distributed within a distance This assumption is confirmed by the density distribution of particles of radius Ci_ at 'X:::.'2.£'~\) I inFigurel8. the macroscopic motion of particles character . <'.12_ For .'2.S), 0 ~ )(;$ l.'2.S>w l I is primarily diffusive in Collisions between particles <lj and particles~ determine the form of the density distribution of particles <Ji in this region . The dispersive effect of collisions for X ~ l.2S A.., 1 may be seen from the following considerations. For transverse range of particles o;:_, A(X), due to collisions at X. be- comes small quite rapidly. the maximum Because of the physical significance of }.. this greatly reduces the collisional dispersion of particle cloud CJi Consequently in this example the dispersion of particles ~ due to -131collisions is governed primarly by particle 6t'"-particle<Ji encounters that occur in the neighborhood of X"''' .2S~u 1 , that is, where and -V(X) This point of view is further re- have maximum values. inforced by observing the variation of K(X) and X ~ On the other hand, for \ex) \::.x)iW in Figure 2.0(iii). .l. '2..S ,\\.I 1 the form of the distribution of particles o;: is governed primarily by the decelera- ti on of the particle cloud as ~-i) ~ 0 is a piling up of particles <J2_ as a consequence of this deceleration, and Ur2- ~ U.(O:i) . There their density increases, a result that is most apparent near the axis where particles 02. are concentrated. Because AOO is small for few particles are scattered beyond L\ 3 Au, )( ft, 3.S A11 1 '\.J • L1 • For as they move downstream from X. AJ .L,25 the relative velocity, uf,- ~2. of particles lfj and particles <:12 nearly zero and the effect of collisions becomes negligible . is The par- ticles reach mechanical equilibrium with the gas and for all practical purposes the density distribution of particles 0.2, freezes. ing distribution is given at density fp'2.. X:::.. 3.94 Ao 1 This limit- The variation of the may be explained on the basis of our collision model, in a similar manner. Although the discussion presented here, in order to be specific, has concerned a particular example, the general ideas set forth may be applied with success to the other examples to be presented here. They provide further physical consequences of the collision process and examples of the novel character of the dispersion of particles downstream from the shock wave. To fix the physical magnitudes, it -132may be noted that when a shock relaxation zone is over 60 cm in length, the cloud of particles 62_ spreads out a distance yv.'2~u 1 ~6."3on. This variation is easily observed experimentally. Let us now examine the effect of changing several important parameters of the problem. This will also provide physical insight to some of the physical conditions under which the theory may be applied. (b) Variation of M1• The effect of the strength of the shock wave on the dispersion of particles Cl2_ within the shock relaxation zone may be studied by changing M and holding all the other physical 1 parameters of the problem fixed . This variation of M, has been carried out and the results presented in Figures lS-2.3. The state of the gas-particle mixture upstream of the shock wave is specified in Figures lS, 19 and Figures Z.L, 22. has been varied from 1. 6 to 3. 2. The shock wave Mach number By comparing Figures 2f) and 23 it appears that the essential effect of increasing rv\ is to increase and thereby to increase the collision frequency and the momentum transferred from particles 01" to particles CJ.i. verse range of particles <Ji. is increased. increasing M 1 • The trans- The subsequent effect of then is to significantly increase the dispersion of particles within the shock relaxation zone. (c) Variation of k1 • When the state of the gas-particle mix- ture upstream of the shock wave is held fixed, except for the variation of kl = fr 1(1) /~C (') , there are two possible effects of importance. The state of the gas-particle Oj wave is altered thus changing mixture downstream of the shock ·u.r,- U.f2. • In addition, the number .3 A .5 k 1 =-.'So , fs/fCi) ==- .Lo 3 , 0"2./Cf-:::::. J./4- , CT(= +.of'-.> !\.,= >..u 1-==7.9cni, C-;;;/cp ==- \0 1 /,\.T1 =- _L , Upstream conditions; Gas-Air, pC1) ::::- J.,3 X 10~ hi,~ \(1) =-2.o°C ) M{· ~.2) ~ =- l.4 fA shock front. .2 Variation of particle CJi fundamental beam density distribution downstream of the .I X/A Figure 24. 0 3 6 12 15 2.1.5 ....... I \.N \.N .2 r/\ X//\ .3 .4 ~ 0 , ~s/pe1)=1-0'3, <Y-i./a;-=-.L/4 , q=4.0f<- ,/\=Avl==-'7.qCrt\, es/cf:::. Ao,/AT,::::: 1. , M, =-~.2. , 't::: l.4 Variation of particle CJ'2. uniform beam density distribution downstream of the shock .I I I I Upstream conditions; Gas-Air, ~(l)= l.'3x.l..C'l/cm,3 TC1):::: 2o c K1=-.'50 front. Figure 25. 0 .4 .6 frF) _fu .8 1 . r------- ------, lo .5 ,_. I ~ \,/.) -135- <i) 8 1.0 .8 .. 6 (. 11) 1"10 -l .6 x. 4r- iJ"' ~ ~:4 ,,( \..J 2 .2 0 0 3 8 <.:fri) 1i 6 2 )..,; >( r-- Ura. ~4 )I. ~ 2 x/f\ Figure 26. Variation of collisional properties of particles <Ji downstream of the shock front for the density distributions described in Figures 24 and 25. tJ ::: l.. 4- shock front. Figure 27. 6 9 .2 ~I\ .3 A- , K 1 = ~25" .5 -~ ~ 0 3 , fs/pC1) == io , <Ji/Oj = .1/4 , CJ= 4.o [<- I I O' VJ 1--' ;A:=Av 1 =/.'lc~ S/cp=)..11./ATj=l. , fVl 1::: 3,2. OZ. fundamental beam density distribution downstream of the .I Upstream conditions; Gas-Air,p<1)::l,3x lo 9/cm, TCO::: 2.0 C Variation of particle 0 3 f'ri~o) 11 Pea.k a.l 37. 5 3.94 = X//\ .2 JI\ .3 1.25 , A -~ 1 , I{= .l.4 fs /?c1> =1.o , er;/°! ::: .1/4- , c:Jj = 4,0 ft- , ;\=- ~1 -:=1.'l er>\ ~/cf' = Ay AT> =- l. 3 . , ft1 = 3 .2 I I-" v.i --l I .5 Variation of particle <Ji fundamental beam density distribution downstream of the shock .I I I Upstream conditions; Gas-Air, fCO=l.'3x10 9/,.M~ TC.lJ =- 2.o aC. K 1= , '2.S front. Figure 28. 0 .2 .125 _4 r t - - - .6 fr~) .lh:-.8 I I I I I 1.0 -----------..., -138- (\) v 2.5 ru1) 2.0 Ul"l-/QCI) U Q. I) 8 .8 .6 YA <.ii1 6 ~Q x 4r- ~ ..<. 'J .2 2 0 0 8 1.5 ( 11 i) 6 \'110 1.0 -4 \v u.rz. x4 ,--.. < .......... .5 62 OL----L-----'-----.L----12-==~~=----"'-3~--l'----_J4° X./A Figure 29. Variation of collisional properties of particles cr2 downstream of the shock front for the density distributions described in Figures 27 and 28. -139density of particles Oj is increased . The variation of k 1 , holding all other gas-particle properties upstream of the shoc&.k fix e d, may be studied by comparing Figures 24-2.6 with Figures 2.7-2'l. The param- eters describing the gas particle mixture in each case are stated in the Figures and /\-= ~ 01 -=- "J~c!W\ is the characteristic length of the shock relaxation zone. with k1 . From Eq. (3. 15) we see that LJ\(oo)/U.C1)varies slowly Cons e quently the gas properties downstream of the shock wave are not significantly effected by doubling k, as we have done. Therefore Up 1 -~-z. is negligibly affected by the variation of K\ The particle ranges are essentially the same in the two examples. On the other hand, by increasing K1 we have increased the -uf2. density of particles~ proportionally and since the variation of Ll~ is negligible the collision frequency has also increased proportionally. Therefore increasing 1<:'1 increases the dispersion of particles OZ . However, because the particle transverse ranges are not significantly altered by the variation of \<1 and the main contribution of collisions comes from X "\J. 2 S ~1.1 1 , the effect is not very important. Be- cause the particle range ~ is such a sharply peaked function doubling the frequency of collisions does not effectively alter the spreading of the particle cloud 02_ . Particles oz have at most one collision while traversing the region where ~attains a maximum . variation of the distribution of particles The qualitative oz. in Figures 2.4-2.5 and 2.7-2.8 may be explained on the basis of our model of the collision process by using ideas pr es ented in section (a). (d) Variation of Oi . Suppose Cli. is varied and all other properties of the gas particle mixture upstream of the shock wave are .I 3.94 .2 =X/!\ )Y;\ :4 4 , ps!rc\) =- .l0 , Oi/~ = /z 1 , °I =-12.3)<- .5 .6 , (Y\ 1 ='3. 2- , 'i:: .t.4 ,J\:::A1.11=8ocrn. , ~/Gp = ,\o1 /AT,-::. l. l.01\!cm: TCI) =- '2-0c .3 ~ I 0 ..... Variation of particle 02_ fundamental beam density distribution downstream of the shock .25 Upstream conditions; Gas-Ai-r, {'Ci)= 1.3x K 1-=. '2.5 face. Figure 30. 10 IJi01 fh. 20 30 ~ .8 0 .2 .4 .6 I .2 xi\ .3 - X//\ .4 .5 .6 fs/~(I') .:= 10+ , c:Jz./a-1 ~ l/2 , 0\ =12.,8 ;-<- Ten::::. 20°C , JY\ 1-:::: 3,2 , ""()-=l.4 , k,==-.25 , , A=Av1-: 80cm, Cs /cp :=. .\..o,/A.T1-::::. .l . Va ri ation of particle a2_ uniform beam density distribution downstream of the shock face. .I Upstream conditions; Gas-Air,rc1)=1.'3x.Lo-4cilc.""~, Figure 31. I' -fr£00) ! fl. I I I I I I l.Or----------1 I I 1--' ..i::- 1--' -142- ~2.5 (l.C1) 2JJ 1.5 1.0 .scs~=:L::==:=:::===r:=====~~==.. .2 .8 .6 L_~-'-~---'-~~-L-~-L..~~-'--~....::1=:::::::::::;;;;;;;;;;:;===10 1.8 1.5 C..ii' ') 14 l.O N 0 -\ AV )(. 1.0 ,...... Kj/\ ~ ~1) Urz. 6 .6 u.ri. .5 .2 IL.~-L~~..J_~--J~~-2~-==~;;;.--~3--------~4° X//\ Figure 32. Variation of collisional properties of particles downstream of the shock front for the density distributions described in Figures 30 and 31. oz. -143- 30 .25 20 X//\ lh. P.(O) I r2. 10 0 Figure 33. .2 .I .3 Variation of particle ell fundamental beam density distri- bution downstream of the shock face . .0(1)=l.'3x1o+a1~ T'C1);;;:::2o c 1 ' P-s /o\ C1) = 104- , cr2./ot == l/4 \ 0 I Cs/ey, :::. A.ol ;\T1-= .L . ' , Upstream conditions; Gas-Air, l'==.l,4 M1=3,2 c:1. ::: 12. .'8"' 1 1- ' k:==.2.5 ' , I /\~AvI .:::. 8Gon , I I .I I 1 .2 i I I I I I rl\ J'....- .3 T:-=:- =:::;;;;::;;;m; e .4 I fs/f>CO = l.O , CJ-i/ ~ :::. l./4- , °{ = .12. 9 j<. :> , f\. 1 =-3.2, '(=.L.4- , k 1 =,25 A= A~==- 80cM, Cs/ Cp == Ao1;iT,-:::. J. . TC1):::.2.0~C. V ariation of particle o; uniform beam density distribution downstream of the shock face. 1 = .></!\ Upstreamco+ditions; Gas-Air,ec,-,~i.3xi.C;/c..'f Figure 34. -- 1r--- o .2 .4 Fr~00) .6 _fu .8 3.94 ------. ....... ~ ~ I -145- 3.0 ( 1) v2.5 CW) 2.0 ·1.5 U.pi./O..<I) • •5'--';_;;,!...:..:.;_;.;o.._____....______.______._______.~-----------...._----~ .20 8 .15 6 r-J ~ 4 0' .10 ? )( '< ~ . ...J .05 2 4 :4 <..11-i) 3 .3 1Y ~ u.ri ~2 .2 ~ )). v .I OL-------L,__,___,L,__,__-L-,__,__2.C:::::...--~----'-3~----'-------J40 X/I\ Figure 35. Variation of collisional properties of particles downstream of the shock front for the density distributions described in Figures 33 and 34. oz. -146unaltered. 30-35. This dependence may be examined by considering Figures Since A D( CG_2 one expects a substantial effect. In addition to the obvious changes, one should note, from Figure 32.(ii) and Figure 35(ii), that increasing 02, broadens the variation of A. This increases the region over which collisions between particles 6; and particles ~ may substantially affect the dispersion of particle cloud a;. is increased, particles When 02 oz have fewer collisions in the ratio ( ~,,,, <~;)~2. <f) \2 within the shock relaxation zone. v_ Because of the broadening in A(x ), however, the dispersion of particles 02, is increased significantly. d . f ;'\) Cf) The details o f the variation o fy-i. an . d . F. ff2. , d ep1cte in :1.gures N '30, 3l., S3, and 34, may be studied by extending ideas presented in section (a). It is especially important to note the large magnitude of the dispersion of particles 02 under the conditions presented in Figures 30 and 31. One very interesting observation may be made from the fact that, as the relative size of the particles ~O, degree of <lisper sion of the cloud of particles was increased, the 02_ was significantly increased. Thus, if there were a distribution of particle <J2 sizes present with <yt '> 01, , there would be a particle size separation as a consequence of collisions . Relative to the characteristic length of the shock relaxation zone the larger particles would be dispersed the most as a consequence of their larger transverse range. Indeed, the magnitudes of the effects studied here suggest they might be accessible to study by an appropriate experiment. By measuring the dispersion of particle clouds using shock waves, under known conditions, one should be able to measure D and V(2.) and thereby study ) -147the particle-particle and particle-gas interaction laws. As indicated, during the calculation of the particle 02, fluxes, the kinetic theory method used here may be extended to include more complicated particle-particle and particle-gas interaction laws provided the particles collide while moving at their local collisionless velocities. (e) Significance of ~-V. To complete this discussion of our results we wi 11 demonstrate the physical significance of the diffusion velocity ~V. As mentioned earlier, because particles scattered forward by collisions with particles CJ2 are always er, , the average con- vective velocity of the cloud of particles a;_ is greater than the local collisionless velocity of particles of a particles 02_ • The average residence time a;, in the shock relaxation is reduced by collisions. Thus the anisotropy of the particle a;:_-particle Oj interaction process tends to reduce the dispersion of particles Oi . In Figures 36 and 37 the diffusion velocity ~-V has been neglected in the calculations. Figures 38 and 39 the effect of for. In AV has been appropriately accounted Clearly the effect of )\-V is to reduce the degree of dispersion of particles D2_ in the shock relaxation zone. \ 1\1) == 0 .2 3 .94 .4 = X//\ *'" .6 .8 LO . . ft A , 1v1 :::; 4-: 8 1 , , /\.-=-Ao\= \9 cm, C.Sfcp-; .Ao1/Ay1 -=-l 0 =1 .3 XLO~~ /c1'1~ Ten = 2.0 C 0--z,/(Jj, = 1./4 , CJj == b.2 J<- Upstream conditions; Gas-Air, fCI) Variation of particle Q2_ fundamental beam density distribution downstream of the shock 0 2 4 6 8 ~ ~ l..+ , K 1 = .25 , fsfe<\" =-163, face, Figure 36. 1 fr 10 rro) 12 14 16 I *"' 00 rl\ .5 .6 .7 .8 .9 ~ =: .l. .4 face, A-U=o . fs I rr; -: :. r 20°c , M, = 4.8, iq c~, Cs ICy =- Aol /,\Tl-=. l.. Upstream conditions; Gas-Air, pc1) =-l,3 xlo-~ /~ Tc I) = 3 ' 1<, =. '2. 5 ' \:CO ~ lo ' . ~/<:Ji -=-V4 ' ~. 2 'J\ :~->.~y: Variation of particle oz_ uniform beam density distribution downstream of the shock 1.0 ~4 0 .3 I .2 I ..... ~ '° .I I I I I I I .2 .4 Figure 3 7. - /f1- .8 I I l.Or------1 .I 3.94 .2 ~ X//\ r/\ .3 .4 .5 I 3 o 1 1 /4-, Upstream conditions; Gas -Air 'lCIJ =- .l..~ X.10 <ttCm , T<.\) = 2.0 C , M1 1 , ):::: =- .25 , f':;,/f<- )=-.LJ, 0--"2.J'°I ==CJ(=-fa. 2/l , A-:::,\'-'\ =-l9cm, -3 =4-.8 Variation of particle OZ fundamental beain density distribution downstream of the cs/cp = Au 1/\r1 = ..L • 'i :> J..<f shock face. Figure 38. 2 6 8 fr2. io f}~O) 12 14 16 I I 0 U1 ....... Figure 39. K1 :::=.. '2. S face . .25 - / .I -- I I I I I 'iV!\ .3 0 .4 .5 , 3 f~f f:C•) =- 1-0 , o;_j <1j = !./4 , <r; = b.2 j{. Variation of particle , A.-;:; Au 1 -:=. J..9cm, .6 Cs/ Cp = Au1 /\.T1 =- l. . <Ji uniform beam density distribution downstream of the shock -3 Upstream conditions; Gas-Air,ecn,,.1.~x101/~ TC\')-= '2o c , M. 1 = 4.B , '6 = l ."'T 0 .2~ .4 F:f2 .6F (<:C) fpi. =YI\ I l.Or---------------1 I 3.94 I .8 ' I \ JI -- -152- 0 1.0 10 .8 8 ,.,.6 ~Q 6x p ,.-..... < 4:<-- ?A ~ (('Q .2 2 0 0 14 3.5 12 3.0 10 2.5 8 2.0 >.v 'I_ ~6 1.5 LZr.i Y.. '- 4 1.0 2 .5 0 Figure 40. 1.0 °)(..//\ 2.0 3.0 4.CP Variation of collisional properties of particles CJi downstream of the shock front for the density distributions described in Figures 36, 37, 38, and 39. -153- v. PARTICLE-PARTICLE SCATTERING IN A SHOCKWAVE; I '52./<rj 'Vl then '>-» 1(,1 ....:'\:V-z../-c.,~L and the relative When motion of particles 4.L. %-vi O[ and cJi will be small as can be seen from Figure Provided the particle radii CJ! and <Ji_ and their velocities are sufficiently different that the elastic collision hypothesis , -Cu '2. >"">L;, 21 , is satisfied, we may suppose the fallowing conditions are satisfied; ~d << << ( 5. 1) < <. ),02.\ The relative motion and the density of the particles is such that they have at mo st one collision during their passage through the shock relaxation zone. Therefore the particles q- and °2. always collide while 1 moving at their local collisionless velocities ~ and U~, respectively, and only binary encounters are important. As in previous calculations in order to elucidate the essential physical features of the collision process, we will assume ~f1) p > '> ftz. Consequently the motion of particles crz has no effect on the gas-particle CJ\ interaction. Up- stream and immediately downstream of the shock wave the state of this gas particle mixture is given by conditions (4. 2) and (4. 3), respectively. The shock relaxation zone is established by the one- dimensional gas-particle Oj flow studied in Chapter III. Therefore the macroscopic motion of the nonuniform distribution of particles CS:- , downstream of the gas dynamic shock, may be viewed as the scattering of a beam of particles scattering centers, particles properties are also known. o;: from a known distribution of 6i", distributed within a gas flow whose Following a collision, a particle a;: may U('2.) Figure 41. "" x"\ "" >...' Collisionless velocity relaxation conditions when ~S.aj_:::o.!1=ft(_co)-=::~~2-- · - - - v <Ji./ Cl ""V 1. .x. I ~ lJ1 ,_, -155move a distance of order )u2. across the gas-particleOj flow field before eventually coming into mechanical and thermal equilibrium with the gas far downstream from the shock wave. Consequently there may be a redistribution of the cloud of particles of radius directions normal to the gas flow. 02. m The characteristic length, ~11'2. , of this redistribution is of the same order of magnitude as the characteristic length of the shock relaxation zone, ~\)I . We should ana- lyze this process in detail supposing that conditions (5. 1) are satisfied throughout the shock relaxation zone and that the particle-gas interaction is governed by Stokes law. To determine the dynamics of the beam of particles 02_ downstream of the shock when particles 82' have at most one collision in this region we may decompose the beam into two components. One component, hereafter called the primary beam of particles a;, , con- sists only of particles~ that have not suffered a collision. The other component, hereafter called the secondary beam of particles consists only of particles that have collided with a particle point within the shock relaxation zone. o;_ , °i at some Far downstream of the relaxa- tion zone both of these streams of particles <Ji are moving at the local gas velocity U(oo) parallel to the X axis which is eri.ented normal to the shock face as in previous chapters. length ~u Since the velocity equilibration 2 of particles <'2_ is on the same order of magnitude as the characteristic length, >..u 1 , of the gas flow field, the complexity of the motion of particles Q2_ prevents a detailed calculation of the properties of the primary and secondary particle £12 beams. However, using the fact that particles~ may only move a finite distance, the transverse -156range of particles az , normal to the gas flow following a collision, it is possible to calculate the final density distribution of particles 02 . The final density distribution is that achieved far downstream of the shock wave where all particles are in mechanical and thermal equilibrium with the gas. Unless specified the notation of this chapter is consistent with the notation of Chapters III and IV. 1. Density distribution of the primary beam. Particles a;_ composing the primary beam move through the shock relaxation zone at their local collisionless velocity ~ given by: (5. 2) where Q.. is the local gaseous sound speed, U. the local gas velocity, is the velocity equilibration length of particles a- and is a constant since we will assume the gas viscosity, 2 {<- , varies as the square root of the gas temperature. Therefore particles composing the primary beam move in straight lines. these circumstances the equation of continuity for particles Under G; is readily established by considering the motion of the primary beam cltd.:r oriented perpendicular to the X. axis and separated by a distance clx. The distances d.x, '4 , and cl:r are small through two areas compared to the macroscopic length scales of our problem. Let the areas be located at X. and x +dx.. where position downstream of the shock wave , Figure 42. second of primary beam particles X... is any The number per 02.. pas sing through the area -157- Figure 42. and the number located at X is per second passing out of the control volume area element I I 0..~ ~""l: at The density of particles (f) )(+ d)c'.. is (~) JxJL\J.?:. through the .I nf2-(X+a..'X J'-\i-=l:) Uf~X. +JK) cl L\ d"C · 02. which compose the primary beam has been denoted by Y\P2 Collisions between primary beam particles 02.. and particles °I within dxcl'-\J=l:: represents a loss to the primary beam and, since particles oz are conserved in collisions, a gain process for the secondary beam. Consequently if number of particles of radius S(x)'-\ l"=t) d.xJ~ d::c is the a; scattered out of the volume element per second the equation of continuity for particles may be written; (f) (p) \'l~><.+~x /F~) ~(x.+c{x )d.L\ d:c - n~ 2 (X/-\ >t) liy2.(x J dL\ c{-=c - ~ )~ )"'r.)dx ~ J~ (5. 3) To first order in small quantities we obtain the differential form of the equation of continuity for particles dz.; - - (5. 4) Since only binary encounters are significant and -C-°2 <<. "'\02 en- 1 counters are described as elastic collisions between two rigid spheres. Under these conditions ~ <~ 2 (~)-= hrf29~)where ~)==iJC(o-l~)nl'l<XiUr,CXrl\~I o;- o;: is the local collision frequency for particle -particle collisions; -I V -::::: ""Cc2.\ Therefore (5. 4), the equation of continuity for primary beam particles, may be written; -159- the density of particles Gj' is known .and so OJ is the local collisionless velocity Ut of particles l1~)(C:_) (5. 5) is an equation for tained in the primary beam. 1 Consequently the density of particles Cp To obtain Y1pz..as a function of Q;_ con- ~ we re- write (5. 5) as; (?) - hr2.c~) lAe~x) (5. 6) >..c ex.) '2.\ where particle °2. -particle Oj collisions. directly since Equation(5. 6) may be integrated Ac Cx) is presumed k n own; 2.I - (p' nr 4 C><.) Uf 2(X) -=. C. 8 rx d:x' J A.ex) 0 ~ ( 5. 7) where C is a constant with respect to X but may be a function of and -=t'. Since particles ~ 6i are unperturbed by their passage through the gas dynamic shock wave, t heir flux is continuous through it ; (\) fe2..c~,=t:) u.<1) ( 5. 8) \'Y\2 where is the density distribution of particles stream of the shock wave as defined in Chapter IV. CJ2 up- The flux of parti- cles composing the secondary beam vanishes immediately downstream -160of the shock wave. Using (5. 8) and (5. 7) we obtain; x c\x' S ,_, eo ~ ()(') (5. 9) x --x:co;+~r-S 'vx',\~~ -1\cL,( or e 0 (5. 10) is the mass density of the where primary beam of particles <Ji. With (5. 9) or (5. 10) and (5. 2) we have a complete description of the primary beam. We note that far down- stream of the shock wave, X ') "> ~'\ ; the gas and particles are in mechanical equilibrium so that 5. 11) Then for X ">") )w 1 x S 1 Y)1>0< J \ 0 1' the integral ~,cxl_ - ..L \ dvx'--_,.> constant ~X.() and consequently the primary beam density reaches a limiting distribu ti on. -161The form of the initial density distribution where co f? 2 is preserved but its magnitude has been reduced due to collisions between primary beam particles ~ and particles 2. er, . Density distribution of the secondary beam. The secondary beam of particles a;_ originates from collisions which occur in the shock relaxation zone. Each member of this beam is presumed to have had only one collision. Now using the notion, established in Chapter II, Section 2, that following a collision a particle D2, can move only a finite distance across the gas flow downC'5.) stream of the shock wave, the final density distribution, fj>2CX00 )~>~), of secondary beam particles may be calculated. Far downstream, all particles are moving with the gas and consequently the mass flux of secondary beam particles is given by; (5. 13) Then if the mass flux of secondary beam particles Ci is known far downstream of the shock wave, the final density distribution of these particles is obtained from (5. 13); CS) ':?P20<cn)~ >~~ ucoo; (5. 14) (S) Let us now calculate kinetic theory. 43. ~ric~ l\) :c ) by the methods of The geometry of this problem is presented in Figure -162- '\ \ \ ' ' '\ ' ,, \ I \I -163Consider an element of area dA oriented parallel to the (Xro}{o)~). shock face and located far downstream of the shock wave at We will assume that ~is so large that all particles scattered in the shock relaxation zone are in mechanical equilibrium with the gas by the time they reach JA. If ~fz.. is the number per second of particles J.i that pass through the area element ~A after having a collision in dV , a volume element located within the shock relaxation zone, then the mass flux of secondary beam particles of radius <l2_ at <:AA is given by (5. 15) JV in which particle °l" -particle eJi collisions scatter particles U-2.. into dA. It The integration in (5. 15) extends over all volume elements should be noted that there are no transit time effects since the gas- dV is stationary with respect to JA and dn Denote cktf'2 as the number per second of particles particle flow is steady and the shock wave. ()2. JV that have a transverse range, i\\ .1., in the I II // l l ti - f + qf '\> - <f + ci Cf Where f , scattered out of I( interval <:0" \ 2 II It are the polar coordinates of // dC1/\ relative to the X,, ~ If d' ~ co- dV. Then from the physical significance of the transverse range of particles Ci2_ we conclude that ~f 2 -: : . £~ ordinate system centered in It should be noted that this result is independent of the details of the shock relaxation zone provided In calculating and therefore let us assume, for simplicity, that Ur,(X) ":> UyfX) throughout the shock -164Since the transverse range of a particle CJ2 is inde- relaxation zone. pendent of the local variation in gas properties along the particle 1 s ~~2 dt r trajectory has the same value relative to any reference frame d\/ at the time of moving parallel to the gas flow whose origin lies in the collision. Therefore, to simplify the calculation, we choose the )( ~ .:f. center of mass frame of the collision. whose origin lies in J.v at the time It is moving parallel to the gas flow with velocity Ye M = 'ffl1 Upi i- m2. Ue<- (5. 16) m, + m2 since the particles collide while moving at their local collisionless velocities Up 1 and Up,. The dynamical conditions for particle Oj - oz encounters are assumed to satisfy the criteria of Chapter particle III, Section 3 to insure that the particles collide as rigid elastic spheres. The geometry of a collision, as viewed in the center of mass frame X L\ ~ , is depicted in Appendix A, Figure A4. from Appendix A, Equation (A4. 3), the number per second of particles of radius n;:') ~rz(&) di scattered out of the primary beam of density in dV into the solid angle JJZ. as viewed in xq=t. 'r) is - - JNrldSi.) ::::: np1 'llrz <ur,- ur2..J(q- + °2 ?- c;;\n~J~J<V dv' Jt Then d -= ~r-i Cd-&d<p ) (5. 17) 4 dt <r> The quantities Ylp, ) ny2. ) Upi >and Ur2 are evaluated at the solid angle is clli -: : . sln.~ J.:frol<f' ~ olV and (p) The den3ity of primary beam particles, (5. 9). Y\.P'2.. , is given by As depicted in Figure A4 the scattering or recoil angle of -165relative to the X axis is{}- and the collision occurs <12. particles in the Cf-:c.onsT azimuthal plane . 0 ~ <:_ ~ 2.1C Figure A5, since We note that 0 <::::. ~ <::::. X and From the velocity diagram for the collision, U.?t > U.p1 all scattering of particles Oi out of d\J , as seen in the rest frame of the shock wave, is forward and contained within an angle l\J ~X . In addition, particles G2._ scattered into angles~ and %2 , relative to the center of mass frame, have the same radial velocity following the collision and consequently the same transverse range relative to d.V. The relationship between f and the scattering angle fY is (I the final transverse displacement obtained from the magnitude of the transverse range of a particle, ~ 1 I (.f) 11!-?'2.. _ \ ~2. and Equations (A3.2-A3.4); , cu.r, - uf'2-) S~n~ \ - m,-r:u2 + II 'rY\ I where "Up 1 and particles Urz.. are evaluated at ,.l/T'\ \I \ -r "-'"'-\' (1) 11 tI dV. The ref ore the number of C\ scattered out of d..V in unit time that have a final transverse displacement relative to 1 (5. 18) VY\ '2. 1 dV in the interval is; + Consequently from (5. 17) using t~ fact that we obtain dn~" Jt J~t dt (5. 20) -166C\ -<'. -v, 0 where )Cl"'~ ..:::: using (5. 18) to express 1J1 < 1' II <: 2. 7("" in terms of . f " we o b ta1n; 0 Then where (5. 22) is the maximum transverse range, relative to dV, of a particle cr2 scattered out of c\\/ by a collision with a particle of radius r -r ;: :,. 0 < If satis · f"ies t h e re l at1on . sequent 1y (5. 21), the number per second, scattered out of dYl di/2.. 11 / \ I\ There f ore, f rom . , of particles of radius <JZ , JV , whose transverse range lies in the interval - q> l( + d ~ II is; i1dV But ~rz. Jt er os sing Oj . Con- (5. 23) d~~ js also the number of particles of radius o;: J.t'- dA per second; dN " - dn - ..l (p> \-'2. 1 dip - J:\;rz. - 2 nr, Ylr' (Ur1-LApz.K~+~) I\. JV \).A (5. 24) y( ..L - A\ -2. Cf)2 , where the element of area ~A is written dA - f ll Jf Cl ~ ll 2 fl Expressing element JV in cylindrical coordinates with origin at the area -167- -~ 'd (d ( j f <I(! ctX I (5. 25) J..V > 0 and we are interested where the minus sign is required since c\ X / < 0 . Furthermore, from Figure 43, we in cases where f =- fll . Therefore using (5. 24) and (5. 25); have I d~r2 clt Where 2 I =- -i co;+oz:) '<lp, '11y2 (tAp, - ~ft-) x-e, Jr'~,ol x dA (f) J Y\.p 1 (p) .L - ).-2 (5. 26) Cf f \ Y1p:z.. ) Ufl ) U.pl.) and I\ are evaluated at the location ) of the source of particles <Ji. the volume elementdV; (See Figure 43) I nr1 -=- np, cXoo + x ~ (~) · ( n p2 = nr2(r) ex°" + x I ) ~ 0 + f'f CC>'S<ff ) ~ + FI S'IV\C\' ) Up 1 -:::: uf 1C.Xoo + K) tipz. ::: °'ft. CXco + x/) (5. 27) Using (5. 15) and ( 5. 26) the net mass flux of particles of (Xco J ~o>~o) due to collisions radius OZ.. in the positive X.. direction at for x < Xoo in the relaxation zone, x~'> >A.", , may be written 211 -x()I \ ~( Xeo + x') ("5) ~~)\,,>~' ;;;,)=-~ (<>t+a;]-f~')dx.' jd~ { Yl ,o~X> + X () f 0 <) 1 ) (5.28) (~) I I t \-2 J Ylfz lXQ:>+-x) ~o+r ~<;~ ) -lo+ f Stvil£ )\Y·e,CXco+XJ-~Y.co+X)) A(\,+X)f I The quantity particles 0-z .\,Y\CX 00 r X. I 1 ) I • ' /: . 0.l - 'A2c~-t-x') (~)21 is the maximum transverse range of scattered out of a volume element located in the plane Con.c;+a.n.t . Consequently; -168- and AW\. is the transverse range of particles corresponding to the limiting scattering angle shock wave) ~ M<L?( observed in the rest frame of the A-= Am.• The region of integration in (5. 28) is sketched in Figur e 44 and the physical reasons for its size and shape are easily seen. Only particles <JZ scattered at points within this region possess enough momentum perpendicular to the gas flow to overc,ome the viscous d.A. force exerted upon them by the gas and thereby reach The net mass flux of secondary beam particles of radius \.0 C-s) i-p2. , at (Xo:, .> io )'\ ') may also be written, using (5. 28) as an integral over cartesian coordinates; 1 (51 - Xoo ~ (Xoo-r >' '> 'fftcx./i•i~J =- ~c"i+ •25'-j~, J<l'i' 2 0 \ - 1'. (X 00 -t" f\1 '2.' Id't, i vti>i< x.. + J X) - [\ '2. , / V/\ ( Xoo+x; ) -l-\ x/) t \) A. (Xoo+X )- ~ 'z-, Introducing the mass densities of particlesO\ and primary beam and (~ . ff ~) z.. .::::.. m2. Ylpt • we obtain; . C£ -169- ,,......._ ,.... 6Nti;:\ I I r-- ">< v_ "- I ;::S I 'J g "' 1~ ~ ~I\ "' )( '.J ,,,-< I , .....,-I ~ I ' '' ' '' ' '' - -- ___' \ (5. 31) where the mass density of primary beam particles is given by (5. 10); X UCL) -X(Q\-t-UiJ \ n" cf) uF,<x > J )) u (X 1) en ff2.C~)~) ~fX) () I ° e l \ dx' I r& (5 . 32) \..0 ('S) ?r} and Tpi must be evaluated numerically. By super- In general position of primary and secondary beams, the final density distribution of particles OZ", using (5. 14) may be written (-s) -t- ~,_(>\,,,) ~,'C) (5. 33) UC~) To simplify the examination of these results we will suppose the distribution of particles~ upstream of the shockwave is uniform (1) (1) in the :Z:. dir e ction; \p~-~l =t;) -=: fpc,C.~). This procedure allows us to establish the physical features of the collisional dispersion of particles a;__ when If <Ji. I ().L -v .L . (1) ft2. is independent of r likewise independent of -1: . then, from (5 . 32), (p) f!'2- Consequently the integration over is I 7l: in -171(5. 31) may be easily carried out. The mass flux of secondary beam particles <Ji_ under these conditions becomes (5) - ~ \CX'OG t" X:') I ~rI~.,,~.),,, _i co;+a;_1\to,:''" ) Jx j.i.~' ( \r•< (.,.+><-') f~r.~><.,+~ ~·'\')} s. 3 41 '"'\ ~2.. o -,\(>\u+x') and then; (5. 35) The domain of integration in (5. 35) is described in Figure 46. again, because of the finite transverse range of particles those particles scattered at )( within a distance Xai , ~o parallel to the shock face may reach (S'> mass flux 3. A(x) Once er;:. , only from ~0 and contribute to the i f2. (Xco) ~o '). Uniform distribution of particles a;_; It is informative to check the validity of the previous calcula- tion by treating a problem whose solution can be readily obtained by C1) other means. When lf'2. ( i) ::C) ::::. f = c.o"-s+o..n.i0 and the particles 0( are uniformly distributed across the gas flow, there can be no redistribution of particles 02_ due to collisions with particles <r, • In addition since particles OZ are conserved in collisions the equation of continuity for particles 02_ may be written ; - (5. 36) -172- x x 00 Figure 45. -173 where 'lrpz...(X) is the local average velocity of particles a;_. Since far downstream of the relaxation zone the particles are in mechanical U(oo) equilibrium with the gas where X'oo >> ~I . Then from (5. 36) we obtain the limiting density of particles "2 when their upstream density is a constant; (5. 37) Let us now compute fpz. (Xoo) L{) =l:;) results of the previous section. (I) fp:i. (_i> t ) 02: when (1) fpi.. =-fo using the Writing (5. 32), (5. 33) and (5. 35) for , the final density distribution of particles is; (S) -\- -~pz.C~/1>i::) (5. 38) L\.(a:>) and Substituting into (5. 40) -174- Xi ocr (S) Y+ACe') •l J :fy,_c X«>i"\,'l: i == ~ C°i-t-<iS'("'•:""·j I~)~,&> .,,ai ~ 1 M1°C-Oi.. 0 L l j (5. 411 ~-\(~) Integrating over ~ and using (5. 35), the definition of \ (~) ~ Jf (S) ~rl~i~ )7:.' ;: 11'. ca-:;-az:. i'- JX ff cx >~ex> ( ~cX" i - ur,sx >iJ (5. 421 I 1 C> Using (5. 44), (5. 39) and (5. 38), the final density distribution of (5. 45) -17 5or fr, C~co ) -=- ( ~ \ P u.coo) I t· 0 (5. 46) Since this result checks with (5. 37) we have demonstrated the consistency of our formulation. 4. Fundamental Beam Solution, During their passage through the relaxation zone of the shock wave, particles ~ interact only with the gas and particles CJ/ . There- fore the particles 02. are non interacting and their density distribution (I) satisfies the principal of superposition. and Cl) Then if frt.C\> t )I Fr~ ci)~ 1 , are two distinct upstream density distribu- tions of particles (}2_ and and are their corresponding final density distributions, respectively, then the final density distribution corresponding to the composite (\) upstream density ~P.(Xoc»L\>~) (() (l) fr-(i>t) -:=. f pz.(i 7=t:) (+ ff2.C~i~)2 is simply = ff2-( Xoo/\,=!.) 1 + fffXooJ~)~~ Therefore the fundam ental beam solution has its usual physical significance, for fr om it and the principle of superposition we can construct the final density distribution corresponding to any upstream density distribu(1) tion \pz.([>~)· This formulation provides an alternative to that of Section 2 . Suppose that the upstream density distribution of particles is given by (5. 47) -176where 00 ~l~) is the Dirac delta function and Nl'2 = ~s fr-C<I)~ -cc is the number of particles 02 contained within a cross section of unit thickness in the X and ~ directions upstream of the shock wave. From (5. 32) the primary beam part of the fundamental beam solution x is simply; ~ rt\2. Nr2.c~ L) \ Ufj>O) -Jt(<rj+o;:~ S11r• (~' - l )d.x' 6clp e C> ri. (5. 48) Sub stituting (5. 47) into (5. 35) the secondary beam mass flux density at io') ( Xco for all 7:: may be written; ) ~t ACX°') ><'«> i;;x~) ~·) = ~ ( +o;f(:~JCY\1 )]JxJ,q ~ fr~"l1cX1 &vJ "1 +1•i, 5. 49) M~z_ 0 ~;ACX) >( where s ~"KC°i+~f npt (~ - l JJ./ = Np. ( urn ) e o up, (5. 50) Uf~X) and AlX.) ~ M 1-Cui_ ( L-Lr1('X.) ~ url_(><.)) (5. 51) \'Y\1-t-~ is the maximum transverse range of particles o;:scattered at X • Because of the singular nature of the integrand care must be taken during the integration in (5. 49). There are essentially four distinct cases corresponding to the value of ~o. variation of u.f1-u.p2. Because of the in the shock relaxation zone A(X) has a -177X-::: X single maximum at and decreases monotonically on either h'\ If we let; side of this maximum. M 1To2.. (ur/xrn) ~ u.rz..CXm)) (5. 52} M 1+Mz... be the maximum value of ACX) in the shock relaxation zone, the integration in (5. 49} is straight forward. ~o < -~""' ~($)( +ri. ) Xoo)~~ or Then, because of the nature of -o = · This result is a simple consequence of the transverse range of particles G'2_. a;:_ Suppose is composed of particles The secondary beam of particles az scattered out of the primary beam which, in the pre sent problem, is distributed in the ~ ~o plane. Therefore, since ~~is the maximum transverse range of particles CZ: scattered within the shock relaxation zone, none of the particles [ ~ 0\ <Ji. scattered out of the primary beam may reach > ~m Consequently the flux of secondary beam particles 02_ vanishes for (~) irz.CX'oa 1 ~o ).::f-O and 0 <. l~ol <~m' From Figure 46 when is given by xflo) ~ c0(-to;:f-C~: )( ~~Afof ~ i1cX') r11 I where and X2...C\ 0 ) :z... (5. 53) \(lo) are points where the curve + ~CX) crosses the X: axis depending on whether or , respectively. Then are solutions of the transcendental relation; and -178- - - - - - - ----::;...----. ~<) )\.(X) :::=. =0 ~ ~Tui. (u.f'C.")(.) ~ Ufi.()(J) Ml+ 1'12. Figure 46. -179 - - ~1-r~ ( uf ,ex)- ufz...C'<.)) (5. 54) M1-t Mi_ It is to be expected that the fundamental beam solution is symmetrical in io. In collisions all impact parameters have been assumed equal- ly probable. Physically (5. 53) results because only between X Ct.to) 1 and sufficiently large that particles 02_ scattered from this region possess enoug h momentum normal to the gas flow to reach ~ 0 at Xoo)) .\.11 . 1 Finally, when 9 .::.Q from 0 (5. 49) we have ~ (S) i pi.()(.,,) j .l ~Co;+ CLf-(~) (:',;:·?jfr•cX) 1(51') ~ ( 5. 55I -=> 0 Collecting results (5. 48), (5. 50, (5 . 53) and (5. 55) to,gether with (5. 33) we may write down the final density distribution, mental beam, since For fFCt) , for the funda- \o was arbitrary. lj\ )j,M. (5. 56) For0~1jl~L\M : • (5. 57) - 180where; "Y\ Cx ) ( 5. 58) are solutions of the transcendental equation - rY11 T" z. ( U.p1 (x.) - Llri..(X) J (5. 59) 'IYl1+ m, When I[. I =O, as an approximation, u M In addition, l '=:! rYlt"IUi.. ( 1-lpt CxYlt) m 1+ M 2 transverse range for particles 5. xi co) .::.a ) X"2..Co)=X00 L\f'-(Xm) ) is the maximum 02" in the shock relaxation zone. General Beam Solution by Superposition Suppose the upstream density distri bu ti on of particles <J2 is (I) given by ~f2. ( ~) . By an appropriate superposition of fundamental (t) beams we can construct the final density distribution for fr2. c~ ') r-. :=-1. in the fundamental beam solution. Then from the contribution to the particle 02 density at X =- Xcc Take m2.N Figure 4-7 at (1) I ~ er2.(l ) elf p:/ tt) Where m2.Nr:l..-:::... J... Summing over all contributions from we (5. 60) Since only when 0 ~ 1~ l < ~W\ , we have -181 - ~I Figure 47. -182- \ + L\W\ Frf ""'J"\) =- } ~~2 cl1) Pt"°"i '\ -~,) 41 (5. 61) ~-iM Introducing the variables (5.6l)as; ~:::::~-~' and de; = -~ we may rewrite 1 ~M r1~ (Xco, •fl = Jf~~ ~ -~) P1'("'5) c\-.;; (5. 62) -~'r'v\ Then substituting (5. 57) and (5. 58) into (5. 60) with m2.. \.Jf2.. ::. .L and using the properties of the Dirac delta function, we obtain P. (Xoo v\) = ~l)C4)(~) eX(<>~:) Ifi .> l I 11 ~ \A CooI "'2.. \ iY\1(1) (S) Sx:oo lfl (~\ -l) olx <> uf z.. . (5. 63) fr'L c!- :.SJ pihc ~ >-.:;) ~s -t- j 1 -~M is the secondary beam contribution to the funda- where A.J x.fi~ll mental beam solution; p0/Xoo>~) - ir C°t+di::) ~" (mrr(2..T;;._ +M2 \ Rex) (ucc~ )e ~ J l fl \ u (~) 2 (S) 2 Ll(oo) -JtC"i+if } Ml 1 I Xl(\~l) J~r.(~ -l)~ 0 f2(5. 64) f~ The evaluation of (5. 63) and (5. 64) must usually be done numerically. Introducing the dimensionless variables defined in Chapter IV, Equation (4. 157) the previous results (5. 63) and (5 . 64) may be written; "'x In this relation (5. 66) and "'\! l\J The quantities \~ \ ~ . x\ (\ ~ L) ~ ~ (I~ l ) and are solutions of (~) f~) (~ r?:.) - ~ ,C".l)) I c.L+ n\2/~ ~ ~01 I \Q(2.} r r We have used the continuity equation for particles Ofto eliminate Finally the characteristic length ~ -;:::. fr) I (5. 68) fft . ~ is + '(Y\ 2 7C(Cl+02_'f ><;fCt) (5. 69) - 184Let us now examine the phy sical significance of the diinensionless parameter \LI \ • ( -J.._') As discussed in Chapter IV, the results of collisions may be interprete d as a dynamical friction force between the two clouds of particles. Since fpi) negligible effect on particles q-. p '> > fp2 this force has a Howeve r, it plays an important role in the macroscopic motion of particles CZ . From Eq. (4.ll+) of Chapter IV the dynamical friction force , FDF ex particle cloud (fz. due to collisions with particles f=DFC~ 1 =ffS)() frlx) 7C (O(+o;_ )-i.. ' exerted on °i' is given by; CLlpiCx)-Uyl>c.f)\~Fx>-Uplx> !(5. 70) ( M(t- rn2-) The relative importance of collisions may be measured by the ratio of \='Of: to Fs , the Stokes drag force exerted on the particle cloud by the gas . Using the definition (5. 69) we obtain (5. 71) than or of order unity except at the two ends of the shock relaxation This is in contrast to the situation of Chapter IV where tk-Upz. was small over most of the shock relaxation zone and the ratio f-0 ~/fs zone. was dominated by the velocity term. <Ii/ <3j /\J J..- Since (~f< /Ke(\)) "\.>.L , when , we find that the magnitude of (}'vJQ.) characterizes the effect of collisions on the macroscopic motion of particle a; This point of view is reinforced if we take the limit corresponding to FDf --..:;> 0, in (5. 65) we find Fs; cloud -18 5(5. 72) the effect of collisions has vanished. In addition, we may write the dynamical friction force ft>~ as; (5. 73) Therefore~ is the characteristic length of the dynamical friction force. Note that X is the characteristic length of the Stokes drag "z. force. Before proceeding with numerical solution of (5. 65)-(5. 69) for a particular upstream density distribution of particles ct let us review the assumptions that underlie the previous calculations; (a) The particles are nearly the same size (5. 74) (b) The particle-gas interaction is governed by Stokes Law; (5. 75) (5. 76) As discussed in Chapter IV this assumption is frequently violated. As a consequence, our results represent an upper bound on the degree of dispersion of particles <J2. due to collisions in the shock relaxation zone. However, our results are expected to describe -186the essential physical features of the collision process. (c) The velocity equilibration time must be large in comparison with the time required for one particle to pass through the Stokes flow field of the other; (~ )( e<lj l~;_- Uf,_\ )( f))) l (5. 77) (5. 78) In the pre sent problem this condition is generally satisfied as required by McLaughlin's results except if Cf is very nearly equal to °2 . When this is the case, however, collisions will be unimportant because of the finite length of the shock relaxation zone. (d) The particles ~have at most one collision during their passage through the shock relaxation zone. Consequently they collide while moving at their local collisionless velocities 1Ay,and U.f2. . The restriction introduced by this assumption may be explored in the following manner. The length \ (X) 2.t is the average distance traveled by a particle <li_ between collisions with particles of radius ~ . If Pcx) is defined as the probability, a particle ()2. will travel a distance X -sx clx_'._ Pc><: ) :::::.. e downstream from the shock wave without collision then 15 '. ( ') o '~i.,x Using Aez.I(X'.) for rigid sphere collisions we have; (5.79) -187- - .L \ ot.x.' sx.Pr ex I ~.exi <~' f:i.. >\ Pcx) 0 l ' u (5.80) In the present problem for the single collision assumption to be valid we must impose a condition which assures that most of the particles <5;_ have only one collision in traversing the region we take this to be t'l 0 ~ X ~ ~CO > 8 -I . Consequently the single collision r(~) rv assumption requires; ( 5. 81) Introducing the dimensionless variables used to obtain (5. 68)-(5. 69) the single collision condition may be rewritten /"I, )( 1 + P11) rK..(~ )i ~· -11 ~x' ~ l (~~)(~ ~ i ur, url. )."a. (5. 82) VYlL or (5. 83) Whether a given solution satisfies this condition must be explored numerically. bound on It is not surprising, however, that it places an upper (A.oz./~ ) . If the dynamical friction force exerted on particle cloud ~ due to collisions with particles OJ is too large, the single collision hypothesis is invalidated. The previous results may be extended to the case Up,< tlpz. -1886. Results By treating several specific examples, our purpose in this section is; (l} to illustrate the physical features of the final density distribution of particles a;:- when the single collision hypothesis is valid and (2} to establish the possible significance of particle-particle interactions when 62./ Oj '\.I l . As a first example, consider the collisional dispersion of particles ()2.. downstream from the shock wave when their densitv upstream is given by; (5. 84} 0 This is chosen to examine the behavior of any distribution of particles a;_ whose width is of the same order as the characteristic length of the shock relaxation zone. The limiting case of very narrow upstream density distributions may be examined by taking -189- and (5. 87) 1\1 The quantities /'\,.; X1(l2$ \) and 1\1 ""' X2.( I~ l ) are solutions of; (5. 88) ~ (t) is the Heaviside step function; (5. 89) and \f2- : : fric°'; rri. =- ( fpf n U(I) /u.cro)) ff2. Because particles 02_ can move at most a distance ~M across the gas flow due to a collision, far downstream of the shock wave all particles 02'.. will be distributed within a distance X. axis. Ci b + L\M) of the This result is independent of the magnitude of thermore if ~b) ~b . Fur- im , only the edge of the beam fy6-~rvi\ <(~ l will be affected by the collisions. Since all impact parameters are equally probable, the density of a finite width uniform beam can only be redistributed within a distance ~~of its sides. When ~wi)L{b the entire beam cross section will be dispersed due to collisions. are illustrated in Figures 48-51. These effects Final particle fs /pen : : ; l o , Oi/az z o. 9 4 . .3 T( L"")-= 20 C , , M 1"" 3. 2 --1 :o .L. 4 , OJ ~ 2 ·~ft- , J\:< A.u,-== S.'}cY"l, c:a/cr = Av1 />..T, =- J. . 0 , Up- k \::::. . 2.S a;, uniform beam density distribution downstream of the shock face. -4 '3 stream conditions·, Gas-Air ' \ocn= .l."2> :X. .1.0 q/cm ) , Figure 48. YA I 4.80 ~ X//\ ~--1 0--~_J__~t.1~~~~21-~~--_i__ 2 .4 .6 fr~OO) _fu_ .8 1.01 J I '° 0 ...... 573 X/;\ stream = Final particle <f2 uniform beam density distribution downstr e am of the shock face. 3 Gas-Air,f"i°)::l.. 3x.1-;;/crt\ !Ct)::. '20°C, M1==3,Z , O 1.4, .1<1 ,25 o--_,_~}.1~~:------!-~~,,,__L___ )1/\ .2 .3 .2 .4 .6 - condit~ns; = fs /pCr) = 10 , CS./o-;, :::. l. L , ~ = 2 .CC fl ,A==Au1..::: 3.°icm , Csj cf' = AuyAT1 = J_ Figure 49 . f~co) jb:_ .8 1.0 Up- ·'......° I ...... Final particle I fs/ O(O = .L.04 ' , <02./a;-1 :::::- c. q ,,~ X/!\ .3 J 3 , , , I , ..L. 4 CIVl , d'-:::. /\=Av.:::: <1.2 M 1 -:: 3 .2.. erl = 4. ~ J-<. T< D=- 20°C • K 1 = 2 SI '°'!" I ...... , Upstream cs- I er :z .Au, />..T =l , a;:- uniform beam density distribution downstream of shock face. conditions· Gas-Air llCl)=l.~xlo4-4 /cwi Figure 50. 4.80 o~~---'-~-t.1~--;o/:-~~.2~~~~--'-~ ~;\ .2 .4 .6 fr~oo) - p, .8 I f2. 1.0t---------- 3 ' .I X//\ \.{/ /II\ .2 p~ I cC.I ==- 1-0 4l ' , <12. I er;= o.4 , , , , 7/ = 1 :t , r<1 =- • 2.5 I ...... -.D VJ Upstream /\=Au,:=: 'f, 2.c1Y\ , Cs/ Cor = Aui AT.-= .L • l 1"\-::: 3.2.. OJ := 4.3 ,H.. TCI)== 20°c Final particle 02, uniform beam density distribution downstream of shock face. conditions; Gas-Air,fCll:: L3x4fc~ Figure 51. 0 .2 I ~ 4.80 = .4~ I I .6~ I I k .8, I \rioo) I i.o..-1 I In Figures 4S and 4q the size of particles <Ji has been varied keeping all other properties of the gas-particle mixture upstream of In Figures 50 and 51 the beam width has the shock wave constant. been varied holding other upstream properties fixed. For strong shock waves the relative velocity of the particles Oj and oz may be large consequently the single collision condition is violated unless <52../<!j 'V l or j.::: 1 is small. Sine e l.\ ("2.) /U.CM) is a slowly varying function of k 1 , in the region of interest, changing has little effect on \m . r\J In Figure 52 ~(S) ft'b, calculated from Eq. (5. 92), has been plotted for upstream gas-particle mixture conditions stated in the figure. A finite width uniform distribution related to this secondary fundamental beam by (5. 91) appears as curve ~~ earlier, r~ must vanish for ~ 1 in Figure 53. As discussed ~ ll-\ l :> L-\m since particles a;: can move only a finite distance across the gas flow following an encounter. manner in which "'\i('S) Pit> goes to zero as 1~"' l --)ll-\m\ may be accounted for /\J by noting that ACX), the transverse range of particles peak at The within the shock relaxation zone. ~, has a single Collisions that ,._,cs) occur in the neighborhood of this peak contribute to l\{s) The sharp peak in Pi\ at particles 02_ and the fact that relaxation zone. Pit near is due to forward scattering of at the ends of the shock The results shown in Figure 52. are representative of those corresponding to more general variations of Consider curves 2 and 3 in Figure 53. 'Ve -s) f'~ For the same up- stream state of the gas-particle mixture , the final density distribution of particles £J2_ has been computed using the single collision formalism, Figure 52. )<(1=-~25 front. Q 2 f"\"'.> t"\A JI ~~ 5 )11'!\ .06 .08 .10 , 2 fs /p<"z"') ~ J.a4 , 02/ CJ( == /3 , ~ = 3. Sf<.. , A= A01 -= (,,O Final particle Cl'1)> I ...... <:~/c p :::. Ai>1/\T1 :::. J.. I \)1 -D 02: secondary beam contribution to fundamental beam downstr earn of shock -4 . Upstream conditions; Gas-Air,f(1)==.l.,"3x.Loj/c&i;~ Ten:=: '2..0°C , M1 ~ _l .G , ~ = l .4- ft (S) 6 7 8 Eqs. (5. 85)-(5. 89), and the multiple collision or diffusion result, In the calculation of curve Eq. (4. 183), of Chapter IV, respectively. -2. 2 it was found that e of the particles 62: passed through the shock relaxation zone without having a collision. model result, curve 3 In this case the diffusion , may well afford the better approximation to the final density distribution. These results are so similar, however, as to suggest that the entire range of collision problems . may be covered by the single collision and diffusion models and that no treatment of the difficult intermediate case is required. On the basis of our results it appears that for upstream gas density greater than atmospheric density and k(" 0 (. 25) , the single collision condition (5. 82) is violated unless particles (f2 and <Ji are very small or vff!ry nearly the same size. Under these condi- tions their relative velocity is so small that the transverse range, \Cx) , of particles 02_ is not large enough to cause a substantial redistribution of the cloud of particles 02_. On the other hand, at lower pressures with the same particle loading, the restrictions on the particle sizes due to (5. 82) are relaxed. The relative velocity of the parti- cles is no longer small and their transverse range is such that important redistributions of particles sion conditions. £J;_ may result under single colli- Further study is needed to examine all possible conditions of physical interest. Rewriting (5. 82} the essential condition for single collisions becomes; .15 ~ .20 .25 r~ .30 ~ Xf/\ -4 3 UpC) \ fs/r<o , M=J...~ , O':; J. .4 , K,=- .25" 2 Oi/c-1 ~ /~ , °l == 3.5 /<- , /\ ==Au1 -;:(,,of- , c:-~f c:p ::: /\v1J:\T -::: .L. 1 P~/~(1)::: .S xlCi , ( (2) Curve 2 and 3; 4 oz uniform beam density distributions downstream of shock front. .10 Final particle .05 -~ stream conditions; Gas-Air; ((Curve I; pC1)=-"2.f,x.to1b; -4- 3 4 f<O~.t..::l.>x 1ci 1 fc..,, == J.o , T<\)=:o 2.d c Figure 53. 0 .2 .4 .6 Pr~J fu. .8 1.0 ... - - - - I '°-.J I ,_. -198- .!:S.·( f-(1) Oj Q(I) \ ( i + 02_ ) .(. <O r-c 1) ) "i ~ -1 sL 5c~vii. -1\ cQ'~)(5. o U:.~IX, r1 i. 90) 1 It appears possible to study experimentally the interaction of a shock wave with a very thin stream of two particles sizes whose density is such that single collision conditions exist. By studying the redistri- bution of these particles due to collisions, it is possible to obtain direct evidence of the particle-particle interaction law. -19.9VI. SUMMARY AND CONCLUSIONS In this thesis the interaction of a normal gas dynamic shock wave with a gas containing a distribution of small solid spherical particles of two distinct radii, Oj and OZ. , was studied. In order to elucidate the essential physical features of the particle-particle and particle-gas interaction processes upstream of the shock wave particles <3j were assumed uniformly distributed with mass density fpi and particles ();_were nonuniformly distributed parallel to the shock face with mass density ~fl.." Ahead of the shock, particles and the gas were in mechanical and thermal equilibrium and fri )f > '> fp2.. · Under these conditions the gas-particle CJ) flow downstream of the shock wave of particles o:;:. is one-dimensional and independent of the presence The usual shock relaxation zone is established by the interaction of particles Oj and the gas downstream of the shock wave. By considering in some detail the effect of the particleO(particle<Ji and particle eJi-gas interactions on the macroscopic motion of particles 02 in the shock relaxation zone, under the previous conditions, it has b e en the three-fold aim of this thesis ( 1) to demonstrate that the methods of kinetic theory can be extended to treat solid particle collision phenomena in multidimensional gas-particle flows; (2) to elucidate some of the essential physical characteristics of the dispersion of particles in a gas-particle flow field due to particleparticle encounters; and (3) to assess the importance of particle collisions in particle-laden gas flows. -200To render the problem tractable, the scope of the investigation was restricted to physical conditions which allowed the collisionalmodel proposed by Marble 3 to be extended to treat the three-dimensional motion of particles~ downstream of the shock wave. particularly important for two reasons. This model is The criteria for its appli- cation are fairly straightforward and easy to handle. Secondly, par- ticularly in the present problem, the model admits a wide class of physically interesting situations. The essential feature of model is that the random velocity imparted to a particle <Ji by a collision is damped by its viscous inter- action with the gas before the particle CJi suffers another collision. The model used here also assumes the particle motion is governed by Stokes law and the viscous flow fields about each particle do not interfere during collision. Within the framework of the kinetic theory method used in this thesis, the last two conditions may be relaxed to include more general particle-particle and particle-gas interaction laws when necessary. Within the previous restrictions the mean free path method of kinetic theory was extended to derive the macroscopic properties of particles 02_ including particle O{-particleOi_ and particle a;_ -gas interactions. The gas-particle Oj' relaxation zone was calculated in manner of references 4 and 6 . Within the restrictions of the colli- sion model it was possible to analyze the macroscopic motion of particles <JZ for the three important limiting cases: 2 (<1?:/Oj J'2.. >"'> J.. , (<S2/OjJ 2 << J_ , and (<12/<11 ) .L . When (02/0j )2. > ) l the length of the shock relaxation zone, f'..J -20 1 - established by the interaction of particles <Jl and the gas, is small compared to the relaxation length of particles CJ;_ • of the shock wave particles Then downstream <Ji are moving through a uniform distri- bution of particles Cfj . Consequently there is essentially no redistri- bution of particles <f normal to the gas flow. The only effect of particleCJ!-particletr2. encounters is a drag force acting to retard particles G""2. • ~/Oj' ) <::.::::l the large difference between velocities 2 When of particles ( eJi. and particles Oj results in many collisions for each particle o;_ while tr aver sing the relaxation zone. As a consequence of these collisions there is a force of interaction between the two particle clouds which opposes the viscous force exerted on the particles the gas and tends to accelerate particles O: . 2. "Z by It was found this force may significantly exceed the viscous force exerted on the particles by the gas. In addition the collisions cause a diffusion of particles across the gas flow. G;_ The characteristic stepsize of this diffusion process is the local maximum transverse range of particles d,2. The transverse range is defined as the maximum distance a particle can move across the gas flow as the result of a local collision. Although the dispersion of particles <ii is inhibited by the viscous force, there exists a wide range of physically interesting conditions for which collisional effects dominate the viscous forces. Under such circum- stances there may be a substantial redistribution of particles 02_ downstream of the shock wave. As an example, consider a normal gas dynamic shock with -202Mach number 3. 2 passing through a gas-particles suspension with the following properties; atmospheric air at 20°C, metal particles 3 with~ =-10 ~ - L p"I_ 2s <G_~ .L and cr, -=8.0k C-p, CTI 4/" l 1- • , If particles a;_ are distributed over a beam of .Lem wide upstream of the shock wave then, far downstream, collisions will have diffused the beam into one about 12. 6 cm wide. The characteristic dimension of the density distribution of particles"2_ has been increased by a factor of 12. 6 due to particle-particle collisions. In addition it has been found that increasing the shock Mach number and the particle loading increases the degree of dispersion of particles <J;_ • However, moderate increases in the particle<Jj loading do not result in corresponding increases in particle a;, dispersion. One of the most signifi- cant factors affecting the degree of dispersion of particles OZ,. is the o;_/q- . When <>i../°i--7>0 the transverse range of particles 02 is significantly reduced. On the other hand, as 62.JOj approaches radius ratio unity but stays different from it, the degree of dispersion of particles OZ. , due to several scattering events in the shock relaxation zone, was found to increase considerably. The appropriate conclusion here is that the collisional dispersion of particles az: is most important when the particle radii (Ji_ and <Ji are the same order of magnitude. Clearly, however, thedispersion vanishes when To examine the case when particles <r;_/c:r(:::_ .L G\':::::.. a;. it was assumed that <J2_ had at mo st one collision during their pas sage through the shock relaxation zone. It was found again that the redistribution of particles ~ due to collision with particles q- was limited by the maximum value of the transverse range of particles 02_ which is -203 - attained within the shock relaxation zone. A rigorous condition was established under which the single collision model is valid. The re- sults of these calculations showed that for lov1 particle Oj loadings or upstream gas densities less than atmospheric, and permissible values of <fi/<Jt , single collision events may result in an important redistribution of particles ~· The general results that have been obtained indicate the significant results associated with particle collisions in dusty gases may be examined by extending the methods of kinetic theory. On the basis of the similarity between shock relaxation flow and other gas-particle flow problems of interest, it is expected that the general kinetic theory calculation procedures established here should have a relatively wide range of applicability. Certainly as our theoretical and experi- mental understanding of the effects of particle-particle interactions on gas-particle flows increases a more comprehensive kinetic theory treatment will evolve. One of the most substantial obstacles to extending our analysis is the lack of good experiment regarding the particle-particle and particle-gas interaction laws. Definitive experiments to establish these laws are needed. The calculations which have been described indicate that the experimental study of shock waves passing through gas-particle suspensions, may be particularly well suited for investigating particleparticle interactions and other non-equilibrium phenomena in solid particle-gas flows. By controlling <:r;./cr, and upstream gas-particle conditions the results obtained make it possible to study both the -204collisional regime, """C"'v:z.. < T.::21 the particle-gas relaxation flow. , and the single collision regime of By measuring the di sp e rsion of the particle <1i_ cloud downstream of the shock wave, the important physical constants for given particle-particle and particle-gas interaction laws may be measured. -205REFERENCES ( 1) Zenz, F. A. and Othmer, D. F ., Fluidization and Fluid-Particle Systems, Reinhold Publishing Corp., New York ( 1960). (2) Soo, S. L . , Fluid Dynamics of Multiphase Systems, Blaisdell Publishing Company, Massachusetts ( 1967). (3) Marble, F. E., "Mechanism of Particle Collision in the OneDimensional Dynamics of Gas-Particle Mixtures," The Physics of Fluids, 7, 1270 (1964). ( 4) Marble, F. E., "Dynamics of a Gas Containing Small Solid Particles, " Proceedings of the Fifth AGARD Combustion and Propulsion Colloquium, Pergamon Press, New York (1963). (5) Carrier, G. F., "ShockWavesinaDustyGas, 11 J. Fluid Mechanics, 376 (1955). (6) Rudinger, G., "Some Properties of Shock Relaxation in Gas Flows Carrying Small Particles," The Physics of Fluids, 7, 658 ( 1964). (7) Kriebel, A. R., "Analysis of Normal Shock Waves in ParticleLaden Gas, " Journal of Basic Engineeri~, Transactions of ASME, 655, (December, 1964) . (8) Hoglund, R. F., "Recent Advances in Gas-Particle Nozzle Flows, 11 J. American Rocket Society, 662 ( 1962). (9) McLaughlin, M. H., "Experimental Study of Particle-Wall Collision Relating to Flow of Solid Particles in a Fluid, " Thesis, California Institute of Technology (1968). (10) Torobin, L. B. and Gauvin, W. H., "Fundamental Aspects of Solids-Gas Flow," Canadian Journal of Chemical Engineering, Part I, p. 129 (Aug.1959); Part II, p. 167 (Oct. 1959); Part III, p. 224 (Dec. 1959); Part IV, p. 142 (Oct. 1960); Part V, p. 189 (Dec. 1960). (11) Fuks, N. A., The Mechanics of Aerosols, Revised Edition, Pergamon Press (1964). (12) Present, R. D., Kinetic Theory of Gases, McGraw-Hill Book Company, Inc. (1958). (13) Chandrasekhar, S., "Stochastic Problems in Physics and Astronomy, 11 Rev. Mod. Phys., _!2, p. 1 (1943). i_, B, (14) Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. II, Interscience, New York (1965). (15) Jeans, J., The Dynamical Theory of Gases, Dover Publications, 4th Edition (I 954), p. 293. (16) Rudinger, G., Multi Phase Symposium, edited by N. J. Lipstein (American Society of Mechanical Engineers, New York, 1963 ), p. 55. -20 7APPENDIX A DYNAMICS OF THE COLLISION PROCESS On the basis of the results presented in Chapter II, Section 3 the collision process has been treated as the interaction of two rigid elastic spheres. The effect of the gas on the particle-particle inter- action has been neglected. In this Appendix we consider the collision of two elastic spheres of radius 0£ and cJi in sufficient detail to establish the results required in the calculations of Chapters IV and V. Prior to the collision the spheres are assumed to be moving parallel to the X axis at the local collisionless velocities, u.f' and 1 ~2 , of particles <1j and o;_ respectively. 1. Collision of particles <:r; and oz as viewed in the local collisionless velocity frame of particles °i . The reference frame, x~'i, is moving at the local collisionless velocity, 11rz, of particles 02 • The appropriate geometrical descrip- tion of the collision is presented in Figure Al. 0- 2 then ~"(ui•-uf-i.) <<::: (up,-~z.J (<Ji+02..) V/'- Since Au1, A11:z., ) ) °t andconsequentlythe velocities of the particles used to analyze the collision process may be evaluated at the same value of x . Prior to the collision, as indi- cated in Figure (Al-i), particle CJ2 is at rest and particle CJ( is x' direction with impact parameter b' approaching from the negative and azimuthal angle that U.p 1(x) <p I > ut'z.Cx ) We have assumed, without loss of generality, The spheres are considered to interact only at the instant of contact. Then the momentum of particle <fZ immediately before impact is where )< is t f I the position of the origin of '><l\'C downstream of the shock wave which -208- Cl) 'I'W\ me!;~+c.l / lo e. .\'.ore col\ I~·l oV'. ( + ~ ~f ~c-1 l\'2.) 1 ; Cl ( <f 1;:.. COY\'ilA"'-+ f\a.ne. \"' ~--~~~.au-~--~--- x' (\trf" ~ - ur2.()(.') ("i-\) lW\~e d.1C.-..fd/ crt+er c0Ui•::10'Y\, <e'= (OTI<;-k"\1- l'~O.Y\.e. .,..., Figure Al. ~, -209is also the center of particle CJ2'. For smooth elastic spheres during the instant of contact, see Figure (Al-ii), the force of interaction is along their line of centers, described by the contact angle 4 and I azimuthal angle 'f . Consequently the total angular momentum of the particles is conserved and the entire collision event takes place I in the <p = constant plane. Furthermore, during the collision the spheres exchange momentum only along their line of centers which, as mentioned, is denoted by ~ I in the ~ I = constant plane. Conse- quently, referring to Figure (Al-ii), after the collision, particle recoils in the direction a;: 'lf and particle er, is scattered into angle {)' I both particles remaining in the C\' = constant plane. By conservation of momentum and energy; Cf> rl. 1 'M 1( up 1(.,o - LAyiX l) = M 1 l.9r 1(x.J \ Cos..\t' + '(V)z. I ~Cx) \ CO'S 'f t (f) (Al. l) (Al. Z) (Al. 3) we obtain three equations in three unknowns; yt) ) U.pz ()() \ ~ • The contact angle 4' is related to the 1 , and c+J 1~p 1 cx > I impact parameter \, by simple geometry, (Al. 4) Since in the calculations we are only interested in the motion of particle~ following the collision if we let: -ZlOCf) (,f) (f) ur[x = '12(~) :? x' + vfz. (x) -'i e' 1 ({) ~ °lAp-i.()() + = <J -r (~1 u~x) '§l'' , where I I I X, ~ , ::t. are unit vectors in the / + Cf J Wp 2 (X) ~/ ~I J e'il ) e=t./ , and and Y' I directions, re spec ti ve- ly, then the solution of Equations (Al. 1-Al. 3) for the velocity of particle oz. after the collision is; (Al. 6) I I The ~ and ~ components of velocity are derivable from the radial l Cf) velocity component and are given by; 'Uyf;) (f) Vy2.(X) .:= ~ CUy,('X) - \kf2.0< )") (Al. 7) t'YI l +Mi.. (Al. 8) Immediately following the collision the tip of the velocity vector of particle /I / <l2_ lies on a sphere in the velocity space, for the X ~ ~ reference frame, that is displaced from the origin by an amount The situation is sketched in Figure AZ. In the present case where 1 forward as viewed in I XL\~ ~p 1 (X.) > L)>~) the scattering is entirely I frame of reference; Figure (AZ-i). However, formulas (Al. 5-Al. 8) are also valid if cf _1 provided 'r , 11 ~>() 1 < U.rz..(.X) <p( and IJoI retain their meaning. When l\p (>0< Uf.2..(x) 1 <;:{) 'lftz..()() (i) Figure AZ. U.rfX) > Uy:J,0<:) 1~\TtYlz. 1 (.ii) u..r oo < Recoil velocity sphere for particles 02_ m, hJ..r/'O -ufz.00 I (-f) ur/") ~.z<x1 ~) 1J'ri-()() I ,__. ,_. N I 1.{~XJ -212and particles °2_. overtake particles OJ in the local collisionless I I I velocity frame X j1:. of particles v;_ the scattering is all backward; Figure (AZ-ii). Statistics of encounter for particle °2 relative to reference frame 2. I I I xi~ . Consider a small volume element dV' centered at the origin of )<1 '11~ the reference frame 1 which is moving at the local collisionless a; . We wish to calculate J"-JP2, the average number of particles of radius a:: that are scattered, due to collisions 2 t - t -t dt and into with particles Oj , out of dV in a time interval velocity of particles I I · r1( I I I I I d\I. The a solid angle o.Q ::z. ~In 'Y cilV ctq> \f ( angles \ and <fI are defined, as in the previous section, as the con- which is centered in tact angle and azimuthal angle, respectively, for a collision between a particle of radius I Variation of t\-1 and oz with a particle er that occurs within dv ' r f I for collision events throughout ~v' is neglected. We suppose that encounters in which more than two particles take part are negligible in number and effect, compared with binary encounters. This is consistent with the previous assumption that the distribution of particles is dilute. life of a particle Collisions occupy only a small fraction of the c;: . I In considering such encounters within JV it will be assumed that both sets of particles are moving at their local collisionless velocities before impact and that they are randomly distributed across ly I • ol Particles a;: are essentially at rest indV while particles approach them along the negative x I direction. During the calculation - 213.- ., / / r -- - I ......... I b J[--L---11~---+--~"< x' I I -----/ -- ,.. / 'i' 't::' Figure 13. Collision diagram / -214U.p,(x) ") ~;z.(X ') we will assume Then there is no correlation between their position ind\( and their velocity. of time Also the interval Ji will be taken to be small compared to the time scales governing the motion of the particles between successive encounters, but large compared to the duration of an encounter which in the present case involving hard spheres is quite short. I lV is related to the impact parameter b' of particle 0\ prior to collision by (Al. 4) then JNf:z. is also I the number of collisions between particles q and ~ within dV in a time t - 1:- +J-b where the geometric encounter variables b' and 'f/ / I \/ t I I of particle oz:_ are located in the range 6 - b +ctb and ~ - ~ + ~. Now since the contact angle This is apparent from Figure A3. Let us now compute dN from this f2 point of view. Consider Figure A3, the number of collisions that occur in time Jt , between a given particle <:s_and particles Oj , is equal to the number of incident particles of radius di whose centers are to be found in the oblique cylinder of base area I "2. . ( 1.' I LL I d~ =-(<J\-t-0~)5'1Y'\~d~~ and slant length (v\p 1CK) - Ll.p~GO )ct.I.. This I number is simply the volume of the cylinder multiplied by the local number density, r'ly\(x") , of particles This gives the number of particles of radius I (~ 1 (X) -u~2(X)) Jt<J:)S4JS 'f. <J\ scattered from a a;- in d\1, during time Jt, such that particle 1 single particle of radius I a;: recoils into solid angle J..Q..; (AZ. 1) -Zl5This follows because particles exchang e mom e ntum only along their ell i1! <; recoil into the angle 1!' '-t- 'i' +d't' 1 1 on each collision whe re particle ClJ has impact parameter b - b +Jb' . line of cente rs and hence particles Multiplying (AZ. 1) by '0yz (~) dV centers, particles 6i_, in Strictly speaking l\rz_(~) / the total number of scattering ciV ' we obtain dN ?'2. ; I should be the density of particles in are essentially at rest awaiting a collision. J..V that This is not the average density since there will be particles of radius az pas sing through that are not elig ible, under assumption < Tcz.1 sion. -Cu"- , for a colli- As an approximation we will neglect this discrepancy and assume is, indeed, the average density of particles o;_ within This approximation becomes less accurate as the velocity equilibration time , -CU.._ , of particles OZ time between successive collisions, Then from (AZ. 2) the number approaches the average -c;;, 1 , for a particle a;: . ffz(.D.) of particles of radius j..Q..." due to collisions in dv' at a position 02:, scattered per unit time X relative to the shock face, is; , appropriate redefinition of as in Figure (AZ-ii) leads to cf' ·<:J'f' and b' -216- I Therefore if I 4 and~ are appropriately defined as the contact angle I and azimuthal angle, respectively, for a collision in JV that in general; Alternatively Eq. (A2. 5} may be expressed in terms of the sion coordinates of particle <72:. pre colli- Using (Al. 4); (A2. 6) I Since the impact parameter f (b )<f) of the collision are the same in I all reference frames moving parallel to the )( or X axis (A2. 6) is the same in these reference frames. 3. Collision of particles Cf and cr as viewed in their center of mass 4 reference frame. Consider the collision process discussed in the last two sections as viewed in the center of mass reference frame X momentum of the colliding particles is zero. qi where the total The velocity of this frame relative to the rest frame of the shock wave is; (A3. 1) The geometry of the collision for Figure A4. U-f 10<) 7 l)z CX) is given in In contrast to normal convention we define the scattering -21 7- r Figure A4. - 218or recoil angle of particle <Ji:as -3" ::::.2qi since in the present case -lJ-' is measured with respect to the positive x axis. We note the collision in this frame preserves the magnitudes of the particle velocities. The velocity components of the particle ~ in this frame after the collision can be written using Figure A4; (-[.) - Uf~X) ~ (~~-Ur2-)Co'>% =:: ~( Up 1 -ur~ 'o"i{r (A3. 2) YV\ 1i- O'\z. q:) '1ff 2 (}() -::::. (vcm- upz.) siri~ co5~ :=_::0J.-(uy,-uy2.) '51'1-B-c:osf m,+W\z.. (A3. 3) (A3. 4) where Cf> ~ The impace parameter is related to the contact angle - b : : : ( ~, .+- <rz. \ si n Lf \P by; (A3. 5) The velocity diagra m for this collision relative to the rest frame of the shock wave is given in Figure A5. 4. Statistice of encounters for particles a;_ as viewed in the center of mass frame. Referring to Section 2 for some of the details the number per second of particles a-2.. scattered in a the center of mass frame is; volume element dV relative to -21 9- G .c. ?~~~--,-~-+----.;~~~~ Figure AS. 3l where -2 20- f,()( ) oo l\ f, (X l rJ f•( ~) l Ur,(;<) -Ur2 <i<)\ bJ6 o\'f dV (b )<t) are the impact variables of the collision. (A4. l) Using (A3. 5) this may be expressed in terms of the contact angle~ as; or in terms of the recoil angle~ and where is the number per second of particles solid angle * d..R.::::. S 1Y\ J.~ ~ . 0 • Then (A4. 3) a;_ scattered out of JV into the -221- APPENDIX B CYLINDRICAL SYMMETRY - ()2/v: << 1 I Within the assumptions set forth in Chapter IV, consider the collisional dispersion of particles downstream from the shock wave when their densities upstream are given by; (B.• 1) and (B. 2) 0 The independent variable Vordinate system ( \") e) x is the radial coordinate of a polar co- ) with the 're plane on the downstream face of the shock wave. When the density of particles <Ti. upstream of the shock wave is given by the delta function distribution (B. 1) then downstream of the shock wave their density is given by the fundamental beam solution, Eq. (4. 149); (B. 3) The notation of Chapter IV is used throughout Appendix B. Introducing the dimensionless variables in Equations (4. 157) and (4. 158) -22 2- (B. 4) The fundamental beam solutions (B. 3) and (B. 4) represent the limiting case of very narrow upstream density distributions whose radial dimensions are much less than ~vl. The density of particles cr2 downstream from the shock wave ffz.. (ul'=t::') , of particles oz. up(1\ for a general density distribution, stream of the shock wave is given by Eq. (4. 150); Changing variables in (B. 5) from cartesian coordinates to cylindrical coordinates and using the upstream density distribution (B. 2) we obtain; p Cr-) \ fz (B. 6) where; '2.7\ \ R.cr»'l"')G) = j e 0 / rr' cosCG -8) /2.~x) c{g/ (B. 7) The integral of B. 7 may be rewritten, using the relation GO -t- 2. ~ Jii-t> cos k.~ (B. 8) l?.= I where "'J;,{1:) is the nth order modified Bessel function. (B. 8) into (B. 7) and carrying out the integration in Substituting E} we obtain; (B. 9) a result that is independent of tions. e as expected by symmetry considera- Substituting (B. 9) into (B. 6) we obtain; -~/+~()(1 ~ fI 7..(X/f.. ) ;;;; frLC1') (\W\ }§.rff"' ) Io (rs' 2.250<) _:~~()() )e r'clr' (B. 10) 2,c;cx:1 0 Introduce dimensionless variables; °"" r D'5(x: J r~ "'2. /4 -r p (; ~) =- ~") ( MI ) e "' \? u.ci> \ vrx>1 2.D ~(R') 2 .) where rz. A> N2j "'\J Jo\ T -o QD~cxJ "'-1 \f2 = \r, cco> )rr... if there are no collisions; "\f 14D'5<x> cE' )e r'd~' -V" The density (B. 11 ) fffco) is the final density frzcoo)-== (uJ1)/ucco)) fri.(1) The density distributions (B. 4) and (B. 11) have been computed numerically and typical solutions are give n in Figures Bl and B2. Except for the upstream density distribution of particles °2. the state of the gas-particle mixture upstream of the shock wave is the same as that used in the calculations for Figures (L8-2.0). Therefore the .I XI/\ .2 ~/\ .3 %:'.: .4 Figure Bl. Variation of particle a;: cylindrical fundamental beam density distribution downstream of 0 the shock front. Upstream conditions; Gas-Air ,f1)-==.l..,"3.X161kl T(\)=0:2o c , M 1 =~:2... , ~ ==l.4 1(1 =:. 25 , fs/f"O = .La"3 , CS. /oi :::: l /4 , Cfj = 8.o f<... /\-= ).v 1 31 crn. , cs {cf ::: ,A .. , IAT\ :== J_ 0 10 20 30 A.CO) \ Fz. 40 _fu 60 I I .... N N K1 =.2S shock front. Figure BZ. 0 .2 .4 .2 I I I fA 3.94 == .3 x / !\ .4 .5 I krl-\~ , fs· /fC 1-:> == ic/', Clz./Oj :::: 1/4, ~::?. 8,o fc Upstream conditions; Gas-Air, f ( I ) == .l.. ."31' lo -3 1 , M ::: 3, 2 , o-= 1.4 -J , A= A\1 1 -:::::. 3lc 01 ) ~/~ = Au~T, :::.1-. 0 Tc1"")::: 2.o c Variation of particle <S. cylindrical uniform beam density distribution downstream of the .I \ffOO) .6 I _fu:_ .8 I I I I LO - - - - - - - - - -· l I , I N N lJ1 -226collision parameters presented in Figure 20 also apply to the present case. Comparison of the results presented in Figures lB and lq with corresponding results plotted in Figures Bl and B2 reveals that the essential physical features of the collisional dispersion of particles OZ. downstream of the shock wave are independent of the cross sectional geometry. The only differences may be accounted for by the fact that, in the cylindrical case, the particles °i are scattered over a larger cross sectional area.