Slender-Body Theory for Stokes Flow and Flagellar Hydrodynamics

Author: Johnson, Robert Edward

Year: 1977

Degree: Dissertation (Ph.D.)

Advisor: Wu, Theodore Yao-tsu

Committee Member: Unknown, Unknown

Option: Engineering

DOI: 10.7907/HYMQ-AH77

Abstract

The singularity method for Stokes flow is used to examine the flow past slender bodies possessing finite centerline curvature, in a viscous, incompressible fluid without any appreciable inertia effects. The motion of a slender toroidal ring in Stokes flow is considered first. The symmetry of the geometry and absence of ends has made an accurate analysis possible; the result of this problem elucidates the general flow characteristics present for bodies moving in an arbitrary manner with a finite centerline curvature. Using the methods developed here it is possible to calculate the force/length to higher orders in the slenderness parameter, ϵ, than has previously been possible. In particular, we find the Stokeslet strength with an error of O(ϵ²). The solution of the torus problem serves as an effective guide in extending the theory to slender bodies of circular cross section with arbitrary centerline configurations and spheroidal ends. In all the cases considered, the no-slip boundary condition is satisfied by distributing appropriate Stokeslets, doublets, rotlets, sources, stresslets, and quadrupoles on the body centerline up to an error term of O(ϵ² lnϵ), which is sufficient for practical application. From the general slender body analysis we find an integral equation which determines the Stokeslet strength up to the term of O(ϵ²). The general theory is then applied to examine the propulsion of flagellated microorganisms, including an approximate solution for the interaction between cell body and flagella. A final brief note is made on the thrust enhancing capabilities of oscillating non-spherical cell bodies.

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