Vol. 7, No. 1, 2012

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
Slender body theory for Stokes flows with regularized forces

Ricardo Cortez and Michael Nicholas

Vol. 7 (2012), No. 1, 33–62
Abstract

Existing slender body theories for the dynamics of a thin tube in a Stokes flow differ in the way the asymptotic errors depend on a small parameter defined as the radius of the body over its length. Examples are the theory of Lighthill, that of Keller and Rubinow, and that of Johnson. Slender body theory is revisited here in the more general setting of forces which are localized but smoothly varying within a small neighborhood of the filament centerline, rather than delta distributions along the centerline. Physically, this means that the forces are smoothly distributed over the cross-section of the body. The regularity in the forces produces a final expression that has built-in smoothing which helps eliminate instabilities encountered in computations with unsmoothed formulas. Consistency with standard theories is verified in the limit as the smoothing parameter vanishes, where the original expressions are recovered. In addition, an expression for the fluid velocity at locations off the slender body is derived and used to compute the flow around a filament.

Keywords
slender body theory, Stokes flow
Mathematical Subject Classification 2000
Primary: 76D07, 76Z99
Milestones
Received: 9 July 2010
Revised: 29 August 2011
Accepted: 5 October 2011
Published: 3 January 2012
Authors
Ricardo Cortez
Mathematics Department
Tulane University
6823 St. Charles Ave.
New Orleans, LA 70118
United States
http://tulane.edu/sse/math/faculty/ricardo-cortez.cfm
Michael Nicholas
Mathematics Department
Tulane University
6823 St. Charles Ave.
New Orleans, LA 70118
United States
http://faculty.carthage.edu/mnicholas