Slender body theory for Stokes flows with regularized forces

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.