Abstract
We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius ϵ>0. Denoting the regularization parameter by δ, we consider regularized SBT based on the most common regularized Stokeslet plus a regularized doublet correction. Given sufficiently smooth force data along the filament, we derive L∞ bounds for the difference between regularized SBT and its classical counterpart in terms of δ, ϵ, and the force data. We show that the regularized and classical expressions for the velocity of the filament itself differ by a term proportional to log(δ/ϵ); in particular, δ=ϵ is necessary to avoid an O(1) discrepancy between the theories. However, the flow at the surface of the fiber differs by an expression proportional to log(1+δ2/ϵ2), and any choice of δ∝ϵ will result in an O(1) discrepancy as ϵ→0. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed slender body PDE which classical SBT is known to approximate. Numerics verify this O(1) discrepancy but also indicate that the difference may have little impact in practice.
Highlights
The method of regularized Stokeslets was introduced by Cortez in [1] to eliminate the need to integrate a singular kernel in boundary integral methods for Stokes flow
The method of regularized Stokeslets has become especially popular for modeling the dynamics of thin fibers in a three-dimensional fluid, providing an alternative way to deal with the singular integrals arising in the classical slender body theories of Lighthill [9], Keller–Rubinow [10], and Johnson [11]
In this paper we aim to compare slender body theory based on regularized Stokeslets [28] to its classical counterpart [9–11,29–31]
Summary
The method of regularized Stokeslets was introduced by Cortez in [1] to eliminate the need to integrate a singular kernel in boundary integral methods for Stokes flow. The method of regularized Stokeslets has become especially popular for modeling the dynamics of thin fibers in a three-dimensional fluid, providing an alternative way to deal with the singular integrals arising in the classical slender body theories of Lighthill [9], Keller–Rubinow [10], and Johnson [11]. Cortez and Nicholas [28] used a matched asymptotic expansion of the regularized SBT velocity field at the filament surface to compare the regularized fiber velocity to the classical Lighthill [9] and Keller–Rubinow [10] slender body theories. Part of the issue stems from balancing the effects of two small parameters, and δ, and trying to relate them in a physically meaningful yet practically useful way This type of issue does not arise for regularized Stokeslets in 2D or over surfaces in 3D. We verify the above differences numerically and provide numerical evidence that the small magnitude and extent of the discrepancy between regularized and classical SBT may mean that their difference is more of a moral issue than a practical one
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call