Abstract
Slender-body approximations have been successfully used to explain many phenomena in low-Reynolds number fluid mechanics. These approximations typically use a line of singularity solutions to represent flow. These singularities can be difficult to implement numerically because they diverge at their origin. Hence, people have regularized these singularities to overcome this issue. This regularization blurs the force over a small blob and thereby removing divergent behaviour. However, it is unclear how best to regularize the singularities to minimize errors. In this paper, we investigate if a line of regularized Stokeslets can describe the flow around a slender body. This is achieved by comparing the asymptotic behaviour of the flow from the line of regularized Stokeslets with the results from slender-body theory. We find that the flow far from the body can be captured if the regularization parameter is proportional to the radius of the slender body. This is consistent with what is assumed in numerical simulations and provides a choice for the proportionality constant. However, more stringent requirements must be placed on the regularization blob to capture the near field flow outside a slender body. This inability to replicate the local behaviour indicates that many regularizations cannot satisfy the no-slip boundary conditions on the body’s surface to leading order, with one of the most commonly used blobs showing an angular dependency of velocity along any cross section. This problem can be overcome with compactly supported blobs, and we construct one such example blob, which can be effectively used to simulate the flow around a slender body.
Highlights
Slender bodies immersed in fluids are frequently studied in biology, polymer mechanics and colloids
The former uses asymptotic methods to match the boundary conditions in slender-body theories (SBTs), while the latter assumes that the regularized singularities will produce the correct flow outside the slender body with a suitable choice of regularization parameter
We compared the asymptotic flow from a line of regularized Stokeslets to that from a slender body
Summary
Slender bodies immersed in fluids are frequently studied in biology, polymer mechanics and colloids. Two major versions of these regularized SBTs exist: one in which both regularized point forces and source dipoles are distributed over the centreline [31,32,36] and one that consists of placing a series of regularized point forces along the centreline of the body [34,35,37,38,39,40] The former uses asymptotic methods to match the boundary conditions in SBT, while the latter assumes that the regularized singularities will produce the correct flow outside the slender body with a suitable choice of regularization parameter. We look at how these conditions apply to typical blob types in Section 5 before we conclude the paper
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