A single-layer based numerical method for the slender body boundary value problem

TL;DR

This paper introduces a novel single-layer potential numerical method for solving the slender body boundary value problem, improving stability and performance especially near fiber intersections.

Contribution

It presents a new completed single-layer potential formulation that enhances the numerical stability of slender body simulations, particularly when fibers are close or intersecting.

Findings

Method demonstrates good conditioning and stability.

Improved performance near fiber intersections.

Applicable to closed fibers, avoiding free-end complications.

Abstract

Fluid flows containing dilute or dense suspensions of thin fibers are widespread in biological and industrial processes. To describe the motion of a thin immersed fiber, or to describe the forces acting on it, it is convenient to work with one-dimensional fiber centerlines and force densities rather than two-dimensional surfaces and surface tractions. Slender body theories offer ways to model and simulate the motion of immersed fibers using only one-dimensional data. However, standard formulations can break down when the fiber surface comes close to intersecting itself or other fibers. In this paper we introduce a numerical method for a recently derived three-dimensional slender body boundary value problem that can be stated entirely in terms of a one-dimensional distribution of forces on the centerline. The method is based on a new completed single-layer potential formulation of fluid velocity which removes the nullspace associated with the unmodified single layer potential. We discretize the model and present numerical results demonstrating the good conditioning and improved performance of the method in the presence of near-intersections. To avoid the modeling and numerical choices involved with free ends, we consider closed fibers.