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.
Bibliographical noteFunding Information:
WM and HB were supported by NSF grant DMS-1907796 . DS was supported in part by NSF grant DMS-2009352 . LO was supported by NSF Postdoctoral Research Fellowship DMS-2001959 . YM was supported by NSF ( DMS-2042144 ) and the Math+X grant from the Simons Foundation . We acknowledge the generous hospitality and computational resources of the IMA, where the project was initiated. We thank the anonymous reviewers for their careful reading of the manuscript and for their helpful insights.
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- Integral equations
- Numerical methods
- Slender body theory
- Stokes flows