Title: Mathematical foundations of slender body theory
Abstract: Slender body theory (SBT) facilitates computational simulations of thin filaments in a 3D viscous fluid by approximating the hydrodynamic effect of each fiber as the flow due to a line force density along a 1D curve. Despite the popularity of SBT in computational models, there had been no rigorous analysis of the error in using SBT to approximate the interaction of a thin fiber with fluid. In this talk, we develop a PDE framework for analyzing the error introduced by this approximation. Given a 1D force along the fiber centerline, we define a notion of `true’ solution to the full 3D slender body problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing. In addition, we perform a complete spectral analysis of the slender body PDE in a simple geometric setting, which sheds light on the use of SBT in approximating the `slender body inverse problem,’ where we instead specify the fiber velocity and solve for the 1D force density. Finally, we introduce a numerical method based on a boundary integral formulation of the slender body PDE and discuss numerical comparisons with slender body theory.
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