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arXiv · 2404.04737

A free boundary problem for an immersed filament in 3D Stokes flow

Laurel Ohm
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Abstract

We consider a simplified extensible version of a dynamic free boundary problem for a thin filament with radius $ε>0$ immersed in 3D Stokes flow. The 3D fluid is coupled to the quasi-1D filament dynamics via a novel type of angle-averaged Neumann-to-Dirichlet operator for the Stokes equations, and much of the difficulty in the analysis lies in understanding this operator. Here we show that the main part of this angle-averaged NtD map about a closed, curved filament is the corresponding operator about a straight filament, for which we can derive an explicit symbol. Remainder terms due to curvature are lower order with respect to regularity or size in $ε$. Using this operator decomposition, it is then possible to show that the simplified free boundary evolution is a third-order parabolic equation and is locally well posed. This establishes a more complete mathematical foundation for the myriad computational results based on slender body approximations for thin immersed elastic structures.

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Laurel Ohm. 2025-09-20. A free boundary problem for an immersed filament in 3D Stokes flow. https://arxiv.org/abs/2404.04737

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