Analysis of the stability of an immersed elastic surface using the method of regularized Stokeslets

TL;DR

This paper performs a linear stability analysis of an immersed elastic surface in viscous fluid using the method of regularized Stokeslets, providing insights into numerical stability and regularization effects.

Contribution

It introduces new doubly periodic regularized Stokeslets and analyzes their stability properties across various models and parameters.

Findings

Eigenvalues determine critical time step for stability.

Power law relates regularization parameter and surface discretization.

Stability properties are consistent across different elastic models.

Abstract

A linear stability analysis of an elastic surface immersed in a viscous fluid is presented. The coupled system is modeled using the method of regularized Stokeslets (MRS), a Lagrangian method for simulating fluid-structure interaction at zero Reynolds number. The linearized system is solved in a doubly periodic domain in a 3D fluid. The eigenvalues determine the theoretical critical time step for numerical stability for a forward Euler time integration, which are then verified numerically across several regularization functions, elastic models, and parameter choices. New doubly periodic regularized Stokeslets are presented, allowing for comparison of the stability properties of different regularization functions. The stability results for a common regularization function are approximated by a power law relating the regularization parameter and the surface discretization for two different elastic models. This relationship is empirically shown to hold in the different setting of a finite surface in a bulk fluid.