2D force constraints in the method of regularized Stokeslets

TL;DR

This paper introduces a computational method for solving 2D Stokes flow problems with net nonzero force by imposing a mean zero velocity condition, addressing Stokes' paradox and enabling applications in biological modeling.

Contribution

It presents a simple, efficient approach to obtain valid 2D Stokes solutions with net force, overcoming Stokes' paradox through a mean zero velocity boundary condition.

Findings

Method effectively handles net nonzero force scenarios.

Applied successfully to cellular motility and blebbing models.

Addresses Stokes' paradox in 2D fluid simulations.

Abstract

For many biological systems that involve elastic structures immersed in fluid, small length scales mean that inertial effects are also small, and the fluid obeys the Stokes equations. One way to solve the model equations representing such systems is through the Stokeslet, the fundamental solution to the Stokes equations, and its regularized counterpart, which treats the singularity of the velocity at points where force is applied. In two dimensions, an additional complication arises from Stokes' paradox, whereby the velocity from the Stokeslet is unbounded at infinity when the net hydrodynamic force within the domain is nonzero, invalidating the solutions. A straightforward computationally inexpensive method is presented for obtaining valid solutions to the Stokes equations for net nonzero forcing. The approach is based on imposing a mean zero velocity condition on a large curve that surrounds the domain of interest. The condition is shown to be equivalent to a net-zero force condition, where the opposite forces are applied on the large curve. The numerical method is applied to models of cellular motility and blebbing.