2D force constraints in the method of regularized Stokeslets

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 any solutions that use the free space Stokeslet. A straightforward computationally inexpensive method is presented for obtaining valid solutions to the Stokes equations for net nonzero forcing. The approach is based on modifying the boundary conditions of the Stokes equations to impose a mean zero velocity condition on a large curve that surrounds the domain of interest. The corresponding Green’s function is derived and used as a fundamental solution in the case of net nonzero forcing. The numerical method is applied to models of cellular motility and blebbing, both of which involve tether forces that are not required to integrate to zero.