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.