  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.
