Numerical methods for Stokes flow

In a number of interesting biological applications, viscous forces are several orders of magnitude greater than inertial forces. For these situations, modelers use the Stokes equations to describe the fluid dynamics. I have been working primarily on an extension to the method of regularized Stokeslets that would allow for improved accuracy of the method when simulating the motion due to forces distributed on 2D surfaces in Stokes flow.

Modeling weak inertia in Stokes flow

Typically, models of fluid dynamics at the microscale neglect all inertial terms. The result of this is the Stokes equations (see above.) An interesting question is “what happens when there is a time scale different from the derived characteristic time scale of the fluid motion?” For example, the time period of one cilia beat may differ from the characteristic time scale of the fluid. We have been interested in modeling situations where this difference is significant enough that it is appropriate to use the unsteady Stokes equations. In this case, the inertia of the fluid plays a role, but it is still much weaker than the viscosity of the fluid. Even so, the incorporation of this “weak inertia” may significantly change the dynamics of models that instead use the steady Stokes equations.