Nature can provide us with important lessons on how to swim efficiently in a variety of situations. Though scallops can swim in the ocean by opening and closing their valves, the scallop theorem shows that this swimming mechanism would be ineffective in highly viscous conditions. In this regime, the Navier-Stokes equations linearize to give the Stokes equations. Since the Stokes equations are time-reversible, microorganisms have developed hair-like structures known as cilia in order to generate propulsion by breaking time-reversal symmetry. The squirmer model uses a slip velocity defined on the microswimmer surface to mimic the effect of beating cilia. There has been previous work on spherical and axisymmetric squirmers, but nature is not necessarily axisymmetric – thus, we explore microswimmers of arbitrary shape. We propose a framework for expressing the slip velocity on an arbitrary surface in terms of tangential basis functions. Using the reciprocal theorem, given a time-independent slip profile, we can determine the rigid body velocities and the resulting swimming trajectory for an isolated microswimmer suspended in free space.


At much larger length scales, flexible swimming sheets can be used to transport solar panels on water and can even model ice sheets on the ocean. In this regime, the inviscid Euler equations hold. Unlike the Stokes equations, the Euler equations are nonlinear and time-dependent. However, the boundary integral techniques that are commonly used in Stokes flow can also be used in this context to track the motion of the water-air interface. We will showcase preliminary results in which we test our numerical methods on linear and nonlinear water waves in the absence of a sheet. We will then discuss how the slithering motion of snakes can help us model the elastic sheets.