On Self-Propulsion by Oscillations in a Viscous Liquid

Let $\mathscr B$ be a body moving in an otherwise quiescent Navier-Stokes liquid, $\mathscr L$, that fills the entire space outside $\mathscr B$. We will consider the case where $\mathscr B$ is prevented from performing rigid rotations around its center of mass $G$, a condition that can be realized by applying a suitable torque on $\mathscr B$. \par

Denote by $\Omega=\Omega(t)$, $t\in\mathbb R$, a one-parameter family of bounded, sufficiently smooth domains of $\mathbb R^3$, each one representing the configuration of $\mathscr B$ at time $t$ with respect to an inertial frame. We assume that there are no external forces acting on the coupled system $\mathscr S:=\mathscr B\cup\mathscr L$ and that the only driving mechanism is a {\em prescribed} change in shape of $\Omega$ with time, in a given precise way. \par

The self-propulsion problem that we would like to address can be thus qualitatively formulated as follows. Suppose that $\mathscr B$ changes its shape in a given time-periodic fashion, so that, for some $T>0$ and all $t\in \real$, $\Omega(t+T)=\Omega(t)$. Then, find sufficient conditions on the map $t\mapsto \Omega(t)$ securing that $\mathscr B$ self-propels, namely, the center of mass $G$ covers any given finite distance in a finite time.

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