On Self-Propulsion by Oscillations in a Viscous Liquid

Suppose that a body $\mathscr B$ can move by translatory motion with velocity $\boldsymbol{\gamma}$ in an otherwise quiescent Navier-Stokes liquid, $\mathscr L$, filling the entire space outside $\mathscr B$. Denote by $\Omega = \Omega(t)$, $t\in\mathbb{R}$, the 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 a frame with the origin at the center of mass $G$ and axes parallel to those of an inertial frame. We assume that there are no external forces acting on the coupled system $\mathscr S := \mathscr B +\mathscr L$ and that the only driving mechanism is a prescribed change in shape of $\Omega$ with time. 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, namely, $\Omega(t+T) = \Omega(t)$, for some $T > 0$ and all $t \in \mathbb{R}$. Then, find necessary and sufficient conditions on the map $t\mapsto \Omega(t)$ securing that $\mathscr B$ self-propels, that is, $G$ covers any given finite distance in a finite time. We show that this problem is solvable, in a suitable function class, provided the amplitude of the oscillations is below a given constant. Moreover, we provide examples where the propelling velocity of $\mathscr B$ is explicitly evaluated in terms of the physical parameters and the frequency of oscillations.