On the Self-Propulsion of a Rigid Body in a Viscous Liquid by Time-Periodic Boundary Data

TL;DR

This paper investigates the conditions under which a rigid body can achieve net motion in a viscous fluid driven by time-periodic boundary velocities, distinguishing between linear and nonlinear cases based on the average boundary velocity.

Contribution

It provides a comprehensive analysis of self-propulsion mechanisms, addressing both linear and nonlinear cases, and establishes sufficient and necessary conditions for net motion.

Findings

In case (a), the problem reduces to a linear Stokes problem for small boundary velocities.

In case (b), self-propulsion involves a nonlinear analysis due to the non-zero average boundary velocity.

Counterexamples show the derived conditions are generally necessary for self-propulsion.

Abstract

Consider a rigid body, $\mathscr B$, constrained to move by translational motion in an unbounded viscous liquid. The driving mechanism is a given distribution of time-periodic velocity field, $\bfv_*$, at the interface body-liquid, of magnitude $\delta$ (in appropriate function class). The main objective is to find conditions on $\bfv_*$ ensuring that $\mathscr B$ performs a non-zero net motion, namely, $\mathscr B$ can cover any given distance in a finite time. The approach to the problem depends on whether the averaged value of $\bfv_*$ over a period of time is (case (b)) or is not (case (a)) identically zero. In case (a) we solve the problem in a relatively straightforward way, by showing that, for small $\delta$, it reduces to the study of a suitable amd well-investigated time-dependent Stokes (linear) problem. In case (b), however, the question is much more complicated, because we show that it {\em cannot} be brought to the study of a linear problem. Therefore, in case (b), self-propulsion is a genuinely nonlinear issue that we solve directly on the nonlinear system by a contradiction argument approach. In this way, we are able to give, also in case (b), sufficient conditions for self-propulsion (for small $\delta$). Finally, we demonstrate, by means of counterexamples, that such conditions are, in general, also necessary.