On Self-Propulsion by Oscillations in a Viscous Liquid

TL;DR

The paper rigorously analyzes self-propulsion of a deformable body in a Navier–Stokes liquid driven solely by time-periodic shape changes. It develops a reformulation in a fixed reference configuration, introduces a generalized div-inversion operator, and combines linear and nonlinear periodic analyses via an invading-domain Galerkin framework to prove existence of $T$-periodic solutions for small deformation amplitude $\delta$. A key outcome is the explicit, order-$\delta^2$ propulsion relation, linking the averaged center-of-mass velocity to deformation data through a thrust functional, with a clear nonlinear mechanism for propulsion. An application to a spherical body with a dipole-like deformation demonstrates nontrivial propulsion features, including an optimal frequency for maximum speed and quantitative predictions beyond linear models.

Abstract

Suppose that a body $\mathscr B$ can move by translatory motion with velocity $\boldsymbolγ$ in an otherwise quiescent Navier-Stokes liquid, $\mathscr L$, filling the entire space outside $\mathscr B$. Denote by $Ω= Ω(t)$, $t\in\mathbb{R}$, 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 $Ω$ 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, $Ω(t+T) = Ω(t)$, for some $T > 0$ and all $t \in \mathbb{R}$. Then, find necessary and sufficient conditions on the map $t\mapsto Ω(t)$ 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.

Figures (2)

• Figure 1: Domain
• Figure 2: Dependence of the the thrust on the Stokes number $h$

Theorems & Definitions (37)

• Remark 2.1
• Remark 2.2
• Lemma 3.1
• Remark 3.1
• Remark 3.2
• Lemma 4.1
• Lemma 4.2
• Remark 4.1
• Remark 4.2
• Remark 4.3