On the Propulsion of a Rigid Body in a Viscous Liquid Under the Action of a Time-Periodic Force
Mher Marc Karakouzian, Giovanni P. Galdi
TL;DR
The paper investigates when a rigid body moving in an unbounded viscous fluid under a time-periodic force with nonzero average achieves a nonzero net propulsion. By formulating the problem in a body-fixed frame and seeking $T$-periodic weak solutions, it analyzes the small-force regime via a $\delta$-scaling that connects propulsion to a shape-dependent hydrodynamic coupling, ultimately reducing to a stationary Stokes problem at leading order. A key result is a necessary and sufficient condition for propulsion at order $\delta$: $\widehat{\bm b}$ must satisfy $\widehat{\bm{b}} \neq \mathbf{B}\cdot(\bm{r}\times \widehat{\bm{b}})$, with propulsion guaranteed if the body is constrained from spinning. The analysis is complemented by an explicit sphere example showing propulsion for homogeneous spheres and possible nonpropulsion for certain nonhomogeneous mass distributions, highlighting how shape and mass distribution govern propulsion via hydrodynamic matrices.
Abstract
A rigid body $\mathcal{B}$ moves in an otherwise quiescent viscous liquid filling the whole space outside $\mathcal{B}$, under the action of a time-periodic force $\boldsymbol{\mathsf{f}}$ of period $T$ applied to a given point of $\mathcal{B}$ and of fixed direction. We assume that the average of $\boldsymbol{\mathsf{f}}$ over an interval of length $T$ does not not vanish, and that the amplitude, $δ$, of $\boldsymbol{\mathsf{f}}$ is sufficiently small. Our goal is to investigate when $\mathcal{B}$ executes a non-zero net motion; that is, $\mathcal{B}$ is able to cover any prescribed distance in a finite time. We show that, at the order $δ$, this happens if and only if $\boldsymbol{\mathsf{f}}$ and $\mathcal{B}$ satisfy a certain condition. We also show that this is always the case if $\mathcal{B}$ is prevented from spinning. Finally, we provide explicit examples where the condition above is satisfied or not. All our analysis is performed in a general class of weak solutions to the coupled system body-liquid problem.