On the Propulsion of a Rigid Body in a Viscous Liquid by Time-Periodic Force with a Zero Average

TL;DR

This paper analyzes how a rigid body can be propelled in a viscous fluid using a time-periodic force with zero average, providing conditions for propulsion and confirming them through numerical simulations.

Contribution

It offers a rigorous proof of conditions for second-order propulsion and explores cases with and without symmetry, supported by numerical validation.

Findings

Propulsion occurs at second order for asymmetric bodies.

Symmetric bodies like ellipsoids do not propel at second order.

Numerical results confirm propulsion in symmetric cases at higher orders.

Abstract

We perform analytical and numerical analyses of the propulsion of a rigid body in a viscous fluid subjected to a periodic force with zero average over a period. This general formulation specifically addresses the significant case, where propulsion is generated by the oscillation of a mass located in an internal cavity of the body. We provide a rigorous proof of the necessary and sufficient conditions for propulsion at the second order of magnitude of the force. These conditions are implemented and confirmed by numerical tests for bodies without fore-and-aft symmetry, while they are silent for bodies with such symmetry, like round ellipsoids. Consequently, in this case, propulsion can only occur at an order higher than the second. This problem is investigated by numerically integrating the entire set of equations, and the result shows that, in fact, propulsion does occur, thus opening new avenues for further analytical studies.