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

TL;DR

The paper analyzes propulsion of a rigid body in a viscous fluid driven by a time-periodic, zero-mean force, notably from an oscillating internal mass. It proves propulsion is genuinely nonlinear and occurs at second order, with $\overline{\bm{\gamma}}=\delta^2\mathbb A\cdot\bm G+o(\delta^2)$, where $\mathbb A$ is SPD and $\bm G$ encodes nonlinear thrust; propulsion requires $\bm G\neq{\bf 0}$ and is sensitive to body fore–and–aft symmetry via the hydrodynamic resistance matrix $\mathbb K$. The work establishes a rigorous weak-solution framework for the nonlinear problem and validates the theory with numerical experiments showing propulsion for non-symmetric shapes and specific internal-mass forcing, while revealing higher-order effects can drive propulsion even in some symmetric cases. Collectively, these results advance understanding of nonlinear propulsion mechanisms in viscous fluids and suggest further analytic and numerical exploration of higher-order contributions and broader geometries.

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.

Figures (5)

• Figure 2.1: Schematic of the Driving Mechanism of $\mathscr{B}$.
• Figure 6.1: Three forces used in the simulation. From left to right: internal motion of the rigid body $y(t)$, relative velocity of the rigid body $\dot{y}(t)$ and its acceleration $\ddot{y}(t)$. From top to bottom: symmetric smooth force $\ddot{y}(t)$, non-symmetric forces $\ddot{y}_2(t)$ and $\ddot{y}_3(t)$.
• Figure 6.2: Ellipsoid with symmetry in the direction of flow (left), drop-like shape (center), and flipped drop (right).
• Figure 6.3: Variation with Stokes number of the average velocity, $\overline{\gamma}$, for the ellipsoid, computed from the full nonlinear problem for forces $\ddot{y}_2$ and $\ddot{y}_3$.
• Figure 6.4: Position of the center of mass of the flipped drop $\mathscr B_{fd}$ ($\int_0^t\gamma(s)\,{\rm d}s$) and net distance covered $(\overline{\gamma}\,t$) vs. time, computed from the full nonlinear problem.

Theorems & Definitions (18)

• Remark 2.1
• Lemma 3.1
• Lemma 3.2
• Theorem 4.1
• Remark 4.2
• Definition 4.3
• Lemma 4.4
• Theorem 4.5
• proof
• Definition 5.1