Arbitrary axisymmetric steady streaming: Flow, force and propulsion

TL;DR

This paper develops a mathematical framework to analyze steady streaming flows around spherical bodies with arbitrary axisymmetric boundary conditions, extending classical results and enabling applications in microscale force generation and propulsion.

Contribution

It introduces a generalized asymptotic method for steady streaming around spheres with arbitrary boundary oscillations, including free surface effects, broadening classical theories.

Findings

Flow solutions for small-amplitude oscillations with no-slip boundaries.

Extension of classical steady streaming results to complex boundary conditions.

Application potential in microscale force generation and synthetic locomotion.

Abstract

A well-developed method to induce mixing on microscopic scales is to exploit flows generated by steady streaming. Steady streaming is a classical fluid dynamics phenomenon whereby a time-periodic forcing in the bulk or along a boundary is enhanced by inertia to induce a non-zero net flow. Building on classical work for simple geometrical forcing and motivated by the complex shape oscillations of elastic capsules and bubbles, we develop the mathematical framework to quantify the steady streaming of a spherical body with arbitrary axisymmetric time-periodic boundary conditions. We compute the flow asymptotically for small-amplitude oscillations of the boundary in the limit where the viscous penetration length scale is much smaller than the body. In that case, the flow has a boundary layer structure and the fluid motion is solved by asymptotic matching. Our results, presented in the case of no-slip boundary conditions and extended to include the motion of vibrating free surfaces, recovers classical work as particular cases. We illustrate the flow structure given by our solution and propose one application of our results for small-scale force-generation and synthetic locomotion.