Hydrodynamic Interactions between Two Forced Objects of Arbitrary Shape: II Relative Translation

TL;DR

This paper investigates how hydrodynamic interactions influence the relative translation of two arbitrarily shaped objects in viscous fluids, highlighting the role of symmetry breaking and boundary effects, with new insights into time-dependent behaviors.

Contribution

It extends previous work by analyzing the time-dependent relative translation of self-aligning objects and explores how symmetry and boundaries affect hydrodynamic interactions.

Findings

Relative translation vanishes in symmetric configurations.

Breaking inversion symmetry by boundaries induces interactions.

Separation increases over time as t^{1/3} or t depending on object type.

Abstract

We study the relative translation of two arbitrarily shaped objects, caused by their hydrodynamic interaction as they are forced through a viscous fluid in the limit of zero Reynolds number. It is well known that in the case of two rigid spheres in an unbounded fluid, the hydrodynamic interaction does not produce relative translation. More generally such an effective pair-interaction vanishes in configurations with spatial inversion symmetry; for example, an enantiomorphic pair in mirror image positions has no relative translation. We show that the breaking of inversion symmetry by boundaries of the system accounts for the interactions between two spheres in confined geometries, as observed in experiments. The same general principle also provides new predictions for interactions in other object configurations near obstacles. We examine the time-dependent relative translation of two self-aligning objects, extending the numerical analysis of our preceding publication [Goldfriend, Diamant and Witten, Phys. Fluids 27, 123303 (2015)]. The interplay between the orientational interaction and the translational one, in most cases, leads over time to repulsion between the two objects. The repulsion is qualitatively different for self-aligning objects compared to the more symmetric case of uniform prolate spheroids. The separation between the two objects increases with time $t$ as $t^{1/3}$ in the former case, and more strongly, as $t$, in the latter.