Shapes optimising grand resistance tensor entries for a rigid body in a Stokes flow

TL;DR

This paper develops a calculus of variations approach to optimize the shapes of rigid bodies in Stokes flow for desired hydrodynamic resistance properties, revealing novel geometries including chiral shapes.

Contribution

It introduces a shape derivative formula and a numerical algorithm for optimizing resistance tensor entries, enabling tailored shape design in low Reynolds number flows.

Findings

Optimized shapes include chiral helical geometries.

The method effectively improves specific resistance tensor components.

Numerical results demonstrate diverse shape adaptations for different objectives.

Abstract

We investigate the optimal shapes of the hydrodynamic resistance of a rigid body set in motion in a Stokes flow. In this low Reynolds number regime, the hydrodynamic drag properties of an object are encoded in a finite number of parameters contained in the grand resistance tensor. Considering these parameters as objective functions to be optimised, we use calculus of variations techniques to derive a general shape derivative formula, allowing to specify how to deform the body shape to improve the objective value of any given resistance tensor entry. We then describe a practical algorithm for numerically computing the optimized shapes and apply it to several examples. Numerical results reveal interesting new geometries when optimizing the extra-diagonal inputs to the strength tensor, including the emergence of a chiral helical shape when maximising the coupling between the hydrodynamic force and the rotational motion. With a good level of adaptability to different applications, this work paves the way for a new analysis of the morphological functionality of microorganisms and for future advances in the design of microswimmer devices.