Locomotion in complex fluids: Integral theorems

TL;DR

This paper develops three integral theorems that connect the swimming motion of organisms to their kinematics in complex, non-Newtonian fluids, aiding understanding of biological and synthetic locomotion.

Contribution

It introduces a unified mathematical framework with three integral theorems for low-Reynolds number locomotion in complex fluids, applicable to various swimmer geometries and fluid models.

Findings

Derived three integral theorems for non-Newtonian fluid locomotion

Applicable to a wide range of swimmer geometries and surface motions

Facilitates quantification of biological and synthetic swimmer movement in complex environments

Abstract

The biological fluids encountered by self-propelled cells display complex microstructures and rheology. We consider here the general problem of low-Reynolds number locomotion in a complex fluid. {Building on classical work on the transport of particles in viscoelastic fluids,} we demonstrate how to mathematically derive three integral theorems relating the arbitrary motion of an isolated organism to its swimming kinematics {in a non-Newtonian fluid}. These theorems correspond to three situations of interest, namely (1) squirming motion in a linear viscoelastic fluid, (2) arbitrary surface deformation in a weakly non-Newtonian fluid, and (3) small-amplitude deformation in an arbitrarily non-Newtonian fluid. Our final results, valid for a wide-class of {swimmer geometry,} surface kinematics and constitutive models, at most require mathematical knowledge of a series of Newtonian flow problems, and will be useful to quantity the locomotion of biological and synthetic swimmers in complex environments.