Propulsion in a viscoelastic fluid

TL;DR

This paper analyzes how viscoelastic stresses affect flagella propulsion, deriving a universal formula for swimming velocity in complex fluids, and showing how material properties can tune locomotion without changing gait.

Contribution

It provides a simple, general model for flagella propulsion in viscoelastic fluids, revealing how fluid properties influence swimming speed and efficiency.

Findings

Swimming velocity depends on Deborah number and fluid viscosities.

Transport can be tuned by changing fluid properties, not gait.

Results are independent of specific nonlinear fluid models.

Abstract

Flagella beating in complex fluids are significantly influenced by viscoelastic stresses. Relevant examples include the ciliary transport of respiratory airway mucus and the motion of spermatozoa in the mucus-filled female reproductive tract. We consider the simplest model of such propulsion and transport in a complex fluid, a waving sheet of small amplitude free to move in a polymeric fluid with a single relaxation time. We show that, compared to self-propulsion in a Newtonian fluid occurring at a velocity U_N, the sheet swims (or transports fluid) with velocity U / U_N = [1+De^2 (eta_s)/(eta) ]/[1+De^2], where eta_s is the viscosity of the Newtonian solvent, eta is the zero-shear-rate viscosity of the polymeric fluid, and De is the Deborah number for the wave motion, product of the wave frequency by the fluid relaxation time. Similar expressions are derived for the rate of work of the sheet and the mechanical efficiency of the motion. These results are shown to be independent of the particular nonlinear constitutive equations chosen for the fluid, and are valid for both waves of tangential and normal motion. The generalization to more than one relaxation time is also provided. In stark contrast with the Newtonian case, these calculations suggest that transport and locomotion in a non-Newtonian fluid can be conveniently tuned without having to modify the waving gait of the sheet but instead by passively modulating the material properties of the liquid.