Fluid transport by individual microswimmers

TL;DR

This paper analyzes the fluid transport and tracer particle displacement caused by microswimmers, deriving a universal formula for the Darwin drift that accounts for swimmer flow fields and wake effects.

Contribution

It introduces a universal expression for the Darwin drift in microswimmer flows, incorporating quadrupolar flow and wake contributions, validated by models and numerical simulations.

Findings

Tracer displacement behavior varies with distance from swimmer path.

The Darwin drift formula includes universal and non-universal terms.

Validation with squirmer and realistic swimmer models confirms the theory.

Abstract

We discuss the path of a tracer particle as a microswimmer moves past on an infinite straight trajectory. If the tracer is sufficiently far from the path of the swimmer it moves in a closed loop. As the initial distance between the tracer and the path of the swimmer $\rho$ decreases, the tracer is displaced a small distance backwards (relative to the direction of the swimmer velocity). For much smaller tracer-swimmer separations, however, the tracer displacement becomes positive and diverges as $\rho \to 0$. To quantify this behaviour we calculate the Darwin drift, the total volume swept out by a material sheet of tracers, initially perpendicular to the swimmer path, during the swimmer motion. We find that the drift can be written as the sum of a {\em universal} term which depends on the quadrupolar flow field of the swimmer, together with a non-universal contribution given by the sum of the volumes of the swimmer and its wake. The formula is compared to exact results for the squirmer model and to numerical calculations for a more realistic model swimmer.