Hydrodynamic force on a small squirmer moving with a time-dependent velocity at small Reynolds numbers

TL;DR

This paper extends the calculation of hydrodynamic forces to active spherical squirmers with time-dependent velocities, incorporating convective and unsteady inertia effects, and explores how swimming gait influences these forces.

Contribution

It generalizes previous passive particle models to active squirmers, accounting for unsteady motion and fluid inertia, and analyzes the impact of swimming gait on hydrodynamic forces.

Findings

Convective inertia alters the history contribution to force.

Hydrodynamic force depends on the swimming gait.

Additional contributions arise from inhomogeneous solutions.

Abstract

We calculate the hydrodynamic force on a small spherical, unsteady squirmer moving with a time-dependent velocity in a fluid at rest, taking into account convective and unsteady fluid inertia effects in perturbation theory. Our results generalise those of Lovalenti and Brady (1993) from passive to active spherical particles. We find that convective inertia changes the history-contribution to the hydrodynamic force, as it does for passive particles. We determine how the hydrodynamic force depends on the swimming gait of the unsteady squirmer. Since swimming breaks the spherical symmetry of the problem, the force is not completely determined by the outer solution of the asymptotic matching problem, as it is for passive spheres. There are additional contributions due to the inhomogeneous solution of the inner problem. We also compute the disturbance flow, illustrating convective and unsteady effects when the particle experiences a sudden start followed by a sudden stop.