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

T. Redaelli, F. Candelier, R. Mehaddi, C. Eloy, B. Mehlig

Published 2022-09-16Version 1

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 brought by the inhomogeneous solution of the inner problem. We also compute the disturbance flow, illustrating convective and unsteady fluid-inertia effects for a sudden start of the centre-of-mass motion, and for swimming with a periodic gait. We discuss the implications of our findings for small motile organisms in a marine environment.