Hydrodynamics of a slip-stick sphere with a non-axisymmetric patch

Abstract

We study the low Reynolds number hydrodynamics of a slip-stick sphere suspended in an arbitrary ambient Stokes flow, whose surface is partitioned into two regions with different slip lengths. The fore-aft symmetry of the sphere breaks due to the varied slip length over the surface, which causes translational and rotational motion of the slip-stick sphere. An analytical solution is developed using the double curl method to evaluate Faxén's formulae for the hydrodynamic drag and torque exerted on the slip-stick sphere for the sub-cases, namely, (a) cap/strip model and (b) patch model. Subsequently, we compute the flow field, velocity, and rotation rate, which strongly depend on the slip lengths and configuration of the patch. As a specific example, we consider the slip-stick sphere immersed in a Poiseuille flow. For the cap/strip model, we find an optimal configuration for which the velocity of the slip-stick sphere is maximum compared to the slip-stick sphere with uniform slip. We also find configurations for which the velocity is independent of the slip lengths. Subsequently, in the patch model, we obtain the optimal azimuthal angles for the maximum rotation rate of the slip-stick sphere. We observe near-field deviations in streamlines due to the heterogeneous nature of the surface of the slip-stick sphere. These findings help design efficient artificial passive swimmers with prescribed slip lengths.