Hydrodynamic Interactions between Two Forced Objects of Arbitrary Shape: I Effect on Alignment

Abstract

We study the properties and symmetries governing the hydrodynamic interaction between two identical, arbitrarily shaped objects, driven through a viscous fluid. We treat analytically the leading (dipolar) terms of the pair-mobility matrix, affecting the instantaneous relative linear and angular velocities of the two objects at large separation. We prove that the instantaneous hydrodynamic interaction linearly degrades the alignment of asymmetric objects by an external time-dependent drive [Moths and Witten, Phys. Rev. Lett. 110, 028301 (2013)]. The time-dependent effects of hydrodynamic interactions are explicitly demonstrated through numerically calculated trajectories of model alignable objects composed of four stokeslets. In addition to the orientational effect, we find that the two objects usually repel each other. In this case the mutual degradation weakens as the two objects move away from each other, and full alignment is restored at long times.