Two-body transient viscous interactions in free space

This article elucidates how unsteady hydrodynamic interactions between two closely situated spheres in a viscous liquid affect their time-dependent motion. The system represents typical Brownian particles for which temporal inertia is always comparable to the viscous forces even though convective inertia is negligible. The analysis quantifies the transient mutual interactions in terms of frequency-dependent friction coefficients of both spheres as well as their temporally varying mobility response to an impulsive force. To this end, a generalization of Stokesian dynamics is formulated, where instead of Stokes equation, linearized unsteady Navier-Stokes is Fourier transformed in frequency space to describe flow fields. Accordingly, two complete sets of basis functions for the Brinkman equation instead of the Stokes equation are constructed in spherical coordinates centered around two particles. The mutual transformations between these two sets enable the enforcement of the no-slip boundary conditions on all solid-liquid interfaces. The resulting algebraic relations provide the frequency-dependent two-body frictions, whereas inverse Fourier transform of these after adding appropriate inertial contributions yields a time-dependent mobility response. The friction and mobility values are validated in limiting cases under short-time and long-time limits. The scaling laws of these quantities are also explored as functions of the separation distance between two solid bodies, revealing important physical insight into the complicated dynamics.