When hard spheres overlap - generalization of the Rotne-Prager-Yamakawa hydrodynamic tensors

The Rotne-Prager-Yamakawa (RPY) approximation is commonly used to model the hydrodynamic interactions between small spherical particles suspended in a viscous fluid at a low Reynolds number. It takes into account long-range contribution to hydrodynamic interactions and yields positive definite diffusion matrix, which is essential for Brownian dynamics modeling. However, when the particles overlap, the RPY tensors lose their positive definiteness, which leads to numerical problems in the Brownian dynamics simulations as well as errors in calculations of the hydrodynamic properties of rigid macromolecules using bead modeling. We extend the RPY approach to the case of overlapping spherical particles of different radii in a consistent way that preserves positive definiteness of diffusion tensors for translational, rotational and dipolar degrees of freedom. Moreover we show how the Rotne--Prager--Yamakawa approximation can be generalized for other geometries and boundary conditions.