The Singular Hydrodynamic Interactions Between Two Spheres In Stokes Flow

TL;DR

This paper derives exact solutions for the hydrodynamic interactions between two spheres in Stokes flow, extending the range of applicability beyond near-contact and far-field approximations, and improves the accuracy of force calculations.

Contribution

It provides a comprehensive analytical framework for the complete range of sphere separations in Stokes flow, surpassing existing approximate methods and confirming asymptotic behaviors.

Findings

Derived exact stream functions for two-sphere interactions.

Obtained asymptotic force behaviors at small and large separations.

Demonstrated improved numerical accuracy and positive definiteness in simulations.

Abstract

We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching each either along or parallel to their line of centres, valid at all separations. This goes beyond the applicable range of existing solutions for singular hydrodynamic interactions (HIs) which, for practical applications, are limited to the near-contact or far field region of the flow. For the normal component of the HI, by use of a bipolar coordinate system, we derive the stream function for the flow as $Re\to 0$ and a formula for the singular (squeeze) force between the spheres as an infinite series. We also obtain the asymptotic behaviour of the forces as the nondimensional separation between the spheres goes to zero and infinity, rigorously confirming and improving upon known results relevant to a widely accepted lubrication theory. Additionally, we recover the force on a sphere moving perpendicularly to a plane as a special case. For the tangential component, again by using a bipolar coordinate system, we obtain the corresponding infinite series expression of the (shear) singular force between the spheres. All results hold for retreating spheres, consistent with the reversibility of Stokes flow. We demonstrate substantial differences in numerical simulations of colloidal fluids when using the present theory compared with existing multipole methods. Furthermore, we show that the present theory preserves positive definiteness of the resistance matrix $\boldsymbol{R}$ in a number of situations in which positivity is destroyed for multipole/perturbative methods.