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 the Reynolds number (Re) tends to zero, 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 R in a number of situations in which positivity is destroyed for multipole/perturbative methods.

We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching along their line of centres, valid at all separations. This goes beyond the applicable range of existing solutions which are limited to the near-contact region of the flow. By use of a bipolar coordinate system, we give, for the first time, the stream function for the flow as Re → 0 and a formula for the hydrodynamic interaction force between the spheres as an infinite series. We also derive the behaviour of the forces as the nondimensional separation goes to zero and infinity, rigorously confirming and improving on known heuristic results relevant to a widely accepted lubrication theory. The results also hold for retreating spheres, consistent with the reversibility of Stokes flow, and we recover the limit of a sphere moving perpendicularly to a plane. We demonstrate how our results give rise to differences in dynamics when implemented in numerical solutions to colloidal flow compared with existing methods. We also provide an efficient way to compute the infinite series as a polynomial interpolant.

No abstract

No abstract