  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.
