Hydrodynamic force between two hard spheres tangentially translating in a power-law fluid

The hydrodynamic interaction between two hard spheres tangentially translating in a power-law fluid is investigated. By considering the gap between the two spheres being sufficiently small such that the Reynoldsʹ lubrication theory applies, an analytical equation to the pressure in the gap is obtained using truncated Fourier series. To a good approximation, the pressure equation can be further simplified. The simplified approximate equation over-predicts the pressure for shear thickening fluid (n>1) but under-predicts the pressure for shear-thinning fluid (n>1). However, the errors in the predicted tangential force and moment are relatively small. In particular, for a Newtonian fluid, the accurate solution and the simplified approximate solution degenerate to the asymptotic solution of Goldman et al. [1967. Slow viscous motion of a sphere parallel to a plane wall-motion through a quiescent fluid. Chemical Engineering Science 22, 637-651.] and OʹNeill and Stewartson [1967. On the slow motion of a sphere parallel to a nearby plane wall. Journal of Fluid Mechanics 27, 705-724.]. Both solutions predict that for shear thickening fluid (n>1), the hydrodynamic force converged in the inner region of the gap between the two spheres and the contribution from the outer region is sufficiently small. For shear thinning fluid (n<1), the contribution from the outer region is also significant.