On the slow motion of a sphere parallel to a nearby plane wall

A new method using a matched asymptotic expansions technique is presented for obtaining the Stokes flow solution for a rigid sphere of radius a moving uniformly in a direction parallel to a fixed infinite plane wall when the minimum clearance ea between the sphere and the plane is very much less than a . An ‘inner’ solution is constructed valid for the region in the neighbourhood of the nearest points of the sphere and the plane where the velocity gradients and pressure are large; in this region the leading term of the asymptotic expansion of the solution satisfies the equations of lubrication theory. A matching ‘outer’ solution is constructed which is valid in the remainder of the fluid where velocity gradients are moderate but it is possible to assume that e = 0. The forces and couples acting on the sphere and the plane are shown to be of the form (α 0 +α 1 e) log e + β 0 + O (e) where α 0 , α 1 and β 0 are constants which have been determined explicitly.