On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere

Extract

The uniform approach of a solid sphere of radius a towards a stationary solid sphere through a viscous fluid is considered when the fluid motion may be regarded as essentially steady with Reynolds number sufficiently small to permit the linearization of the Navier–Stokes equations by neglect of the inertia terms. Discussion is given when the minimum clearance εa between the spheres is arbitrary and a detailed analysis is made of the asymptotic behaviour of the solution as the minimum clearance tends to zero by constructing matched asymptotic expansions. The forces on the spheres are shown to be of the form as a₀ε⁻¹ + b₀logε + c₀ + O(εlogε) when ε is small where a₀, b₀ and c₀ are constants. The values of a₀ and b₀ are determined explicitly in the general case of arbitrary sized spheres and c₀ is also determined explicitly when the stationary sphere degenerates into a plane. For this case, it is shown that the asymptotic expression for the force is useful in practice even when ε is not very small, the error being less than 3% when ε ∼ 0·5.