On the drag force of a viscous sphere with interfacial slip at small but finite Reynolds numbers

We investigate the hydrodynamic drag force on a viscous sphere in a fluid of different viscosities at small but finite Reynolds numbers when interfacial slip is present at the surface of the sphere. The sphere is small enough for it to retain its spherical shape, as is the case with most small droplets. By using a singular perturbation method, the exterior flow field of the droplet is decomposed into an inner region, where the viscous effects dominate, and an outer region, where the inertia is important. The interior flow of the viscous sphere is also solved analytically. By applying appropriate boundary conditions to the surface of the viscous sphere and matching the conditions between the inner and outer flow fields, stream functions up to the order of Re2 log Re for both the exterior and the interior flow are obtained. Thus, an analytical expression for the drag force coefficient of the viscous droplet is derived. This general expression yields, as special cases, several other expressions that are applicable to spheres that translate rectilinearly under more restrictive conditions. One of the practical conclusions from this study is that the presence of interfacial slip can significantly reduce the drag force on a droplet.