Slow motion of a droplet between two parallel plane walls

Abstract

A combined analytical–numerical study for the creeping flow caused by a fluid sphere translating in a second, immiscible fluid parallel to two flat plates at an arbitrary position between them is presented. To solve the Stokes equations for the fluid velocity fields inside and outside the spherical droplet, a general solution is constructed from fundamental solutions in both rectangular and spherical coordinate systems. Boundary conditions are enforced first at the plane walls by the Fourier transforms and then on the droplet surface by a collocation technique. Numerical results for the hydrodynamic drag force acting on the droplet are obtained with good convergence for various relative viscosities of the droplet and separation distances between the droplet and the walls. For the motion of a solid sphere (droplet with infinite viscosity) parallel to a single plane wall or to two walls, our drag results are in perfect agreement with the available solutions in the literature for all particle-to-wall spacings. The boundary-corrected drag force exerted on the droplet normalized by the value in the absence of the walls is found to increase monotonically with an increase in the internal-to-external viscosity ratio for any given geometry.