Translational motion of a spheroidal drop in a viscous fluid

Abstract

The problem of translational motion of a spheroidal drop along its axis of revolution in a viscous incompressible fluid is investigated semi-analytically. The flow fields in the exterior and interior of the drop are governed by the Stokes equations. Stream function formulation is adopted to solve the hydrodynamic equations in both regions. The general solution for the stream function in prolate and oblate spheroidal coordinates is expressed in an infinite-series form of semi-separation of variables. The leading order coefficients in the stream function are obtained using suitable boundary conditions. The hydrodynamic drag force experienced by the spheroidal drop is numerically evaluated with adequate convergence behavior for various values of the internal-to-external viscosity ratio and axial-to-radial aspect ratio of the drop. The numerical values of the drag force for the infinite and infinitesimal viscosity ratios agree with the available corresponding results for the slow translation of a slip spheroidal particle in the limiting conditions of no slip and full slip, respectively. At intermediate values of the viscosity ratio, the hydrodynamic force may not be a monotonic function of the aspect ratio. For a spheroidal drop with a fixed aspect ratio, its drag force increases monotonically with an increase in the viscosity ratio.