Slow translation of a slightly deformed spherical fluid drop in an arbitrary unsteady viscous flow

Abstract

The present study investigates the translational motion of a slightly deformed spherical fluid drop suspended in an arbitrary unsteady viscous fluid. The analysis is conducted under the assumption of a negligible Reynolds number, indicating a scenario where the induced stresses are slightly higher than the interfacial tension. Consequently, the drop undergoes a slight deformation but remains intact without breaking. The flow fields in both the interior and exterior of the drop are governed by the unsteady Stokes equations, which are solved asymptotically using a method of regular perturbation expansions under appropriate boundary conditions. The deviation from the spherical shape is quantified by a small parameter referred to as the deformation parameter, which is taken as the perturbation parameter. A complete general solution to the unsteady Stokes equations is employed to solve the equations governing the fluid flow. The hydrodynamic forces on the drop are then determined and expressed in terms of Faxén’s law for an arbitrary ambient flow field. The hydrodynamic problem is tackled up to the first order of the deformation parameter, disregarding higher-order terms. Closed-form expressions for the hydrodynamic drag force acting on the drop are derived for the specific scenarios of prolate and oblate spheroidal drops. The hydrodynamic forces obtained in the present study agree with the respective hydrodynamic forces experienced by prolate and oblate spheroidal drops in the limiting case of steady flow, as existing in the literature.