Slow translation of a slightly deformed spherical fluid drop

Abstract

A drop of one fluid moving in another immiscible fluid causes shear, the flow-induced stress tends to deform the drop, and the interfacial tension between the phases resists this deformation. The present article deals with the analytical treatment of the problem of steady translational motion of a slightly deformed spherical fluid drop suspended in an immiscible viscous fluid under the consideration of vanishing Reynolds number. This is the case when the induced stress is slightly higher than the interfacial tension so that the drop is slightly deformed but does not break. The flow fields in both the interior and exterior of the drop are governed by the steady Stokes equations that are solved asymptotically using a method of perturbed expansions under suitable boundary conditions. The deformation from spherical shape is characterized by a small parameter called the deformation parameter, and the hydrodynamic boundary value problem is solved up to the second order of the deformation parameter by neglecting the higher-order terms. The effect of deformation parameter is observed by means of force expression. The explicit expressions for the hydrodynamic drag force exerted on the drop are obtained for the special cases of prolate and oblate spheroids. In the limiting cases of the drop behaving as a solid particle and a gas bubble, the force expressions agree with the corresponding formulas for the slow translation of a slightly deformed slip sphere in the limiting conditions of no slip and full slip, respectively.