The instability of a moving viscous drop

Abstract

The deformation of a moving spherical viscous drop subject to axisymmetric perturbations is considered. The problem is formulated using two different variations of the boundary integral method for Stokes flow, one due to Rallison & Acrivos, and the other based on an interfacial distribution of Stokeslets. An iterative method for solving the resulting Fredholm integral equations of the second kind is developed, and is implemented for the case of axisymmetric motion. It is shown that in the absence of surface tension, a moving spherical drop is unstable. Prolate perturbations cause the ejection of a tail from the rear of the drop, and the entrainment of a thin filament of ambient fluid into the drop. Oblate perturbations cause the drop to develop into a nearly steady ring. The viscosity ratio plays an important role in determining the timescale and the detailed pattern of deformation. Filamentation of the drop emerges as a persistent but secondary mechanism of evolution for both prolate and oblate perturbations. Surface tension is not capable of suppressing the growth of perturbations of sufficiently large amplitude, but is capable of preventing filamentation.