The effect of interfacial slip on the dynamics of a drop in flow: Part I. Stretching, relaxation, and breakup

Abstract

Using a numerical method based on the boundary-integral technique, we assess the impact of interfacial slip on the dynamics of deformation and breakup of a single drop subjected to a uniaxial extensional flow under creeping-flow conditions. Interfacial slip is incorporated in our continuum development as a jump in the tangential velocity across the interface. This velocity jump is shown to reduce to the Navier-slip boundary condition to leading order and is characterized by a dimensionless slip coefficient α=(dI/μI)(μ/R), where dI is thickness of the diffuse interface between the liquids, μI is the viscosity of the interfacial region, μ is the viscosity of the suspending fluid, and R is the drop radius. A key contribution of this paper is the development of a stable, boundary-integral formulation to incorporate interfacial slip into existing, no-slip boundary-integral frameworks for drop deformation. Slip has a fourfold impact on the drop stretching, relaxation, and breakup phenomena. First, when the capillary number is small, the steady deformation of the drop with slip is smaller than the no-slip result, and the difference increases with the viscosity ratio and the capillary number. Slip thus leads to larger critical capillary numbers beyond which the drop stretches continuously in the extensional flow. Second, for capillary numbers greater than the critical value, we find that the shape of the deformed drop for the same drop elongation is relatively insensitive to the slip coefficient, but the time required to reach this deformation is a strong function of the slip coefficient—slip slows down the deformation process. Third, the end-pinch mechanism of drop breakup leads to a different number and sizes of droplets with the inclusion of slip. Finally, slip causes the capillary-instability mechanism of drop breakup to produce larger drops at faster rates relative to the no-slip case. In addition to the above results, we also show that slip moderates the viscosity and normal stress differences in a sheared, dilute emulsion. Our study has important implications in the area of blending of immiscible polymers and indicates that the drop size distribution, which ultimately governs the material properties of the blend and composites prepared from it, is influenced strongly by interfacial slip.