Effect of interfacial slip on the cross-stream migration of a drop in an unbounded Poiseuille flow

Abstract

We analyze the motion and deformation of a buoyant drop suspended in an unbounded fluid which is undergoing a quadratic shearing flow at small Reynolds number in the presence of slip at the interface of the drop. The boundary condition at the interface is accounted for by means of a simple Navier slip condition. Expressions for the velocity and the shape deformation of the drop are derived considering small but finite interface deformation, and results are presented for the specific cases of sedimentation, shear flow, and Poiseuille flow with previously reported results as the limiting cases of our general expressions. The presence of interfacial slip is found to markedly affect axial as well as cross-stream migration velocity of the drop in Poiseuille flow. The effect of slip is more prominent for drops with larger viscosity wherein the drop velocity increases. The presence of significant interface slippage always leads to migration of a deformed drop towards the centerline of the channel for any drop-to-medium viscosity ratio, which is in contrast to the case of no slip at the interface, which allows drop migration towards or away from the centerline depending on the viscosity ratio. We obtain the effect of slip on the cross-stream migration time scale, which quantifies the time required to reach a final steady radial position in the channel. The presence of slip at the drop interface leads to a decrease in the cross-stream migration time scale, which further results in faster motion of the drop in the cross-stream direction. Gravity in the presence of Poiseuille flow is shown to affect not only the axial motion, but also the cross-stream migration velocity of the drop; interfacial slip always increases the drop velocities.