Effects of a spherical slip cavity filled with micropolar fluid on a spherical viscous droplet

Abstract

An analytical study for the quasisteady flow caused by a spherical viscous fluid droplet translating at a concentric position in a second immiscible micropolar fluid within a spherical cavity with a slip surface is presented. Attention is focused on the case when one of the two fluid phases has a microstructure nature (micropolar fluid). The droplet translates along a diameter connecting their centres under the conditions of low Reynolds numbers. To solve the Stokes equations for the velocity fields inside and outside the droplet, general solutions are obtained in terms of spherical coordinates based on the concentric position. For the case of the viscous droplet within a micropolar fluid, a boundary condition related to vorticity with microrotation is used. The wall correction factors acted on the viscous droplet for the different cases are represented through graphs for various values of relative viscosity, radii ratio of droplet and cavity, and the parameter that connects the vorticity with microrotation. The wall-corrected drag force is found to be a monotonically increasing function of the ratio of drop-to-cavity radii. The present work is important because of its applications in various natural, industrial, and biological processes, such as raindrop formation, the study of blood flow, liquid–liquid extraction, the prediction of atmospheric conditions, sedimentation phenomena, and the rheology of emulsions.