Mathematical modelling of couple stress fluid flow around a semi‐permeable sphere enclosing a solid core

Abstract

The focus of this article is to study theoretically the steady‐axisymmetric creeping flow dynamics of a couple stress fluid external to a semi‐permeable sphere containing a solid core. This problem is motivated by emulsion hydrodynamics in chemical engineering where rheological behaviour often arises in addition to porous media effects. The non‐Newtonian Stokes’ couple stress fluid model features couple stresses and body couples that are absent in the classical Navier–Stokes viscous model. It provides a robust framework for simulating emulsions, complex suspensions and other liquids which possess microstructure. The physical regime is delineated into two zones – the interior of the semi‐permeable zone (region II) which engulfs the solid core and the external couple stress fluid zone (region I). The model is formulated using a spherical polar coordinate system in terms of the stream function ψ. The Brinkman‐extended Darcy model is deployed for the porous medium hydrodynamics and isotropic permeability is considered. Analytical expressions are derived for dimensionless pressure, tangential stress and the couple stress components using the method of separation of variables and Gegenbauer functions of the first kind. The integration constants are evaluated with appropriate boundary conditions on the inner and outer boundary of the semi‐permeable zone with the aid of Mathematica symbolic software. Solutions for the drag force exerted by the couple stress fluid on the semi‐permeable sphere and volumetric flow rate are also derived with corresponding expressions for the drag coefficient and non‐dimensional volumetric flow rate. The influence of permeability (k), separation parameter (l), couple stress viscosity coefficient (η), couple stress inverse length dependent parameter (λ = √(μ/η)) and couple stress viscosity ratio (τ = η/η/) on all key variables is studied graphically. Additionally, streamline contours are computed for a range of parameters including inverse permeability parameter (α = 1/k). The computations show that increasing couple stress inverse length dependent parameter (λ) greatly reduces the dimensionless volumetric flow rate in particular at high values of separation parameter (l). Flow rate is, however, markedly enhanced with permeability (k) and also couple stress viscosity parameter (η). In the presence of the solid core, a much greater drag coefficient is observed at all values of couple stress inverse length dependent parameter (λ) relative to the case without a solid core. A significant distortion in streamlines is computed with increasing separation parameter (l) with a dual vortex structure emanating. With greater couple stress viscosity parameter (τ), no tangible modification is observed in the streamline contours. The present work generalizes the earlier study of Krishnan and Shukla to consider a semi‐permeable (porous medium) outer sphere and furthermore presents detailed streamline visualizations of the creeping flow for a range of emerging parameters of relevance to non‐Newtonian chemical engineering processes including emulsion droplet dynamics in porous materials.