Creeping flow of Jeffrey fluid through a swarm of porous cylindrical particles: Brinkman–Forchheimer model

Abstract

The majority of the previous studies analyzed the flow of fluids with constant viscosity through membranes composed of porous cylindrical particles using the particle-in-cell approach with the Brinkman equation governing the flow through porous media. However, a slight variation in temperature affects the viscosity of the fluids and hence affects the filtration process of fluids through membranes. The motivation of this problem came from the fact that viscosity is concentration dependent due to presence of impurities and contaminants in the fluids and hence can be taken as function of position or temperature. The present work is a theoretical attempt to investigate the impact of temperature-dependent viscosity on the creeping flow of Jeffrey fluid through membrane consisting of the aggregates of the porous cylindrical particles. The flow pattern of the Jeffrey fluid is taken along the axial direction of the cylindrical particles, and the cell model approach is utilized to formulate the governing equations driven by a constant pressure gradient. The flow regime is divided into two-layer form, one is inside the porous cylindrical particle enclosing a solid core, which is governed by the Brinkman–Forchheimer equation, and another one is outside of the porous cylindrical particle, which is governed by the Stokes equation. Being a nonlinear equation, an analytical solution of the Brinkman–Forchheimer equation is intractable. To overcome this difficulty, the regular and singular perturbation methods have been employed to solve the Brinkman–Forchheimer equation under the assumption of temperature-dependent viscosity for small and large permeability of the porous medium, respectively; however, an analytical approach is utilized to solve the Stokes equation. The analytical expressions for velocity in different regions, hydrodynamic permeability of the membrane, and Kozeny constant are derived. The impact of various control parameters such as viscosity parameter, Forchheimer number, permeability of the porous medium, and Jeffrey fluid parameter on the above quantities are discussed and validated with previously published works on the Newtonian fluid in the limiting cases. The present work is in good agreement with the previously published work on Newtonian fluid under constant viscosity assumptions where the porous media flow was governed by the Brinkman equation. The remarkable observation of the present study is that higher viscosity and Jeffrey fluid parameters lead to enhanced velocity profile and hence the hydrodynamical permeability of the membranes. However, a decay in the Kozeny constant is observed with the increasing viscosity and Jeffrey fluid parameters. The coating of porous layer can be attributed to adsorption of polymers on the solid particles and further makes the present model to be more relevant in understanding the membrane filtration process.