Development of a predictive mathematical model for coupled Stokes–Darcy flows in cross-flow membrane filtration

Free flow regimes accompanied by porous walls feature commonly in a variety of natural processes and industrial applications such as groundwater flows, packed beds, arterial blood flows and cross-flow and dead-end filtrations. Cross-flow microfiltration or ultrafiltration processes are generally employed in a range of industrial situations ranging from oil to medical applications. The coupled free/porous fluid transport phenomenon plays an equally important role along with the particle transport mechanisms concerning the separation efficiency of cross-flow membrane filtration. To provide a theoretical background for the experimental outcomes of cross-flow filtration, a mathematically sound model is desired which can reliably represent the interfacial boundary whilst maintaining the continuity of flow field variables across the interface between the free and porous flow regimes. Notwithstanding the numerous attempts reported in the literature, the development of a generic mathematical model for coupled flows has been prohibited by the complexities of interactions between the free and the porous flow systems. Henceforth, the aim of present work is to gain a better mathematical understanding of the interfacial phenomena encountered in coupled free and porous flow regimes applicable to cross-flow filtration systems. The free flow dynamics can be justifiably represented by the Stokes equation whereas the non-isothermal, non-inertial and incompressible flow in a low permeability porous medium can be handled by the Darcy equation. Solutions to the system of partial differential equations (PDE’s) are obtained using the finite element method employing mixed interpolations for the primary field variables which are velocity and pressure. A nodal replacement scheme previously developed by the same authors has been effectively enforced as the boundary constraint at the free-porous interface for coupling the two physically different flow regimes in a single mathematical model. A series of computational experiments for permeability values of the porous medium ranging between 10-6 -10-12 m2 have been performed to examine the susceptibility of the developed model towards complex and irregular shaped geometries. Our results indicate that at high permeability values, the discrepancy in mass balance calculations is observed to be significant for a curved porous surface, which may be attributed to the inability of the Darcy equation to represent the flow dynamics in a highly permeable medium. At a low permeability, a very small amount of fluid permeated through the free/porous interface as most of the fluid leaves the domain through the free flow exit. The geometry and permeability of the free/porous interface are found to affect the amount of fluid passing through the porous medium significantly. All the numerical solutions that are presented have been theoretically validated for their accuracy by computing the overall mass continuity across the computational domains.