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Free flow channel confined by porous walls is a feature of many of the natural and industrial settings. Viscous flows adjacent to saturated porous medium occur in cross-flow and dead-end filtrations employed primarily in pharmaceutical and chemical industries for solid-liquid or gas-solid separations. Various mathematical models have been put forward to describe the conjugate flow dynamics based on theoretical grounds and experimental evidence. Despite this fact, there still exists a wide scope for extensive research in numerical solutions of these coupled models when applied to problems with industrial relevance. The present work aims towards the numerical analysis of coupled free/porous flow dynamics in the context of industrial filtration systems. The free flow dynamics has been expressed by the Stokes equations for the creeping, laminar flow regime whereas the flow behaviour in very low permeability porous media has been represented by the conventional Darcy equation. The combined free/porous fluid dynamical behaviour has been simulated using a mixed finite element formulation based on the standard Galerkin technique. A nodal replacement technique has been developed for the direct linking of Stokes and Darcy flow regimes which alleviates specification of any additional constraint at the free/porous interface. The simulated flow and pressure fields have been found for flow domains with different geometries which represent prototypes of actual industrial filtration equipment. Results have been obtained for varying values of permeability of the porous medium for generalised Newtonian fluids obeying the power law model. A series of numerical experiments has been performed in order to validate the coupled flow model. The developed model has been examined for its flexibility in dealing with complex geometrical domains and found to be generic in delivering convergent, stable and theoretically consistent results. The validity and accuracy of the simulated results has been affirmed by comparing with available experimental data.

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Transdermal drug delivery using microneedles depends on the rate of drug transport through the viable epidermis. Therefore minimizing the distance between the drug loaded surface and the microcirculation in the dermis where the drug is absorbed into the body is significant in improving drug delivery efficiency. A quantifiable relationship between microneedle design parameters and skin diffusion properties is therefore desirable, which is what this study aims to achieve. A framework is presented to quantitatively determine the effects of design parameters on drug diffusion through skin, where the effects of compressive strain on skin due to insertion of microneedle are considered. The model is then used to analyse scenarios of practical importance. For all scenarios analysed, predicted steady state flux was found to be lower when effect of microneedle strain on diffusion coefficient was accounted for. For example simulations results indicated increasing tip radius from 5μm to 20μm flux increased from 6.56x10 -6 mol/m 2 /s to 7.02x10 -6 mol/m 2 /s for constant diffusion coefficient. However if the effect of strain on diffusion coefficient is considered, the calculated flux increases from 5.30x10 -6 mol/m 2 /s to a peak value of 5.32x10 -6 mol/m 2 /s (at 10μm) and decreases to 5.29x10 -6 mol/m 2 /s. This paper contributes by reporting a framework to relate microneedle geometry to permeability with inclusion of the possible effects the microneedle design may pose on the diffusion coefficient.

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a b s t r a c tFree 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 (PDEs) 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 and 10 −12 m 2 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.

Despite the widespread use of the Darcy equation to model porous flow, it is well known that this equation is inconsistent with commonly prescribed no slip conditions at flow domain walls or interfaces between different sections. Therefore, in cases where the wall effects on the flow regime are expected to be significant, the Darcy equation which is only consistent with perfect slip at solid boundaries, cannot predict velocity and pressure profiles properly and alternative models such as the Brinkman equation need to be considered. This paper is devoted to the study of the flow of a Newtonian fluid in a porous medium between two impermeable parallel walls at different Darcy parameters (Da). The flow regime is considered to be isothermal and steady. Three different flow regimes can be considered using the Brinkman equation: free flow (Da > 1), porous flow (high permeability, 1 > Da > 10 −6 ) and porous flow (low permeability Da < 10 −6 ). In the present work the described bench mark problem is used to study the effects of solid walls for a range of low to high Darcy parameters. Both no-slip and slip conditions are considered and the results of these two cases are compared. The range of the applicability of the Brinkman equation and simulated results for different cases are shown.

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