Asymptotic analysis of contact problems between an elastic material and thin-rigid plates

We consider an elastic material in contact with a three-dimensional rigid plate of varying thickness. We suppose that a perfect adhesion occurs along thin zones disposed in a self-similar way on the interface between the two materials. We suppose that the elasticity coefficients in the plate depend on its thickness and tend to infinity as this thickness tends to zero. We derive the effective material properties using Γ-convergence methods.     

Darcy's laws for non‐stationary viscous fluid flow in a thin porous medium

We consider a non‐stationary Stokes system in a thin porous medium Ω_? of thickness ? which is perforated by periodically solid cylinders of size a _? . We are interested here to give the limit behavior when ? goes to zero. To do so, we apply an adaptation of the unfolding method. Time‐dependent Darcy's laws are rigorously derived from this model depending on the comparison between a _? and ? . Copyright © 2016 John Wiley & Sons, Ltd.     

Derivation and analysis of an effective model for biofilm growth in evolving porous media

The present article deals with the growth of biofilms produced by bacteria within a saturated porous medium. Starting from the pore‐scale, the process is essentially described by attachment/detachment of mobile microorganisms to a solid surface and their ability to build biomass. The increase in biomass on the surface of the solid matrix changes the porosity and impedes flow through the pores. Using formal periodic homogenization, we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients provided by the evolving microstructure at the pore‐scale. Assuming, that the underlying pore geometry may be described by a single parameter, for example, porosity, the level set equation locating the biofilm‐liquid interface transforms into an ordinary differential equation (ODE) for the parameter. For such a setting, we state significant analytical and algebraic properties of these effective parameters. A further objective of this article is the analytical investigation of the resulting coupled PDE–ODE model. In a weak sense, unique solvability either global in time or at least up to a possible clogging phenomenon is shown. Copyright © 2016 John Wiley & Sons, Ltd.     

Creeping flow analysis of slightly non-Newtonian fluid in a uniformly porous slit

This paper provides the analysis of the steady, creeping flow of a special class of slightly viscoelastic, incompressible fluid through a slit having porous walls with uniform porosity. The governing two dimensional flow equations along with non-homogeneous boundary conditions are non-dimensionalized. Recursive approach is used to solve the resulting equations. Expressions for stream function, velocity components, volumetric flow rate, pressure distribution, shear and normal stresses in general and on the walls of the slit, fractional absorption and leakage flux are derived. Points of maximum velocity components are also identified. A graphical study is carried out to show the effect of porosity and non-Newtonian parameter on above mentioned resulting expressions. It is observed that axial velocity of the fluid decreases with the increase in porosity and non-Newtonian parameter. The outcome of this theoretical study has significant importance both in industry and biosciences.     

Homogenization of a two‐phase incompressible fluid in crossflow filtration through a porous medium

We study the homogenization of a slow viscous two‐phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid‐fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two‐phase fluid in the free domain, a coupled Darcy law connecting two‐phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.     

Derivation of a quasi‐stationary coupled Darcy–Reynolds equation for incompressible viscous fluid flow through a thin porous medium with a fissure

We consider a non‐stationary Stokes system in a thin porous medium of thickness ε that is perforated by periodically distributed solid cylinders of size ε, and containing a fissure of width η_ε. Passing to the limit when ε goes to zero, we find a critical size in which the flow is described by a 2D quasi‐stationary Darcy law coupled with a 1D quasi‐stationary Reynolds problem. Copyright © 2017 John Wiley & Sons, Ltd.     

Combined effect of magnetic field and heat absorption on unsteady free convection and heat transfer flow in a micropolar fluid past a semi-infinite moving plate with viscous dissipation using element free Galerkin method

The fully developed electrically conducting micropolar fluid flow and heat transfer along a semi-infinite vertical porous moving plate is studied including the effect of viscous heating and in the presence of a magnetic field applied transversely to the direction of the flow. The Darcy-Brinkman-Forchheimer model which includes the effects of boundary and inertia forces is employed. The differential equations governing the problem have been transformed by a similarity transformation into a system of non-dimensional differential equations which are solved numerically by element free Galerkin method. Profiles for velocity, microrotation and temperature are presented for a wide range of plate velocity, viscosity ratio, Darcy number, Forchhimer number, magnetic field parameter, heat absorption parameter and the micropolar parameter. The skin friction and Nusselt numbers at the plates are also shown graphically. The present problem has significant applications in chemical engineering, materials processing, solar porous wafer absorber systems and metallurgy.     

Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods

This article deals with development and analysis of a numerical method for a coupled system describing miscible displacement of one incompressible fluid by another through heterogeneous porous media. A mixed finite element (MFE) method is employed to discretize the Darcy flow equation combined with a conservative finite volume (FV) method on unstructured grids for the concentration equation. It is shown that the FV scheme satisfies a discrete maximum principle. We derive L^∞ and BV estimates under an appropriate CFL condition. Then we prove convergence of the approximate solutions to a weak solution of the coupled system. Numerical results are presented to see the performance of the method in two space dimensions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008