Search

In this work, a Lattice Boltzmann model for multi-component fluids augmented with repulsive near-contact interactions is employed to simulate the dynamics of flowing emulsions within idealized pores. The model is firstly tested on experimental data of droplets' self-assembly in diverging-converging micro-channels from literature and then used to investigate the trapping/mobilization of the dispersed phase of an emulsion in an idealized series of pores, as influenced by both the Capillary number and the solid walls wettability. Both parameters may vary as the result of an injection of surfactants, a procedure commonly adopted in soil remediation and Enhanced Oil Recovery applications. The analysis shows that the proposed model is able to reproduce correctly the experimental data and gives interesting insights on the trapping/mobilization phenomenon resulting from a modification of the flow conditions caused by the injection of surfactants. Article Highlights: A LBM multiphase model is employed to analyze a two-phase flow in a simplified pore geometry. Capillary number and solid wettability are varied to simulate the injection of surfactants. The mobilization of the previously trapped mass is quantified and related to the modified flow conditions. [ABSTRACT FROM AUTHOR]

We show that the return‐point memory of cyclic macroscopic trajectories enables the derivation of a thermodynamic framework for quasistatically driven dissipative systems with multiple metastable states. We use this framework to sort out and quantify the energy dissipated in quasistatic fluid‐fluid displacements in disordered media. Numerical computations of imbibition–drainage cycles in a quasi‐2D medium with gap thickness modulations (imperfect Hele‐Shaw cell) show that energy dissipation in quasistatic displacements is due to abrupt changes in the fluid‐fluid configuration between consecutive metastable states (Haines jumps), and its dependence on microstructure and gravity. The relative importance of viscous dissipation is deduced from comparison with quasistatic experiments. Plain Language Summary: Fluid flow into a porous material filled with another is not only an everyday process (gardening, stains in fabrics, or printing) but is also a key process affecting the water cycle, contamination in soils and storage of energy or hazardous waste in the subsurface. These flows are controlled by the energy of the fluids, and its dissipation during their advancement, making the knowledge of energy dissipation crucial to our ability to predict these phenomena. However, to date there is no rigorous way to evaluate this energy. This paper describes a novel method that overcomes this challenge, explaining how the properties of the medium affect dissipation and showing why even for very slow flows the viscous energy (i.e., related to rapid fluid motion) still makes a difference. Key Points: Rigorous account of the microscopic physics allows to compute the energy dissipated between consecutive two‐phase configurationsWe link the microscopic origins of hysteresis and dissipation to the macroscopic pressure‐saturation behaviorQuasistatic pressure‐driven experiments point to a secondary contribution of viscous dissipation during Haines jumps [ABSTRACT FROM AUTHOR]

Abstract We show that the return‐point memory of cyclic macroscopic trajectories enables the derivation of a thermodynamic framework for quasistatically driven dissipative systems with multiple metastable states. We use this framework to sort out and quantify the energy dissipated in quasistatic fluid‐fluid displacements in disordered media. Numerical computations of imbibition–drainage cycles in a quasi‐2D medium with gap thickness modulations (imperfect Hele‐Shaw cell) show that energy dissipation in quasistatic displacements is due to abrupt changes in the fluid‐fluid configuration between consecutive metastable states (Haines jumps), and its dependence on microstructure and gravity. The relative importance of viscous dissipation is deduced from comparison with quasistatic experiments.

We investigate a fully nonlinear degenerate parabolic system describing variable-density flows of a two-component mixture through porous media. Using the Rothe method of semidiscretization in time we prove the global-in-time existence of weak solutions to the associated initial–boundary value problem under physically reasonable hypotheses on the data, the exponential constitutive equation and the mixed boundary conditions. [ABSTRACT FROM AUTHOR]

Electromagnetic (EM) heating is an emerging method for storing renewable energy, such as photovoltaic solar and wind electric power, into aquifers. We investigate how the captured energy increases the temperature of a prototypical deep aquifer for a six-month period and then to which extent the stored energy can be recovered during the consecutive six months. Water injected at a constant flow rate is simultaneously heated using a high-frequency electromagnetic microwave emitter operating at the water natural resonance frequency of 2.45 GHz. The coupled reservoir flow and EM heating are described using Darcy's and the energy balance equations. The latter includes a source term accounting for the EM wave propagation and absorption, modeled separately using Maxwell's equations. The equations are solved numerically by the Galerkin least-squares finite element method. The approach was validated using EM-heating input data obtained from controlled laboratory experiments and then was applied to the aquifer. We found that after six years of alternate storage and recovery, up to 77% of the injected energy is recovered when considering realistic heat losses estimated from field data. Even when heat losses are increased by a factor of two, up to 69% of the injected energy is recovered in this case. This shows that down-hole EM heating is a highly effective method for storing renewable energies, capable of helping to solve their inherent intermittency. [ABSTRACT FROM AUTHOR]

In this paper, we consider a liquid–gas system with two components: water and hydrogen flow model in heterogeneous porous media with periodic microstructure taking into account kinetics in the mass transfer between the two phases. The particular feature in this model is that chemistry effects are taken into account. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the Darcy–Muskat and capillary pressure laws and the mass exchange is modeled as a source term in the equations. The problem is written in the terms of the phase formulation; i.e. the saturation of one phase, the pressure of the second phase, and the concentration of dissolved hydrogen in the liquid phase. The mathematical model consists in a system of partial differential equations: two degenerate nonlinear parabolic equations and one diffusion–convection equation. The major difficulties related to this new model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized coupled system of three coupled partial differential equations with effective coefficients which are computed via solving cell problems. We give a rigorous mathematical derivation of the effective model by means of the two-scale convergence. [ABSTRACT FROM AUTHOR]

Electromagnetic (EM) heating is an emerging method for storing renewable energy, such as photovoltaic solar and wind electric power, into aquifers. We investigate how the captured energy increases the temperature of a prototypical deep aquifer for a six-month period and then to which extent the stored energy can be recovered during the consecutive six months. Water injected at a constant flow rate is simultaneously heated using a high-frequency electromagnetic microwave emitter operating at the water natural resonance frequency of 2.45 GHz. The coupled reservoir flow and EM heating are described using Darcy’s and the energy balance equations. The latter includes a source term accounting for the EM wave propagation and absorption, modeled separately using Maxwell’s equations. The equations are solved numerically by the Galerkin least-squares finite element method. The approach was validated using EM-heating input data obtained from controlled laboratory experiments and then was applied to the aquifer. We found that after six years of alternate storage and recovery, up to 77% of the injected energy is recovered when considering realistic heat losses estimated from field data. Even when heat losses are increased by a factor of two, up to 69% of the injected energy is recovered in this case. This shows that down-hole EM heating is a highly effective method for storing renewable energies, capable of helping to solve their inherent intermittency.

The problem of the numerical modeling of nonisothermal water and vapor flows in a porous medium under near-critical thermodynamic conditions is considered. It is shown that the equations of single-phase and two-phase flows in porous media become degenerate at the critical point of water. Some disadvantages of flow modeling methods that are able to exclude degeneracy by selecting the pressure and enthalpy as independent model variables are discussed. A new method of modeling under near-critical thermodynamic conditions in the standard pressure–temperature variables is proposed. This method is based on an approxinate description of the fluid phase diagram with linear splines that eliminates the degeneracy of conservation laws at the critical point and enables the application of equations of state for predicting the fluid properties in pressure and temperature variables. [ABSTRACT FROM AUTHOR]

In this work, we introduce an algorithmic approach to generate microvascular networks starting from larger vessels that can be reconstructed without noticeable segmentation errors. Contrary to larger vessels, the reconstruction of fine‐scale components of microvascular networks shows significant segmentation errors, and an accurate mapping is time and cost intense. Thus there is a need for fast and reliable reconstruction algorithms yielding surrogate networks having similar stochastic properties as the original ones. The microvascular networks are constructed in a marching way by adding vessels to the outlets of the vascular tree from the previous step. To optimise the structure of the vascular trees, we use Murray's law to determine the radii of the vessels and bifurcation angles. In each step, we compute the local gradient of the partial pressure of oxygen and adapt the orientation of the new vessels to this gradient. At the same time, we use the partial pressure of oxygen to check whether the considered tissue block is supplied sufficiently with oxygen. Computing the partial pressure of oxygen, we use a 3D‐1D coupled model for blood flow and oxygen transport. To decrease the complexity of a fully coupled 3D model, we reduce the blood vessel network to a 1D graph structure and use a bi‐directional coupling with the tissue which is described by a 3D homogeneous porous medium. The resulting surrogate networks are analysed with respect to morphological and physiological aspects. [ABSTRACT FROM AUTHOR]

We present the first mathematical model of flow-mediated primary hemostasis in an extravascular injury which can track the process from initial deposition to occlusion. The model consists of a system of ordinary differential equations (ODEs) that describe platelet aggregation (adhesion and cohesion), soluble-agonist-dependent platelet activation, and the flow of blood through the injury. The formation of platelet aggregates increases resistance to flow through the injury, which is modeled using the Stokes--Brinkman equations. Data from analogous experimental (microfluidic flow) and partial differential equation models informed parameter values used in the ODE model description of platelet adhesion, cohesion, and activation. This model predicts injury occlusion under a range of flow and platelet activation conditions. Simulations testing the effects of shear and activation rates resulted in delayed occlusion and aggregate heterogeneity. These results validate our hypothesis that flow-mediated dilution of activating chemical adenosine diphosphate hinders aggregate development. This novel modeling framework can be extended to include more mechanisms of platelet activation as well as the addition of the biochemical reactions of coagulation, resulting in a computationally efficient high throughput screening tool of primary and secondary hemostasis. [ABSTRACT FROM AUTHOR]

The effective slip length at the interface between pure fluid flow and porous media composed of packed spheres has been accurately characterized. In this study, as the first part of a two-part series, the slip length is obtained by matching the flow rate over the actual packed spheres from a direct simulation with that over an effective smooth surface using the Navier-slip boundary condition from analytic solution. Three classical packing structures, e.g., simple cubic (SC), face-centered cubic (FCC), and body-centered cubic (BCC), are employed. The accuracy of the slip length is validated by comparing the velocity field of flow over the actual porous architecture and over the effective smooth surface. We report that the slip length is best described as a function of the free slip area, rather than conventional variables such as solid volume fraction and packing structure, with the error less than 7.5%. Then, the effective smooth surface with the slip length is applied to describe two flow problems: a stick–slip–stick flow and channel flow. The slip velocity as well as its slope at the interface and the velocity profile within the pure fluid channel is accurately reproduced. In Part 2, effective slip length will be employed to characterize optimal effective viscosity and stress jump coefficient in the Stokes–Brinkman approach, which can be applied for industrial and natural flows in dual-scale porous media in predicting flow solutions inside and outside the porous media. [ABSTRACT FROM AUTHOR]

We propose a model for multicomponent flow of immiscible fluids in a deformable porous medium accounting for capillary hysteresis. Oil, water, and air in the soil pores offer a typical example of a real situation occurring in practice. We state the problem within the formalism of continuum mechanics as a slow diffusion process in Lagrange coordinates. The balance laws for volumes, masses, and momentum lead to a degenerate parabolic PDE system. In the special case of a rigid solid matrix material and three fluid components, we prove under further technical assumptions that the system is mathematically well posed in a small neighborhood of an equilibrium. [ABSTRACT FROM AUTHOR]

We define a reiterated unfolding operator for a doubly periodic domain presenting two periodicity scales. Then we show how to apply it to the homogenization of both linear and nonlinear problems. The main novelty is that this method allows the use of test functions with one scale of periodicity only and it considerably simplifies the proofs of the convergence results. We illustrate this new approach on a Poisson problem with Dirichlet boundary conditions and on the flow of a power law fluid in a doubly periodic porous medium. [ABSTRACT FROM PUBLISHER]

In this paper a theoretical model has developed for the dispersion problem in porous media in which the flow is one dimensional and the average flow is unsteady. The Solution of the dispersion problem is presented by means of a new approach to Bäcklund transformations of nonlinear partial differential equations.

• The inherent fundamentals and roles of clusters in fines migration are elucidated. • Two clogging mechanisms of fines at pore-scale are essentially unveiled. • A constant saturation volume fraction of fines in the entrance is confirmed. • An exponential correlation is obtained by growth rate of cluster size. • A probabilistic model is derived for predicting clogging performance. The migration of fines suspended in the liquid through porous media represents a key challenge in many chemical engineering processes, yet the inherent fundamentals and roles of clusters were not well elucidated. In this work, we numerically study the formation, growth, and connection of clusters in a porous medium and unveil the clogging mechanism of fines at pore-scale, i.e., bridging and locking essentially. A saturation volume fraction of fines in the entrance occurs at the critical transition point from depth filtration to caking, where the volume fraction is calculated by the spherical cap method. An inter-particle contact pair method is developed to quantify the cluster size growth. Inherently, an exponential correlation between clogging ability and size ratios is obtained using the cluster information - growth rate of clusters. A probabilistic model of clogging ability and performance is derived and used to verify the exponential formula. This work provides an effective method to quantify clogging ability and could be applied to more complex systems. [ABSTRACT FROM AUTHOR]

We propose a model for unsaturated poro-plastic flow derived from the thermodynamic principles. For the isothermal case, the problem consists of a degenerate coupled system of two PDEs with two independent hysteresis operators describing hysteresis phenomena in both the solid and the pore fluids. Under natural hypotheses, we prove the existence of a global strong solution for this system. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider a simple model describing the motion of a two-component mixture through a porous medium. We discuss well posedness of the associated initial-boundary value problem, in particular, with respect to the choice of boundary and far-field conditions. The existence of global-in-time solutions is proved in the ideal case when the fluid occupies the whole physical space. Finally, similar results are obtained also for the boundary value problems in the simplified 1-D geometry. [ABSTRACT FROM AUTHOR]

In this paper the free boundary problem for groundwater phreatic surface is represented in the form of a variational principle. It is proved that the flow domain Ω that solves the problem is a minimizer of some functional Λ(Ω). Weak solutions are introduced as minimizers of the lower semi-continuous regularization of Λ(⋅). Within this approach the existence of weak solutions is proved for a wide class of input data. [ABSTRACT FROM AUTHOR]

We propose a scaling rule for the flow mobility prediction of a power-law fluid in unidirectional fibrous porous media by separating rheological contribution from geometrical factor. Starting from the analogy to a simple circular pipe flow, we find mobility dependence on pressure drop with the help of lubrication theory for flows of a power-law fluid through arrays of square and hexagonal packing cylinders. There appears a nearly unique intersection point independent of rheological parameters in the mobility and pressure drop space, once scaled with the consistency index, and the intersection point coordinate is expressed by rheological and geometrical factors separately. Further simplification is achieved by approximating the rheological contribution by a constant that scales linearly with the consistency index. The proposed mobility estimation scheme facilitates a simple but reasonably accurate estimation and it has been verified by extensive numerical simulations for various pressure drops, rheological parameters, porosities and fibrous porous architectures. [ABSTRACT FROM AUTHOR]

Abstract: As a remedy to the instability of the Galerkin finite element formulation, symmetric stabilization techniques such as the continuous interior penalty, the subgrid and local projection methods were proposed and analyzed by Burman and Hansbo (2006) [10], Badia and Codina (2009) [11], Becker and Braack (2001) [12], and Nafa and Wathen (2009) [13]. In this work we consider a coupled Stokes–Darcy problem, where in one part of the domain the fluid motion is described by Stokes equations and for the other part the fluid is in a porous medium and described by Darcy law and the conservation of mass. Such systems can be discretized by heterogeneous finite elements in the two parts, such as Taylor-Hood or MINI elements for the Stokes domain, and mixed elements of Raviart–Thomas elements type for the Darcy domain. Here, we discretize by standard equal-order finite elements enriched with bubbles functions and use local projection stabilization technique (LPS) to stabilize the method and control the fluctuation of the velocity divergence vector on the Darcy region. We also suggest a way to control the natural velocity. [Copyright &y& Elsevier]

Water flow in plant tissues takes place in two different physical domains separated by semipermeable membranes: cell insides and cell walls. The assembly of all cell insides and cell walls are termed symplast and apoplast, respectively. Water transport is pressure driven in both, where osmosis plays an essential role in membrane crossing. In this paper, a microscopic model of water flow and transport of an osmotically active solute in a plant tissue is considered. The model is posed on the scale of a single cell and the tissue is assumed to be composed of periodically distributed cells. The flow in the symplast can be regarded as a viscous Stokes flow, while Darcy’s law applies in the porous apoplast. Transmission conditions at the interface (semipermeable membrane) are obtained by balancing the mass fluxes through the interface and by describing the protein mediated transport as a surface reaction. Applying homogenization techniques, macroscopic equations for water and solute transport in a plant tissue are derived. The macroscopic problem is given by a Darcy law with a force term proportional to the difference in concentrations of the osmotically active solute in the symplast and apoplast; i.e. the flow is also driven by the local concentration difference and its direction can be different than the one prescribed by the pressure gradient. [ABSTRACT FROM AUTHOR]

Flow behaviors of a non-Newtonian fluid in spherical microstructures have been studied by a direct numerical simulation. A shear-thinning (power-law) fluid through both regular and randomly packed spheres has been numerically investigated in a representative unit cell with the tri-periodic boundary condition, employing a rigorous three-dimensional finite-element scheme combined with fictitious-domain mortar-element methods. The present scheme has been validated for the classical spherical packing problems with literatures. The flow mobility of regular packing structures, including simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC), as well as randomly packed spheres, has been investigated quantitatively by considering the amount of shear-thinning, the pressure gradient and the porosity as parameters. Furthermore, the mechanism leading to the main flow path in a highly shear-thinning fluid through randomly packed spheres has been discussed. [ABSTRACT FROM AUTHOR]

We assume that Ω is a domain in ℝ or in ℝ with a non-compact boundary, representing a generally inhomogeneous and anisotropic porous medium. We prove the weak solvability of the boundary-value problem, describing the steady motion of a viscous incompressible fluid in Ω. We impose no restriction on sizes of the velocity fluxes through unbounded components of the boundary of Ω. The proof is based on the construction of appropriate Galerkin approximations and study of their convergence. In Sect. 4, we provide several examples of concrete forms of Ω and prescribed velocity profiles on ∂Ω, when our main theorem can be applied. [ABSTRACT FROM AUTHOR]

Different models of compressible fluid filtration are considered. Unlike the classical system of equations, the continuity equation is modified with allowance for the minimum scale of space averaging and for the internal relaxation time of the system. Three-level explicit finite difference schemes are proposed that are convenient for high-performance parallel implementation. The transition from the parabolic to the hyperbolic system of equations makes the stability requirements for them less stringent than for the two-level schemes. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the identification of the space variable time derivative coefficient u in a degenerate fast diffusion differential inclusion. The function u is vanishing on a subset strictly included in the space domain Ω. This problem is approached as a control problem (P) with the control u. An approximating control problem (P) is introduced and the existence of an optimal pair is proved. Under certain assumptions on the initial data, the control is found in W(Ω), with m> N, in an implicit variational form. Next, it is shown that a sequence of optimal pairs $(u_{\varepsilon }^{\ast },y_{\varepsilon }^{\ast })$ of (P) converges as ε goes to 0 to a pair ( u, y) which realizes the minimum in (P), and y is the solution to the original state system. An alternative approach to the control problem is done by considering two controls related between them by a certain elliptic problem. This approach leads to the determination of simpler conditions of optimality, but under an additional restriction upon the initial data of the direct problem. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with the study of the well-posedness of diffusion equations with homogeneous Neumann boundary conditions. Three cases of singular diffusion equations, fast, slow and very fast, with nonlinear transport terms are considered. The fast and slow cases involve differential inclusions. The problem with homogeneous Neumann boundary conditions has been still open for the multivalued singular cases of diffusion analyzed here and especially when transport terms are present. Particular hypotheses imposed to the transport vector, initial conditions and other problem data in order to allow the existence and uniqueness proofs are discussed in each case. The proofs are based on the existence results for an approximating problem with certain Robin boundary conditions, indexed on a small parameter . Appropriate estimates independent of are established, and a passing to the limit technique, which is delicate due to the fact that the functional spaces in the approximating Robin problem depend on , is performed. Results are given for both nondegenerate and degenerate cases. [Copyright &y& Elsevier]

Abstract: We describe a class of finite volume schemes for systems of conservations laws based on a “local” decomposition of the system into a series of single conservation laws but with discontinuous coefficients. The resulting schemes are based on Godunov type solvers of the reduced equations. These schemes are very easy to implement since they do not use detailed information about the eigenstructure of the full system. We illustrate the efficiency of the schemes on a variety of numerical experiments focusing on three-phase flows in porous media and show that they are robust and approximate the flow very well, even in the presence of gravity. [Copyright &y& Elsevier]

Abstract: In this paper we study the model of heat transfer in a porous medium with a critical diffusion. We obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in Besov spaces with by the method of modulus of continuity and Fourier localization technique. [Copyright &y& Elsevier]

The correct numerical modeling of free-surface hydrodynamic problems often requires to have the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In doing so, one may fulfill relevant physical constraints. The existence, the uniqueness, and two constructive iterative methods to solve a piecewise linear system of the form max[l, min(u, x)] + Tx = b are analyzed. The methods are shown to have a finite termination property; i.e., they converge to an exact solution in a finite number of steps and, actually, they converge very quickly, as confirmed by a few numerical tests, which are derived from the mathematical modeling of flows in porous media. [ABSTRACT FROM AUTHOR]

Abstract: This work deals with the study of periodic solutions to a degenerate fast diffusion equation. The existence of the periodic solution to an intermediate problem restraint to a period T is proved first and then the result is extended by the data periodicity to all time real space. The approach involves an appropriate approximating problem whose periodic solution is proved via a fixed point theorem. Next, a passing to the limit procedure leads to the existence of the solution to the original problem on a time period. Finally, the behavior at large time of the solution to a Cauchy problem with periodic data is characterized. [Copyright &y& Elsevier]

We study the unsaturated flow of an incompressible liquid carrying a bacterial population through a porous medium contaminated with some pollutant. The biomass grows feeding on the pollutant and affecting at the same time all the physics of the flow. We formulate a mathematical model in a one-dimensional setting and we prove an existence theorem for it. The so-called fluid media scaling approach, often used in the literature, is discussed and its limitations are pointed out on the basis of a specific example. [ABSTRACT FROM AUTHOR]

We are concerned with an implicit scheme for the finite difference solution to a nonlinear parabolic equation with a multivalued coefficient that describes the fast diffusion in a porous medium. The boundary conditions contain the multivalued function as well. We prove the stability and the convergence of the scheme, emphasizing the precise nature of convergence in this specific case, and compute the error level of the approximating solution. The method is aimed to simplify the numerical computations for the solutions to equations of this type, without performing an approximation of the multivalued function. The theory is illustrated by numerical results. [ABSTRACT FROM AUTHOR]

We consider a simplified model of a two-phase flow through a heterogeneous porous medium. Focusing on the capillary forces motion, a nonlinear degenerate parabolic problem is approximated in a domain shared in two homogeneous parts, each of them being characterized by its relative permeability and capillary curves functions of the phase saturations. We first give a weak form of the conservation equations on the whole domain, with a new general expression of the conditions at the interface between the two regions. We then propose a finite volume scheme for the approximation of the solution, which is shown to converge to a weak solution in one-, two-, or three-dimensional domains. We conclude with some numerical tests. [ABSTRACT FROM AUTHOR]

The dispersion relation for surface waves on an impermeable boundary of a fully saturated poroelastic medium is investigated numerically over the whole range of applicable frequencies. To this aim a linear simplified model of a two-component poroelastic medium is used. Similarly to the classical Biot’s model, it is a continuum mechanical model but it is much simpler due to the lack of coupling of stresses. However, results for bulk waves following for these two models agree very well indeed which motivates the application of the simplified model in the analysis of surface waves. In the whole range of frequencies there exist two modes of surface waves corresponding to the classical Rayleigh and Stoneley waves. The numerical results for velocities and attenuations of these waves are shown for different values of the bulk permeability coefficient in different ranges of frequencies. In particular, we expose the low and high frequency limits, and demonstrate the existence of the Stoneley wave in the whole range of frequencies as well as the leaky character of the Rayleigh wave. [ABSTRACT FROM AUTHOR]

Discusses the spectral stability of convective rolls in porous media. Results of filtrating thermal currents; Condition where bifurcating rolls are said to be spectrally unstable; Steps toward characterizing unstable rolls.

In this paper, we deal with an optimal shape design approach for the numerical realization of the flow problem of incompressible liquid through an inhomogeneous porous medium (say dam), with a class of permeability coefficients. Our interest is in studying numerical approximation of the solution of the optimal shape design problem proposed in [C. R. Acad. Sci. Paris Se´r. I Math. 329 (10) (1999) 933–938]. We consider discretization of this problem based on linear finite elements. We prove the existence of the solution of the discrete problem. We study the convergence of the discrete problem. We present some results on numerical experiments including comparisons to the results obtained by a variational approach. [Copyright &y& Elsevier]