Convergence to the Barenblatt Solution for the Compressible Euler Equations with Damping and Vacuum

We study the asymptotic behavior of a compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcys law. In this paper, we give a definite answer to this conjecture without any assumption on smallness or regularity for the initial data. We prove that any L weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges, strongly in L^p with decay rates, to matching Barenblatts profile of the porous medium equation. The density function tends to the Barenblatts solution of the porous medium equation while the momentum is described by Darcys law.This revised version was published in April 2005. The volume number has now been inserted into the citation line.     

Theoretical derivation of the conservation equations for single phase flow in porous media: a continuum approach

This paper deals with the theoretical derivation of the conservation equations for single phase flow in a porous medium. The derivation is obtained within the framework of the continuum mechanics and classical thermodynamics. The adopted procedure provides the conservation equations of mass, momentum, mechanical energy, total energy, internal energy, entropy, temperature, enthalpy, Gibbs free energy and Helmholtz free energy. The obtained results highlight the connection between the basic equations of fluid mechanics and of fluid flow in porous media, as well as the restrictions and the limitations of Darcy’s law and Richards’ equation.     

Compressible Euler Equations¶with General Pressure Law

We study the hyperbolic system of Euler equations for an isentropic, compressible fluid governed by a general pressure law. The existence and regularity of the entropy kernel that generates the family of weak entropies is established by solving a new Euler-Poisson-Darboux equation, which is highly singular when the density of the fluid vanishes. New properties of cancellation of singularities in combinations of the entropy kernel and the associated entropy-flux kernel are found. We prove the strong compactness of any sequence that is uniformly bounded in L ^∞ and whose corresponding sequence of weak entropy dissipation measures is locally H ^-1 compact. The existence and large-time behavior of L ^∞ entropy solutions of the Cauchy problem are established. This is based on a reduction theorem for Young measures, whose proof is new even for the polytropic perfect gas. The existence result also extends to the p-system of fluid dynamics in Lagrangian coordinates. Accepted: December 16, 1999     

On Large-Time Behavior of Solutions to the Compressible Navier-Stokes Equations in the Half Space in R 3

 The Navier-Stokes equation for compressible viscous fluid is considered on the half space in R ³ under the zero-Dirichlet boundary condition for the momentum with initial data near an arbitrarily given equilibrium of positive constant density and zero momentum. Time decay properties in L ² norms for solutions of the linearized problem are investigated to obtain the rate of convergence in L ² norms of solutions to the equilibrium when initial data are sufficiently close to the equilibrium in . Some lower bounds are derived for solutions to the linearized problem, one of which indicates a nonlinear phenomenon not appearing in the case of the Cauchy problem on the whole space. (Accepted May 8, 2002) Published online October 18, 2002 Communicated by T.-P. LIU     

Poiseuille Flow of a Third Grade Fluid in a Porous Medium

Effects of porous medium have been investigated on the steady flow of a third grade fluid between two stationary porous plates. The continuity and momentum equations along with modified Darcy??s law are used for the development of mathematical problem. The governing nonlinear problem is solved by a homotopy analysis method. The dimensionless velocity and shear stresses at the plates are analyzed.     

L'équation de Navier–Stokes avec mémoire et le transfert de quantité de mouvement dans un milieu poreuxThe Navier–Stokes equation with memory,and momentum transfer in a porous medium

This paper is concerned with the momentum transfer in a porous medium. The equations for the continuous equivalent medium are written by the averaging method but the closure is obtained using an extended thermodynamics. The resulting model corresponds to the Navier–Stokes equation, in which the force exerted by the solid matrix on the fluid satisfies of first order differential equation. This Navier–Stokes equation model with memory generalises the Darcy model with an integro-differential term. To cite this article: O. Séro-Guillaume, D. Calogine, C. R. Mecanique 330 (2002) 383–389.     

Prediction of Non-Darcy Coefficients for Inertial Flows Through the Castlegate Sandstone Using Image-Based Modeling

Near wellbore flow in high rate gas wells shows the deviation from Darcy??s law that is typical for high Reynolds number flows, and prediction requires an accurate estimate of the non-Darcy coefficient (?? factor). This numerical investigation addresses the issues of predicting non-Darcy coefficients for a realistic porous media. A CT-image of real porous medium (Castlegate Sandstone) was obtained at a resolution of 7.57???m. The segmented image provides a voxel map of pore-grain space that is used as the computational domain for the lattice Boltzmann method (LBM) based flow simulations. Results are obtained for pressure-driven flow in the above-mentioned porous media in all directions at increasing Reynolds number to capture the transition from the Darcy regime as well as quantitatively predict the macroscopic parameters such as absolute permeability and ?? factor (Forchheimer coefficient). Comparison of numerical results against experimental data and other existing correlations is also presented. It is inferred that for a well-resolved realistic porous media images, LBM can be a useful computational tool for predicting macroscopic porous media properties such as permeability and ?? factor.     

A Phenomenological Approach to Modeling Transport in Porous Media

The objective of this article is to make use of the phenomenological approach to construct models for the transport of extensive quantities, such as mass of a fluid phase, mass of a component of a fluid phase, momentum of a phase and energy, in porous medium domains. Special attention is devoted to express the fluxes of these extensive quantities, especially the non-advective ones, as functions of their relevant driving forces, obeying the principle of minimum entropy production. It is shown that for each extensive quantity, we have a linear diffusive flux term, a non-linear diffusive term, and a dispersive flux term. The latter is shown to be proportional to the velocity squared. In each case, the number of moduli that describe fluid and porous matrix properties is determined. The momentum balance equation for a porous medium domain, which is the “motion equation,” is analyzed and simplified for special cases, leading to Darcy’s law and to Brinkman’s equation.     

Entropy Solutions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript> for the Euler Equations in Nonlinear Elastodynamics and Related Equations

A compactness framework is established for approximate solutions to the Euler equations in one-dimensional nonlinear elastodynamics by identifying new properties of the Lax entropies, especially the higher order terms in the Lax entropy expansions, and by developing ways to employ these new properties in the method of compensated compactness. Then this framework is applied to establish the existence, compactness, and decay of entropy solutions in L for the Euler equations in nonlinear elastodynamics with a more general stress-strain relation than those for the previous existence results. This compactness framework is further applied to solving the Euler equations of conservation laws of mass, momentum, and energy for a class of thermoelastic media, and the equations of motion of viscoelastic media with memory.     

Numerical Approach of the Permeability of a Macroporous Bioceramic with Interconnected Spherical Pores

Manufacturing a hybrid bone substitute requires a dynamic culture of the cells preliminarily seeded in a scaffold through a flow of physiological fluid. The velocity, pressure, and the distribution of fluid flow in this kind of macroporous medium are the important keys. Because of the difficulties in determining these parameters by experiment, a numerical approach has been chosen. One of the primary step of this study consists in the determination of permeability K. In this article, two types of structure of macroporous bioceramics are concerned. One is the interconnected pore spheres arranged either simple cubic, body-centered cubic or face-centered cubic systems. The other is the interconnected pore spheres randomly arranged. Based on Darcy??s law, the permeability K was calculated for many cases (type, porosity) by simulating the fluid flow through a small representative volume. These results are compared with some previous models such as Ergun, Carman?CKozeny, Rumpf?CGupte, and Du Plessis. The limits of Darcy??s law and the above-mentioned models have been determined using numerical simulation. The result showed that the porous media with spherical interconnected pores of BCC systems can be used to replace a complex random system in a range of porosity from 0.71 to 0.76 (i.e., porosity of our scaffolds). This assumption is validated for a pressure gradient lower the 1,000?Pa m^?C1 and a simple polynomial relation linking permeability and porosity (0.71?C0.76) has been established.     

The Acoustic Limit for the Boltzmann Equation

The acoustic equations are the linearization of the compressible Euler equations about a spatially homogeneous fluid state. We first derive them directly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish this limit for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L ¹) to a unique limit governed by a solution of the acoustic equations for all time, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L ² initial data of the acoustic equations. The associated local conservation laws are recovered in the limit. Accepted: October 22, 1999     

On the Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes Equation on the Whole Space

The existence of a time periodic solution of the compressible Navier–Stokes equation on the whole space is proved for a sufficiently small time periodic external force when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-T-map associated with the linearized problem around the motionless state with constant density in some weighted L ^∞ and Sobolev spaces. The time periodic solution is shown to be asymptotically stable under sufficiently small initial perturbations and the L ^∞ norm of the perturbation decays as time goes to infinity.     

Entropy Generation in Double-Diffusive Convection in a Square Porous Cavity using Darcy–Brinkman Formulation

The article reports a numerical study of entropy generation in double-diffusive convection through a square porous cavity saturated with a binary perfect gas mixture and submitted to horizontal thermal and concentration gradients. The analysis is performed using Darcy–Brinkman formulation with the Boussinesq approximation. The set of coupled equations of mass, momentum, energy and species conservation are solved using the control volume finite-element method. Effects of the Darcy number, the porosity and the thermal porous Rayleigh number on entropy generation are studied. It was found that entropy generation considerably depends on the Darcy number. Porosity induces the increase of entropy generation, especially at higher values of thermal porous Rayleigh number.     

Convergence Rate for Compressible Euler Equations with Damping and Vacuum

 We study the asymptotic behavior of L ^∞ weak-entropy solutions to the compressible Euler equations with damping and vacuum. Previous works on this topic are mainly concerned with the case away from the vacuum and small initial data. In the present paper, we prove that the entropy-weak solution strongly converges to the similarity solution of the porous media equations in L ^p (R) (2≤p<∞) with decay rates. The initial data can contain vacuum and can be arbitrary large. A new approach is introduced to control the singularity near vacuum for the desired estimates. (Accepted August 31, 2002) Published online January 9, 2003 Communicated by C. M. Dafermos     

Finite Element Modelling of Flow Through a Porous Medium Between Two Parallel Plates Using The Brinkman Equation

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.     

Continuous Dependence of Entropy Solutions to the Euler Equations on the Adiabatic Exponent and Mach Number

We establish the L ¹-estimates for continuous dependence of entropy solutions to the full Euler equations away from the vacuum on two physical parameters: the adiabatic exponent γ → 1 that passes from the non-isentropic to isothermal Euler equations and the Mach number that passes from the compressible to incompressible Euler equations. Our analysis involves the effective approach developed in our earlier work and additional new techniques that generalize this approach to the setting of the full Euler equations.     

Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u t = Δϕ(u)

We investigate the long time asymptotics in L¹₊(R) for solutions of general nonlinear diffusion equations u_t = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d₂. In the particular case of ϕ(u) ~ u^m at first order when u ~ 0, we also obtain an optimal rate of convergence in L¹ towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation.     

Numerical Simulation of Forced Convective Heat Transfer Past a Square Diamond-Shaped Porous Cylinder

Fluid flow and heat transfer around and through a porous cylinder is an important issue in engineering applications. In this paper a numerical study is carried out for simulating the fluid flow and forced convection heat transfer around and through a square diamond-shaped porous cylinder. The flow is two-dimensional, steady, and laminar. Conservation laws of mass, momentum, and heat transport equations are applied in the clear region and Darcy–Brinkman–Forchheimer model for simulating the flow in the porous medium has been used. Equations with the relevant boundary conditions are numerically solved using a finite volume approach. In this study, Reynolds and Darcy numbers are varied within the ranges of $1<Re<45$ and $10^{-6}<Da<10^{- 2}$ , respectively. The porosity $(\varepsilon )$ is 0.5. This paper presents the effect of Reynolds and Darcy numbers on the flow structure and heat transfer characteristics. Finally, these parameters are compared among solid and porous cylinder. It was found that the drag coefficient decreases and flow separation from the cylinder is delayed with increasing Darcy number. Also the size of the thermal plume decreases by decreasing Darcy number.     

Counter-examples to Concentration-cancellation

We study the existence and the asymptotic behavior of large amplitude high-frequency oscillating waves subjected to the two-dimensional Burger equation. This program is achieved by developing tools related to supercritical WKB analysis. By selecting solutions which are divergence free, we show that incompressible or compressible two-dimensional Euler equations are not locally closed for the weak L ² topology.