Pair correlations in a multicomponent fluid placed in a porous medium

We consider a multicomponent fluid placed in a porous medium. The Ornstein—Zernike approximation is used to calculate the pair correlation functions for density fluctuations in the mixture components. We show that light scattering in the neighborhood of the critical state of the system is determined (in the single-scattering approximation) by the commonly known Ornstein—Zernike formula. We investigate the shift in the critical parameters due to the porous medium. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 525–528, March, 2006.     

Existence and Nonexistence of Global Classical Solutions to Porous Medium and Plasma Equations with Singular Sources

Let Ω be a bounded or unbounded domain in R^n. The initial-boundary value problem for the porous medium and plasma equation with singular terms is considered in this paper. Criteria for the appearance of quenching phenomenon and the existence of global classical solution to the above problem are established. Also, the life span of the quenching solution is estimated or evaluated for some domains.     

Coupled deformation,filtration, and heat-conduction processes in media with disoriented quasi-spheroidal pores

Using the random function method we determine the effective thermoelastic properties of a saturated porous medium with disoriented quasi-spheroidal pores. On the basis of the functional dependence obtained for the parameters averaged over the macrovolume and phases we construct the equations of coupled deformation, filtration, and heat-conduction processes in a saturated porous medium. Bibliography: 8 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 71–80, 1991.     

Ideals of porous sets in the real line and in metrizable topological spaces

Porous sets and spherically porous sets of a metric space are studied. In particular, porous, superporous, and equiporous sets of a metric space (X, π) are characterized from the topological point of view. Metrizable spaces that are spherically porous in themselves with respect to some metric generating the topology are characterized. Some relations between the ideals of the classes of porous sets in the real line are established. Bibliography: 13 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 221–242.     

A Steady Weak Solution of the Equations of Motion of a Viscous Incompressible Fluid through Porous Media in a Domain with a Non-Compact Boundary

We assume that Ω is a domain in ℝ² or in ℝ³ 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.     

On methods of deriving equations describing effective models of layered media

Methods of deriving equations describing effective models of layered periodic media are presented. Elastic and fluid media, as well as porous Biot media, may be among these media. First, effective models are derived by a rigorous method, and then some operations in the derivation are replaced by simpler ones providing correct results. As a consequence, a comparatively simple and justified method of deriving equations of an effective model is established. In particular, this method allows us to simplify to a degree and justify the derivation of an effective model for media containing Biot layers; this method also produces equations of an effective model of a porous layered medium intersected by fractures with slipping contacts. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998 pp. 219–243. Translated by L. A. Molotkov.     

Linear,hydrodynamic flow in a rotating saturated porous medium

Linear, steady, axisymmetric flow of a homogeneous fluid in a rigid, bounded, rotating, saturated porous medium is analyzed. The fluid motions are driven by differential rotation of horizontal boundaries. The dynamics of the interior region and vertical boundary layers are investigated as functions of the Ekman number E(=v/ΩL ²) and rotational Darcy 3 numberN(=kΩ/v) which measures the ratio between the Coriolis force and the Darcy frictional term. IfN≫E ^−1/2, the permeability is sufficiently high and the flow dynamics are the same as those of the conventional free flow problem with Stewartson'sE ^1/3 andE ^1/4 double layer structure. For values ofN≤E ^−1/2 the effect of porous medium is felt by the flow; the Taylor-Proudman constraint is no longer valid. ForN≪E ^−1/3 the porous medium strongly affects the flow; viscous side wall layer is absent to the lowest order and the fluid pumped by the Ekman layer, returns through a region of thicknessO(N ⁻¹). The intermediate rangeE ^−1/3≪N≪E ^−1/2 is characterized by double side wall layer structure: (1)E ^1/3 layer to return the mass flux and (ii) (NE)^1/2 layer to adjust the interior azimuthal velocity to that of the side wall. Spin-up problem is also discussed and it is shown that the steady state is reached quickly in a time scaleO(N).     

Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones

The boundary value problem for the similar stream function f = f(η;λ) of the Cheng–Minkowycz free convection flow over a vertical plate with a power law temperature distribution T_w(x) = T_∞ + Ax^λ in a porous medium is revisited. It is shown that in the λ-range − 1/2 < λ < 0 , the well known exponentially decaying “first branch” solutions for the velocity and temperature fields are not some isolated solutions as one has believed until now, but limiting cases of families of algebraically decaying multiple solutions. For these multiple solutions well converging analytical series expressions are given. This result yields a bridging to the historical quarreling concerning the feasibility of exponentially and algebraically decaying boundary layers. Owing to a mathematical analogy, our results also hold for the similar boundary layer flows induced by continuous surfaces stretched in viscous fluids with power-law velocities u_w(x)∼ x^λ. (Received: June 7, 2005)     

Stabilization of flows through porous media

In this article, we study the motion of an incompressible homogeneous Newtonian fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed layer having an external source (with an injection rate b), and above by a free surface moving under the influence of gravity. The flow is governed by Darcy’s law. If b(c) = 0 for some c > 0 then the system admits (u, f) ≡ (c, c) as an equilibrium solution. We shall prove that the stability properties of this equilibrium are determined by the slope of b in c : The equilibrium is unstable if b′(c) < 0, whereas b′(c) > 0 implies exponential stability. Zhaoyong Feng: He is grateful to the DFG for financial support through the Graduiertenkolleg 615 “Interaction of Modeling, Computation Methods and Software Concepts for Scientific-Technological Problems”.     

Strong relaxation limit of multi-dimensional isentropic Euler equations

This paper is devoted to study the strong relaxation limit of multi-dimensional isentropic Euler equations with relaxation. Motivated by the Maxwell iteration, we generalize the analysis of Yong (SIAM J Appl Math 64:1737–1748, 2004) and show that, as the relaxation time tends to zero, the density of a certain scaled isentropic Euler equations with relaxation strongly converges towards the smooth solution to the porous medium equation in the framework of Besov spaces with relatively lower regularity. The main analysis tool used is the Littlewood–Paley decomposition.     

Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones

The boundary value problem for the similar stream function f  =  f(η;λ) of the Cheng–Minkowycz free convection flow over a vertical plate with a power law temperature distribution T_w(x)  =  T_∞ + Ax^λ in a porous medium is revisited. It is shown that in the λ-range  − 1/2  < λ  <  0 , the well known exponentially decaying “first branch” solutions for the velocity and temperature fields are not some isolated solutions as one has believed until now, but limiting cases of families of algebraically decaying multiple solutions. For these multiple solutions well converging analytical series expressions are given. This result yields a bridging to the historical quarreling concerning the feasibility of exponentially and algebraically decaying boundary layers. Owing to a mathematical analogy, our results also hold for the similar boundary layer flows induced by continuous surfaces stretched in viscous fluids with power-law velocities u_w(x)∼ x^λ.     

On the ray method for Biot media

This paper is devoted to an investigation of wave propagation in a Biot porous medium, which consists of elastic and fluid phases. The space-time ray expansion of solutions of dynamic equations for a Biot medium is constructed (in the anisotropic inhomogeneous case). In the inhomogeneous isotropic case, a Rytov law analog is derived similarly to elasticity theory. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 112–131.     

A conforming mixed finite-element method for the coupling of fluid flow with porous media flow

Salim Meddahi ^{We consider a porous medium entirely enclosed within a fluidregion and present a well-posed conforming mixed finite-elementmethod for the corresponding coupled problem. The interfaceconditions refer to mass conservation, balance of normal forcesand the Beavers–Joseph–Saffman law, which yieldsthe introduction of the trace of the porous medium pressureas a suitable Lagrange multiplier. The finite-element subspacesdefining the discrete formulation employ Bernardi–Raugeland Raviart–Thomas elements for the velocities, piecewiseconstants for the pressures and continuous piecewise-linearelements for the Lagrange multiplier. We show stability, convergenceand a priori error estimates for the associated Galerkin scheme.Finally, we provide several numerical results illustrating thegood performance of the method and confirming the theoreticalrates of convergence.}     

Homogenization of the Stokes equations with a random potential

Homogenization of the Stokes equations in a random porous medium is considered. Instead of the homogeneous Dirichlet condition on the boundaries of numerous small pores, used in the existing work on the subject, we insert a term with a positive rapidly oscillating potential into the equations. Physically, this corresponds to porous media whose rigid matrix is slightly permeable to fluid. This relaxation of the boundary value problem permits one to study the asymptotics of the solutions and to justify the Darcy law for the limit functions under much fewer restrictions. Specifically, homogenization becomes possible without any connectedness conditions for the porous domain, whose verification would lead to problems of percolation theory that are insufficiently studied. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 504–520, April, 1996. The work of the first author was supported by the INTAS under grant No. 93-2716.     

Modeling flow in porous media with fractures; Discrete fracture models with matrix-fracture exchange

This article is concerned with a numerical model for flow in a porous medium containing fractures. The fractures are modeled as (d − 1)-dimensional surfaces inside the d-dimensional matrix domain, and a mixed finite element method containing both d and (d − 1) dimensional elements is used. The method allows for fluid exchange between the fractures and the matrix. The method is defined for single-phase Darcy flow throughout the domain and for Forchheimer flow in the fractures. We also consider the case of two-phase flow in a domain in which the fractures and the matrix are of different rock type.     

Chemical waves and localized chaos in a model of heterogeneous catalytic reaction in a porous particles

A mathematical model of an oscillatory chemical reaction in a porous catalyst particle is considered. The model describes an oscillatory medium uniformly distributed throughout the volume of a spherical particle. The dynamical interaction of the reaction with the diffusive flow of the gaseous reagent inside the pores generates nonstationary dissipative structures in the oscillatory medium on the surface of the catalyst. Depending on the pressure in the gaseous phase, the model produces specific chemical waves and localized spatio-temporal chaos. The study was partially supported by the Russian Foundation for Basic Research (grant No. 96-03-34427a). Translated from Chislennye Metody i Vychislitel'nyi Eksperiment, Moscow State University, pp. 31–43, 1998.     

Hölder estimates for solutions of the Cauchy problem for the porous medium equation with external forces

We study the interior Hölder regularity problem for weak solutions of the porous medium equation with external forces. Since the porous medium equation is the typical example of degenerate parabolic equations, Hölder regularity is a delicate matter and does not follow by classical methods. Caffrelli-Friedman, and Caffarelli-Vazquez-Wolansky showed Hölder regularity for the model equation without external forces. DiBenedetto and Friedman showed the Hölder continuity of weak solutions with some integrability conditions of the external forces but they did not obtain the quantitative estimates. The quantitative estimates are important for studying the perturbation problem of the porous medium equation. We obtain the scale invariant Hölder estimates for weak solutions of the porous medium equations with the external forces. As a particular case, we recover the well known Hölder estimates for the linear heat equation.     

Asymptotic behaviour of viscoelastic composites with almost periodic microstructures

In this paper, we study the acoustic properties of porous media saturated by an incompressible viscoelastic fluid. The model considered here consists of a linear deformable porous skeleton having memory that is surrounded by a viscoelastic Oldroyd fluid. Assuming the microstructures to be almost periodically distributed and under the almost periodicity hypothesis on the coefficients of the governing equations, we determine the macroscopic equivalent medium. To achieve our goal, we use some very recent tools about the sigma convergence of convolution sequences. Copyright © 2017 John Wiley & Sons, Ltd.     

Effective models of stratified media containing porous biot layers

Periodic stratified media in which either two porous Biot layer, or an elastic and a porous layers, or a fluid and a porous layer alternate are considered. The effective models of these media are constructed and investigated. In the case of alternating porous layers, the effective model is a generalized transversely isotropic Biot medium. In this medium, the density of the fluid phase and the mean density acquire tensor character. It is shown that the effective model of a porous-fluid medium is, on the one hand, a generalized transversely isotropic Biot medium of special type and, on the other hand, a generalization of the effective model of a stratified elastic-fluid medium.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 140–163.This work was supported by the Russian Foundation for Basic Research under grant Nos. 96-01-00666 and 96-05-66207.     

Two-dimensional filtration problem for solutions in a nonhomogeneous porous medium

We present a mathematical model of underground leaching by solutions filtering through a porous medium. The model describes the motion of solutions from injection to extraction boreholes, as well as dissolution and secondary deposition in reduced-pH regions. A numerical algorithm has been developed for solving the problem on a plane in the general case of a homogeneous medium that contains regions with various nonhomogeneities. The algorithm combines triangulation of the region with the finite element method. The model also allows slow variation over time of some of the process parameters, such as porosity and the filtration coefficient. Numerical results are reported for various cases. The model qualitatively describes the main regularities of underground leaching and can be used to study and understand the detailed dynamics of these processes. The model also fills gaps in geological prospecting data, and extraction curves constructed for different wells can be applied to determine the approximate location of a particular nonhomogeneity. Mathematical modeling can help to optimize mineral extraction by underground leaching. Translated from Prikladnaya Matematika i Informatika, No. 29, pp. 29–55, 2008.