Bingham flow in porous media with obstacles of different size

By using the unfolding operators for periodic homogenization, we give a general compactness result for a class of functions defined on bounded domains presenting perforations of two different size. Then we apply this result to the homogenization of the flow of a Bingham fluid in a porous medium with solid obstacles of different size. Next, we give the interpretation of the limit problem in terms of a nonlinear Darcy law. Copyright © 2017 John Wiley & Sons, Ltd.     

Homogenized elliptic equations and variational inequalities with oscillating parameters. Application to the study of thin flow behavior with rough surfaces

Double scale homogenization is used to average stationary equation or inequalities in which both highly oscillating variables and parameters appear. We demonstrate how the limit is obtained using a two-stage procedure, firstly by carrying out a classical homogenization process by freezing the oscillating parameter, then by averaging the result with respect to this parameter. These results allow us to average the pressure for a newtonian fluid in a narrow gap between two rough unstationary surfaces. Numerical results are given.     

Stochastic homogenization of Hamilton–Jacobi and degenerate Bellman equations in unbounded environments

We consider the homogenization of Hamilton–Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton–Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a program proposed by Lions and Souganidis that previously yielded homogenization in probability. The class of problems we study is strongly motivated by Sznitman?s study of the quenched large deviations of Brownian motion interacting with a Poissonian potential, but applies to a general class of problems which are not amenable to probabilistic tools.     

Homogenization of a stationary navier-stokes flow in porous medium with thin film

The article studies the homogenization of a stationary Navier-Stokes fluid in porous medium with thin film under Dirichlet boundary condition.At the end of the article,＂Darcy＇s law＂is rigorously derived from this model as the parameter ε tends to zero,which is independent of the coordinates towards the thickness.     

Asymptotic study of the macroscopic behaviour of a solid-fluid mixture

In the framework of homogenization theory we study a mixture of an elastic solid and a viscous compressible fluid with periodic structure and its limit behaviour as the period tends to zero Existence, uniqueness and convergence theorems are given. The limit behaviour is viscoelastic.     

Polynomial approximations of a class of stochastic multiscale elasticity problems

We consider a class of elasticity equations in \({\mathbb{R}^d}\) whose elastic moduli depend on n separated microscopic scales. The moduli are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger–Reissner mixed problem that allows for computing the stress directly and the multiscale mixed problem with a penalty term for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real parameters which represent the probabilistic law of the stochastic equations. We study the multiscale homogenized problems that contain all the macroscopic and microscopic information. The solutions of these multiscale homogenized problems are written as generalized polynomial chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin approximating problems that project into the spaces of functions with only a finite number of N gpc modes. Assuming summability properties for the coefficients of the elastic moduli’s expansion, we deduce bounds and summability properties for the solutions’ gpc expansion coefficients. These bounds imply explicit rates of convergence in terms of N when the gpc modes used for the Galerkin approximation are chosen to correspond to the best N terms in the gpc expansion. For the mixed problem with a penalty term for nearly incompressible materials, we show that the rate of convergence for the best N term approximation is independent of the Lamé constants’ ratio when it goes to \({\infty}\). Correctors for the homogenization problem are deduced. From these we establish correctors for the solutions of the parametric multiscale problems in terms of the semidiscrete Galerkin approximations. For two-scale problems, an explicit homogenization error which is uniform with respect to the parameters is deduced. Together with the best N term approximation error, it provides an explicit convergence rate for the correctors of the parametric multiscale problems. For nearly incompressible materials, we obtain a homogenization error that is independent of the ratio of the Lamé constants, so that the error for the corrector is also independent of this ratio.     

Homogenization of the oscillating Dirichlet boundary condition in general domains

We prove the homogenization of the Dirichlet problem for fully nonlinear uniformly elliptic operators with periodic oscillation in the operator and in the boundary condition for a general class of smooth bounded domains. This extends the previous results of Barles and Mironescu (2012) [4] in half spaces. We show that homogenization holds despite a possible lack of continuity in the homogenized boundary data. The proof is based on a comparison principle with partial Dirichlet boundary data which is of independent interest.     

Nguetseng’s two-scale convergence method for filtration and seismic acoustic problems in elastic porous media

A linear system is considered of the differential equations describing a joint motion of an elastic porous body and a fluid occupying a porous space. The problem is linear but very hard to tackle since its main differential equations involve some (big and small) nonsmooth oscillatory coefficients. Rigorous justification under various conditions on the physical parameters is fulfilled for the homogenization procedures as the dimensionless size of pores vanishes, while the porous body is geometrically periodic. In result, we derive Biot’s equations of poroelasticity, the system consisting of the anisotropic Lamé equations for the solid component and the acoustic equations for the fluid component, the equations of viscoelasticity, or the decoupled system consisting of Darcy’s system of filtration or the acoustic equations for the fluid component (first approximation) and the anisotropic Lamé equations for the solid component (second approximation) depending on the ratios between the physical parameters. The proofs are based on Nguetseng’s two-scale convergence method of homogenization in periodic structures.     

Individual homogenization of nonlinear parabolic operators

In this article, we prove an individual homogenization result for a class of almost periodic nonlinear parabolic operators. The spatial and temporal heterogeneities are almost periodic functions in the sense of Besicovitch. The latter allows discontinuities and is suitable for many applications. First, we derive stability and comparison estimates for sequences of G-convergent nonlinear parabolic operators. Furthermore, using these estimates, the individual homogenization result is shown.     

Basic Homogenization Results for a Biconnected ε-Periodic Structure

We present, in a bounded domain, a model of an l -periodic structure composed of two phases, both being connected but only one reaching the boundary of the domain, avoiding in this way the local type convergences of the homogenization process. In this framework we revise some basic tools of the homogenization theory in porous media: the extension and the restriction operators, the Ne ) as inequality. Moreover, we obtain some compacity properties which reduce the proof of the pressure type convergences from the homogenization of fluid flows through porous media to the expected procedure of a priori estimations and two-scale convergences. As all the properties can be proved without much technical difficulties, avoiding annoying hypotheses and the use of Kolmogorov's criterion of compacity, the present structure seems one of the most convenient realistic models of porous media that can be studied with the methods of homogenization.     

一类非线性偏微分方程组的解析解

首先,利用共轭算子的性质,将张鸿庆等提出的求伴随算子对的方法推广到了求一类非线性(即部分非线性的)算子矩阵的伴随算子向量．其次,利用机械化的构造方法给出了求解一类非线性(即,部分非线性的,且以所有线性的为其特例)非齐次微分方程组的统一理论,即通过齐次化和三角化求得恰当的变换,从而将原方程组化为较简单的形式,一般为对角化的．最后利用该方法求得了一些弹性力学方程组的解析解．     

Numerical investigation of the effective Skempton coefficient in fluid-saturated fractured rock

Characterization of hydro-mechanical processes in reservoir rocks is an essential issue for many geo investigations such as characterization of subsurface fluid flow or geothermal exploitation. For geothermal applications, the role of fractures as storage and transport components of a hydraulic system are highly important. In the present contribution we focus on investigating the effective Skempton coefficient of a damaged porous rock analyzing a modified Cryer problem, which provides a simple model of a porous rock containing a storage and transport pat. The effective Skempton coefficient is defined as the ratio of the increase in mean pore pressure induced by change in confining pressure for undrained boundary conditions. Using approaches from computational homogenization, we evaluate the confining pressure as the negative volume average of the total mean stress. Similarly, we compute the effective fluid pressure in terms of the volume-averaged fluid pressure in the rocks. We compare the numerical results to those from typical experiments and highlight the problems with the latter. The proposed concept for determination of an effective Skempton coefficient based on numerically evaluated volume averages helps to generate a better understanding of the process-inherent constituents. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the convergence of bloch eigenfunctions in homogenization problems

We study the convergence of continuous spectrum eigenfunctions for differential operators of divergence type with ε-periodic coefficients, where ε is a small parameter. Two cases are considered, the case of classical homogenization, where the coefficient matrix satisfies the ellipticity condition uniformly with respect to ε, and the case of two-scale homogenization, where the coefficient matrix has two phases and is highly contrast with hard-to-soft-phase contrast ratio 1: ε².     

A link between sets of tensors stable under lamination and quasiconvexity

A link is found between quasiconvexity and the conditions for a set L of conductivity or elasticity tensors to be stable under lamination. These conditions, derived in the companion paper, are shown here to be equivalent to the condition that for every point B on the boundary of the set L an operator T_B dependent on the tangent plane and curvature of the set at B is a quasiconvex translation operator. A separate class of quasiconvex translation operators is obtained which are candidates for proving that L is stable under homogenization. The region stable under homogenization associated with any one of these operators shares a common boundary point and tangent plane with the set L and has curvature at that point not greater than the curvature of L. The conditions under which there exists a representative subclass of these operators such that the associated regions stable under homogenization wrap around L remains unresolved. It is proved that L can be characterized by minimizations of sums and dual energies in much the same way that convex sets can be characterized by their Legendre transforms. © 1994 John Wiley & Sons, Inc.     

穿孔区域中双曲-抛物方程均匀化问题的一个注记

In this paper, we study a class of hyperbolic-parabolic problems in periodically perforated domains with a homogeneous Neumann condition on the boundary of holes. We focus on the homogenization of these equations, which generalizes those achieved by BensoussanLions-Papanicolau and Migorski. The proof is based on the periodic unfolding method in perforated domains.     

The role of microvascular tortuosity in tumor transport phenomena and future perspective for drug delivery

We tackle the fluid transport problem in vascularized tumors by solving a double Darcy model obtained via multiscale homogenization. The hydraulic conductivities of the capillary and interstitial compartments are computed solving classical problems on the representative periodic cell, which encodes details of the microvasculature. Microvascular tortuosity leads to a dramatic decrease of the capillary hydraulic conductivity, and the corresponding lowering in the pressure drop impairs tumor blood flow and consequently advection of molecules. Further perspectives for anti-cancer agents delivery are illustrated. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

带不完美界面的双曲问题均匀化的一个注记

本文研究了一类二分区域上的具有非周期系数的双曲问题.利用周期Unfolding方法,得到了均匀化及其矫正结果,推广了Donato,Faella和Monsurrò的工作.     

Proof of Monotone Loss Rate of Fluid Priority-Queue with Finite Buffer

This paper studies a fluid queueing system that has a single server, a single finite buffer, and which applies a strict priority discipline to multiple arriving streams of different classes. The arriving streams are modeled by statistically independent, identically distributed random processes. A proof is presented for the highly intuitive result that, in such a queueing system, a higher priority class stream has a lower average fluid loss rate than a lower priority class stream. The proof exploits the fact that for a work-conserving queue, the fluid loss rate for a given class is invariant of what queueing discipline is applied to all arriving fluid of this particular class. AMS subject classification: 60K25, 68M20