In this paper, we study Brinkman's equations with microscale properties that are highly heterogeneous in space and time. The time variations are controlled by a stochastic particle dynamics described by an SDE. The particle dynamics can be thought as particle deposition that often occurs in filter problems. Our main results include the derivation of macroscale equations and showing that the macroscale equations are deterministic. The latter is important for our (also many other) applications as it greatly simplifies the macroscale equations. We use the asymptotic properties of the SDE and the periodicity of the Brinkman's coefficient in the space variable to prove the convergence result. The SDE has a unique invariant measure that is ergodic and strongly mixing. The macro scale equations are derived through an averaging principle of the slow motion (fluid velocity) with respect to the fast motion (particle dynamics) and also by averaging the Brinkman's coefficient with respect to the space variable. Our results can be extended to more general nonlinear diffusion equations with heterogeneous coefficients.
We consider the homogenization to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. This problem has been studied by the same authors in [A. Giunti and R.M. Höfer, Homogenization for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes] where convergence of the fluid velocity field towards the solution of the Brinkman equations has been established. In the present we consider the pressure associated to the solution of the Stokes equation…
We consider the Stokes equations on a bounded perforated domaincompleted with non-zero constant boundary conditions on the holes. We investigate configurations forwhich the holes are identical spheres and their number N goes to infinity while their radius1/N tends to zero. We prove that, under the assumption that there is no concentrationin the distribution of holes, the solution is well approximated asymptotically by solving aStokes-Brinkman problem.
We prove the homogenization to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. The fluid satisfies a no-slip boundary condition at the holes. The balls generating the holes have centres distributed according to a Poisson point process and i.i.d. unbounded radii satisfying a suitable moment condition. We stress that our assumption on the distribution of the radii does not exclude that, with overwhelming probability, the holes contain clusters made by many overlapping balls. We show that the f…
We study a nonlocal evolution equation that involves a pseudo-parabolic third-order term. The equation models almost uni-directional two-phase flow in Brinkman regimes. We prove the existence of weak solutions for this equation. We also give a series of numerical examples that demonstrate the ability of the equation to support overshooting and explore the behavior of solutions in various limit regimes.
In this paper, we consider the Stokes equations in a perforated domain. When the number of holes increases while their radius tends to 0, it is proven in [L. Desvillettes, F. Golse and V. Ricci. The mean field limit for solid particles in a Navier-Stokes flow. J. Stat. Phys. 131: 941-967, 2008], under suitable dilution assumptions, that the solution is well-approximated asymptotically by solving a Stokes-Brinkman equation. We provide here quantitative estimates in L p-norms of this convergence.
For a fully-coupled Darcy-Stokes system describing the exchange of fluid and stress balance across the interface between a saturated porous medium and an open very narrow channel, the limiting problem is characterized as the width of the channel converges to zero. It is proven that the limit problem is a fully-coupled system of Darcy flow in the porous medium with Brinkman flow in tangential coordinates of the lower dimensional interface.
We consider three-dimensional (3D) Boussinesq convection system of an incompressible fluid in a closed sample of a porous medium. Specifically, we introduce and analyze a 3D Brinkman-Forchheimer-Bénard convection problem describing the behavior of an incompressible fluid in a porous medium between two plates heated from the bottom and cooled from the top. We show the existence and uniqueness of global in-time solutions, and the existence of absorbing balls in $L^2$ and $H^1$. Eventually, we comment on the applicability of a data assimilation algorithm to our system.
In this note, we consider the homogenization of the compressible Navier-Stokes equations in a periodically perforated domain in $\mathbb{R}^3$. Assuming that the particle size scales like $\varepsilon^3$, where $\varepsilon>0$ is their mutual distance, and that the Mach number decreases fast enough, we show that in the limit $\varepsilon\to 0$, the velocity and density converge to a solution of the incompressible Navier-Stokes equations with Brinkman term. We strongly follow the methods of Höfer, Kowalczik and Schwarzacher [arXiv:2007.09031], where they proved convergence to Darcy's law for th…
We study the process of two-phase flow in thin porous media domains of Brinkman-type. This is generally described by a model of coupled, mixed-type differential equations of fluids' saturation and pressure. To reduce the model complexity, different approaches that utilize the thin geometry of the domain have been suggested. We focus on a reduced model that is formulated as a single nonlocal evolution equation of saturation. It is derived by applying standard asymptotic analysis to the dimensionless coupled model, however, a rigid mathematical derivation is still lacking. In this paper, we prov…
The paper deals with homogenization of a model problem describing an immiscible compressible two-phase flow in random statistically homogeneous porous media. We derive the effective (macroscopic) problem and prove the convergence of solutions. Our approach relies on stochastic two-scale convergence techniques, the realization-wise notion of stochastic two-scale convergence being used. Also, we exploit various a priori estimates as well as monotonicity and compactness arguments.
Slightly compressible Brinkman-Forchheimer equations in a bounded 3D domain with Dirichlet boundary conditions are considered. These equations model fluids motion in porous media. The dissipativity of these equations in higher order energy spaces is obtained and regularity and smoothing properties of the solutions are studied. In addition, the existence of a global and an exponential attractors for these equations in a natural phase space is verified.
Brinkman's term for the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in perforated pipes
Fakultät für Mathematik, Universität Regensburg（雷根斯堡大学）；Institut de Mathématiques de Jussieu - Paris Rive Gauche, Université Paris Cité（巴黎西岱大学）；Facultad de Matemáticas, Pontificia Universidad Católica de Chile（智利天主教Pontificia大学）
We prove homogenization for viscous Hamilton-Jacobi equations with a Hamiltonian of the form $G(p)+V(x,ω)$ for a wide class of stationary ergodic random media in one space dimension. The momentum part $G(p)$ of the Hamiltonian is a general (nonconvex) continuous function with superlinear growth at infinity, and the potential $V(x,ω)$ is bounded and Lipschitz continuous. The class of random media we consider is defined by an explicit hill and valley condition on the diffusivity-potential pair which is fulfilled as long as the environment is not ``rigid''.
The paper deals with the homogenization of reaction-diffusion equations with large reaction terms in a multi-scale porous medium. We assume that the fractures and pores are equidistributed and that the coefficients of the equations are periodic. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defined on a fixed domain and is of convection-diffusion-reaction type.
We study in this paper compression effects in heterogeneous media with maximal packing constraint. Starting from compressible Brinkman equations, where maximal packing is encoded in a singular pressure and a singular bulk viscosity, we show that the global weak solutions converge (up to a subsequence) to global weak solutions of the two-phase compressible/incompressible Brinkman equations with respect to a parameter $\varepsilon$ which measures effects close to the maximal packing value. Depending on the importance of the bulk viscosity with respect to the pressure in the dense regimes, memory…
We determine the asymptotic behavior of the solutions to the linear elastodynamic equations in a stratified medium comprising an alternation of possibly very stiff layers with much softer ones, when the thickness of the layers tends to zero. The limit equations may depend on higher order terms, characterizing bending effects. A part of this work is set in the context of non-periodic homogenization and an extension to stochastic homogenization is presented.
In this article we are interested in quantitative homogenization results for linear elliptic equations in the non-stationary situation of a straight interface between two heterogenous media. This extends the previous work [Josien, 2019] to a substantially more general setting, in which the surrounding heterogeneous media may be periodic or random stationary and ergodic. Our main result is a quantification of the sublinearity of a homogenization corrector adapted to the interface, which we construct using an improved version of the method developed in [Fischer and Raithel, 2017]. This quantific…
We consider the homogenization of the Hele-Shaw problem in periodic media that are inhomogeneous both in space and time. After extending the theory of viscosity solutions into this context, we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous Hele-Shaw-type problem with a general, possibly nonlinear dependence of the free boundary velocity on the gradient. Moreover, the free boundaries converge locally uniformly in Hausdorff distance.
We obtain the first quantitative stochastic homogenization result for reaction-diffusion equations, for ignition reactions in dimensions $d\le 3$ that either have finite ranges of dependence or are close enough to such reactions, and for solutions with initial data that approximate characteristic functions of general convex sets. We show algebraic rate of convergence of these solutions to their homogenized limits, which are (discontinuous) viscosity solutions of certain related Hamilton-Jacobi equations.
Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a "cut and project" procedure. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift an almost periodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularized version of the degenerate equation. Homogenized coefficients for quasiperiodic media are determined in terms of the f…
In this paper we establish a homogenization result for a doubly nonlinear parabolic system arising from the hygro-thermo-chemical processes in porous media taking into account memory phenomena. We present a meso-scale model of the composite (heterogeneous) material where each component is considered as a porous system and the voids of the skeleton are partially saturated with liquid water. It is shown that the solution of the meso-scale problem is two-scale convergent to that of the upscaled problem as the spatial parameter goes to zero.
We study a concentration and homogenization problem modelling electrical conduction in a composite material. The novelty of the problem is due to the specific scaling of the physical quantities characterizing the dielectric component of the composite. This leads to the appearance of a peculiar displacement current governed by a Laplace-Beltrami pseudo-parabolic equation. This pseudo-parabolic character is present also in the homogenized equation, which is obtained by the unfolding technique.
In this paper we extend the homogenization results obtained in (G. Allaire, A. Mikelić, A. Piatnitski, J. Math. Phys. 51 (2010), 123103) for a system of partial differential equations describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid periodic porous medium, to the case of random disperse porous media. We present a study of the nonlinear Poisson-Boltzmann equation in a random medium, establish convergence of the stochastic homogenization procedure and prove well-posedness of the two-scale homogenized equations. In addition, after separating scal…
In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties here is the determination of the fixed boundary data corresponding to the limiting (homogenized) problem. This issue has been addressed in recent papers by D. Gérard-Varet and N. Masmoudi, and by C. Prange, however, not much is known about the regularity of this fixed data. The main objective of this note is to initiate a study of this problem, and to prove several regularity results in this connect…
We consider a multicontinuum model in porous media applications, which is described as a system of coupled flow equations. The coupling between different continua depends on many factors and its modeling is important for porous media applications. The coefficients depend on particle deposition that is described in term of a stochastic process solution of an SDE. The stochastic process is considered to be faster than the flow motion and we introduce time-space scales to model the problem. Our goal is to pass to the limit in time and space and to find an associated averaged system. This is an av…
In this paper we study the deterministic homogenization problems for unsteady Navier-Stokes type equations, on one hand in an open set Ω of R^{N}, on the other hand in porous media Ω^{ε}. In the second case, the equations are classical unsteady Navier-Stokes one, and the porous media are periodic
We consider random walks on marked simple point processes with symmetric jump rates and unbounded jump range. We prove homogenization properties of the associated Markov generators. As an application, we derive the hydrodynamic limit of the simple exclusion process given by multiple random walks as above, with hard-core interaction, on a marked Poisson point process. The above results cover Mott variable range hopping, which is a fundamental mechanism of phonon-induced electron conduction in amorphous solids as doped semiconductors. Our techniques, based on an extension of two-scale convergenc…
By using dimension reduction and homogenization techniques, we study the steady flow of an incompresible viscoplastic Bingham fluid in a thin porous medium. A main feature of our study is the dependence of the yield stress of the Bingham fluid on the small parameters describing the geometry of the thin porous medium under consideration. Three different problems are obtained in the limit when the small parameter $\varepsilon$ tends to zero, following the ratio between the height $\varepsilon$ of the porous medium and the relative dimension $a_\varepsilon$ of its periodically distributed pores. …
The paper deals with periodic homogenization problem for a para\-bo\-lic equation whose elliptic part is a convolution type operator with rapidly oscillating coefficients. It is assumed that the coefficients are rapidly oscillating periodic functions both in spatial and temporal variables and that the scaling is diffusive that is the scaling factor of the temporal variable is equal to the square of the scaling factor of the spatial variable. Under the assumption that the convolution kernel has a finite second moment and that the operator is symmetric in spatial variables we show that the studi…
This paper analyzes two classes of second order level set PDE in periodic media in the parabolic scaling. First, we study fully nonlinear geometric operators under general assumptions in dimension $d = 2$ and prove that the associated equations homogenize in this case. Next, we treat a class of quasi-linear geometric operators in arbitrary dimensions $d \geq 2$. In this setting, by adapting arguments form the study of oscillating boundary value problems, we prove that the effective coefficients are generically discontinuous in all dimensions $d \geq 3$. This necessitates a study of level set P…
We prove, under some assumptions, the existence of correctors for the stochastic homoge-nization of of " viscous " possibly degenerate Hamilton-Jacobi equations in stationary ergodic media. The general claim is that, assuming knowledge of homogenization in probability, correctors exist for all extreme points of the convex hull of the sublevel sets of the effective Hamiltonian. Even when homogenization is not a priori known, the arguments imply existence of correctors and, hence, homogenization in some new settings. These include positively homogeneous Hamiltonians and, hence, geometric-type eq…
We investigate a reaction-diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter $ε$. On the interface between the components we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk-domains are given by a reaction-diffusion equation for the traces of the bulk-solutions, including nonlinear reaction-kinetics depending on th…
We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a non-monotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapour state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn-Hilliard equation. Crucial for the analys…
