Two-scale Convergence Method Research Articles

A corrector result for an electrostatic model of interdigitated combs

In this article, we consider two opposite and intergitated 2D-combs, each one with ε-periodically distributed fingers, each finger having cross-section of order ε and fixed height. Via an asymptotic analysis based on the two-scale convergence method, we study, as ε vanishes, the electrical potential in the vacuum between these two combs when one comb is grounded while the voltage in the other one is assumed of order ε, and we prove a corrector result.

An optimal control problem for diffusion–precipitation model

We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method.

Homogenization of ferrofluid flow models in porous media with Langevin magnetization law

The paper is concerned with the homogenization of the equations describing the flow of a ferrofluid through a heterogeneous porous medium Ω in the presence of an applied magnetic field. We discuss two models where the magnetization M is parallel to the magnetic field H. In the first one M and H satisfy the relation M=λ01ΩfH in Ω, where λ0 is a positive constant and 1Ωf is the characteristic function of Ωf (the pore space). In the second model, M and H satisfy the Langevin magnetization law M=MsL(b1|H|)|H|1ΩfH, where L is the Langevin function given by L(x)=1tanh⁡x−1x, Ms is the saturation magnetization and b1 is a positive physical constant. The velocity and the pressure satisfy the Stokes equation with a Kelvin magnetic force. We perform the homogenization of the equations of each of the two models. Using the two-scale convergence method, we rigorously derive the homogenized equation for the magnetic potential and determine the asymptotic limit of the magnetization. Then we rigorously derive a Darcy law.

A homogenization approach to the effect of surfactant concentration and interfacial slip on the flow past viscous drops

In this paper, the collective motion of viscous drops is investigated in terms of a porous medium setting. We assume that the interface of each of the drops is fully covered with a stagnant layer of surfactant in an arbitrary unsteady Stokes flow with low surface Péclet number limit. The effect of the interfacial slip coefficient on the behavior of the flow fields is also considered. The modeling of the physical processes leads to a system of parabolic partial differential equations (PDEs) with interfacial boundary conditions. We have proposed a model in a micro–macro scale setting. We have shown the existence of a unique positive weak solution of such system. We have also obtained a homogenized (upscaled) model of the system via two-scale convergence and boundary unfolding method.

Homogenization of a mineral dissolution and precipitation model involving free boundaries at the micro scale

In this work we present the homogenization of a reaction-diffusion model that includes an evolving microstructure. Such type of problems model, for example, mineral dissolution and precipitation in a porous medium. In the initial state, the microscopic geometry is a periodically perforated domain, each perforation being a spherical solid grains. A small parameter ϵ is characterizing both the distance between two neighboring grains, and the radii of the grains. For each grain, the radius depends on the unknown (the solute concentration) at its surface. Therefore, the radii of the grains change in time and are model unknowns, so the model involves free boundaries at the micro scale. In a first step, we transform the evolving micro domain to a fixed, periodically domain. Using the Rothe-method, we prove the existence of a weak solution and obtain a priori estimates that are uniform with respect to ϵ. Finally, letting ϵ→0, we derive a macroscopic model, the solution of which approximates the micro-scale solution. For this, we use the method of two-scale convergence, and obtain strong compactness results enabling to pass to the limit in the nonlinear terms.

Homogenization of MHD flows in porous media

The paper is concerned with the homogenization of a nonlinear differential system describing the flow of an electrically conducting, incompressible and viscous Newtonian fluid through a periodic porous medium, in the presence of a magnetic field. We introduce a variational formulation of the differential system equipped with boundary conditions. We show the existence of a solution of the variational problem, and derive uniform estimates of the solutions depending on the characteristic parameters of the flow. Using the two-scale convergence method, we rigorously derive a two-scale equation for the two-scale current density, and a two-pressure Stokes system. We derive, in the case of constant magnetic permeability, an explicit relation expressing the macroscopic velocity as a function of the macroscopic Lorentz force, the pressure gradient, the external body force, and the macroscopic current density, via two permeability filtration tensors. When the magnetic field is absent, this relation reduces to the Darcy law.

English

The goal of this paper is to present a different approach to the homogenization of the Dirichlet boundary value problem in porous medium. Unlike the standard energy method or the method of two-scale convergence, this approach is not based on the weak formulation of the problem but on the very weak formulation. To illustrate the method and its advantages we treat the stationary, incompressible Navier-Stokes system with the non-homogeneous Dirichlet boundary condition in periodic porous medium. The nonzero velocity trace on the boundary of a solid inclusion yields a non-standard addition to the source term in the Darcy law. In addition, the homogenized model is not incompressible.

Homogenization of nonisothermal immiscible incompressible two-phase flow in double porosity media

In this paper, we establish a homogenization result for a nonlinear degenerate system arising from two-phase flow through fractured porous media with periodic microstructure taking into account the temperature effects. The mathematical model is given by a coupled system of two-phase flow equations, and an energy balance equation. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy–Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation, i.e. the saturation of one phase, the pressure of the second phase and the temperature are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being rapidly oscillating discontinuous function. Over the matrix domain, the permeability is scaled by ε2, where ε is the size of a typical porous block. Furthermore, we will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The model involves highly oscillatory characteristics and internal nonlinear interface conditions accounting for discontinuous capillary pressures. We then show by a rigorous mathematical argument that the solution of this microscopic problem converges as ε tends to zero to the solution of a double-porosity model of the global macroscopic flow. Our techniques make use of the two-scale convergence method combined to extension and dilation operators in the homogenization context. The memory effects of usual double porosity media are reproduced by this model. We show how the effective coefficients of the porous medium are determined in a precise way by certain physical and geometric features of the microscopic fracture domain, the microscopic matrix blocks, and the interface between them.

Homogenization of Nondilute Suspension of Viscous Fluid with Magnetic Particles

This paper seeks to carry out the rigorous homogenization of a particulate flow consisting of a non-dilute suspension of a viscous Newtonian fluid with magnetizable particles. The fluid is assumed to be described by the Stokes flow, while the particles are either paramagnetic or diamagnetic, for which the magnetization field is a linear function of the magnetic field. The coefficients of the corresponding partial differential equations are locally periodic. A one-way coupling between the fluid domain and the particles is also assumed. The homogenized or effective response of such a suspension is derived, and the mathematical justification of the obtained asymptotics is carried out. The two-scale convergence method is adopted for the latter. As a consequence, the presented result provides a justification for the formal asymptotic analysis of Levy and Sanchez-Palencia for particulate steady-state Stokes flows.

Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands

Multiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems involving convex functionals, converges to the minimizers of a homogenized problem with a suitable convex function.

Existence and two-scale convergence of the generalised Poisson–Nernst–Planck problem with non-linear interface conditions

The paper is devoted to the existence and rigorous homogenisation of the generalised Poisson–Nernst–Planck problem describing the transport of charged species in a two-phase domain. By this, inhomogeneous conditions are supposed at the interface between the pore and solid phases. The solution of the doubly non-linear cross-diffusion model is discontinuous and allows a jump across the phase interface. To prove an averaged problem, the two-scale convergence method over periodic cells is applied and formulated simultaneously in the two phases and at the interface. In the limit, we obtain a non-linear system of equations with averaged matrices of the coefficients, which are based on cell problems due to diffusivity, permittivity and interface electric flux. The first-order corrector due to the inhomogeneous interface condition is derived as the solution to a non-local problem.

A two-scale asymptotic expansion method for periodic composite Euler beams

This paper develops a two-scale asymptotic expansion solution method for periodic composite Euler beams for the first time. In this method, a two-scale solution in the asymptotic expansion form is achieved for the fourth-order uniformly elliptic differential equation with periodic oscillating coefficients, and the first-order perturbed displacement is zero. The analytical solutions of unit cell problems are found with the help of four normalization conditions proposed in this work, and it follows that the homogenized elastic modulus of periodic composite beam is exactly the harmonic mean of material moduli. Another contribution of this work is to mathematically verify the convergence of the two-scale asymptotic expansion solution through the two-scale convergence method. In addition, the paper reveals the effects of higher-order expansion terms on the two-scale displacements and concludes that the second order or even higher-order perturbed displacements are necessary for obtaining accurate curvatures relevant to micro stresses. Finally, numerical experiments validate that the present method is rigorous in mathematics and physically acceptable.

Periodic reiterated homogenization of stationary Navier–Stokes type equations

This paper is devoted to the reiterated homogenization of stationary Navier–Stokes type equations in a periodic structure. The usual Laplace operator in the classical Navier–Stokes equations is here replaced by an elliptic linear differential operator of order two, in divergence form with periodically oscillating coefficients. Our approach is the well known two-scale convergence method. One convergence theorem is proved and we derive de macroscopic homogenized model.

Polarization and relaxation effects induced by two-scale homogenization

Motivated by the two-scale convergence method introduced by G. Nguetseng and G. Allaire, we study the polarization and relaxation effects induced by two-scale homogenization. By virtue of Boltzmann–Poisson system, we characterize the microscopic properties of electrons in dielectric materials. The homogenized equations are obtained which describe the mean behaviors. By the homogenized equation, we get the polarization and relaxation effects which lead to the modified Gauss law; moreover, the dielectric function and the conductivity are also obtained.

Homogenization in the energy variable for a neutron transport model

his article addresses the homogenization of linear Boltzmann equation when the optical parameters are highly heterogeneous in the energy variable. We employ the method of two-scale convergence to arrive at the homogenization result. In doing so, we show the induction of a memory effect in the homogenization limit. We also provide some numerical experiments to illustrate our results. In the Appendix, we treat a related case of the harmonic oscillator.

Homogenization and correctors for linear stochastic equations via the periodic unfolding methods

In this paper, we use the periodic unfolding method and Prokhorov’s and Skorokhod’s probabilistic compactness results to obtain homogenization and corrector results for stochastic partial differential equations (PDEs) with periodically oscillating coefficients. We show the convergence of the solutions of the original problems to the solutions of the homogenized problems. In contrast to the two-scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems

Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

In this paper we deal with the homogenization of stochastic nonlinear hyperbolic equations with periodically oscillating coefficients involving nonlinear damping and forcing driven by a multi-dimensional Wiener process. Using the two-scale convergence method and crucial probabilistic compactness results due to Prokhorov and Skorokhod, we show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a nonlinear damped stochastic hyperbolic partial differential equation. More importantly, we also prove the convergence of the associated energies and establish a crucial corrector result.

Усредненная модель распространения малых возмущений в конфигурации упругое тело – пороупругая среда для двухскоростного континуума

The article continues the series of the authors’ papers devoted to averaging of mathematical models describing the isothermal acoustic processes in a heterogeneous medium with two components separated by a common boundary. One of these components is an elastic body, and the other is a poroelastic continuum. Poroelastic medium is understood to mean an elastic body permeated with a system of pores filled with liquid. The exact mathematical model constructed proceeding from the classical laws of continuum mechanics is used as the initial model. The model’s differential equations contain rapidly oscillating coefficients that appear in making transition to dimensionless variables. It is assumed that there are finite or infinite limits of these coefficients as the small parameter “epsilon” tends to zero. The small parameter “epsilon” is taken equal to the ratio of the average pore size to the characteristic size of the region under consideration. It should be noted that the coefficients of differential equations and the geometry of the considered region both depend on these parameters. Various averaged (limit) models that do not contain rapidly oscillating coefficients have been derived. For the possibility of using the averaging theory and the known averaging results, simplifying geometric assumptions about the periodicity and connectivity of the pore space and elastic skeleton are added. Averaged models are understood to mean such boundary value problems for equations or systems with relatively slowly changing characteristics that the solutions of boundary value problems for the initial models converge (in a sense) to the solution of the corresponding equations for the averaged model when the period e of the considered periodic structure tends to zero. Depending on the characteristics of the continuum (whether the fluid is viscous, low-viscous, compressible, or incompressible; whether the skeleton is highly deformable, elastic, perfectly rigid, etc.), different limit modes are obtained. One of cases involving weakly compressible and low viscous liquid and a weakly deformable elastic skeleton in one region and an elastic body in the other region is investigated. The original mathematical model reflects the real physical process in a fairly accurate manner, but is so complex that the standard averaging scheme does not work for it. Therefore, the two-scale convergence method is used as the main tool. On the one hand, it is often impossible to calculate the model’s limit modes even in terms of weak convergence, but it is possible to do so in terms of two-scale convergence. On the other hand, the sequence of solutions is usually bounded but not compact, and in this case the sequence weak limit is not a satisfactory approximation to the solution of the initial mathematical model, and it is more preferable to use a two-scale limit. The results for an individually taken poroelastic region or for a region occupied by an elastic body were presented in the authors’ previous papers. In the case considered, the joint motion of an elastic body and porous elastic medium is studied, and the main problem lies in deriving the conditions at the common boundary of the elastic and poroelastic regions.

A Brinkman law in the homogenization of the stationary Navier–Stokes system in a non-periodic porous medium

We study the asymptotic behavior of the Navier–Stokes system with Dirichlet boundary conditions posed in a domain Ω ε = Ω ∖ T ε . Here Ω ⊂ R 3 is a bounded open set and T ε is the union of many disjoint closed sets of size ε 3 and density of order 1 ∕ ε 3 , with ε a small positive parameter. Similarly to the periodic case we get a limit system corresponding to a Brinkman flow. The difference is that now the Brinkman matrix is not homogeneous, it depends on the density of the closed sets composing T ε . The result is obtained through an adaptation of the two-scale convergence method.

Homogenization of the Equations of Filtration of a Viscous Fluid in Two Porous Media

A homogenized model of filtration of a viscous fluid in two domains with common boundary is deduced on the basis of the method of two-scale convergence. The domains represent an elastic medium with perforated pores. The fluid, filling the pores, is the same in both domains, and the properties of the solid skeleton are distinct.