Asymptotic analysis of a double porosity model with thin fissures

SUMMARYThe aim of the paper is to study the asymptotic behaviour of solutions of second-order elliptic and parabolic equations, arising in modelling of flow in cavernous porous media, in a domain ε weakly connected by a system of traps P ε , where ε is the parameter that characterizes the scale of the microstructure. Namely, we consider a strongly perforated domain2 are non-intersecting subdomains strongly connected with respect to , P ε is a system of traps and meas W ε → 0 as ε → 0. Without any periodicity assumption, for a large range of perforated media and by means of variational homogenization, we find the homogenized models. The effective coefficients are described in terms of local energy characteristics of the domain ε associated with the problem under consideration. The resulting homogenized problem in the parabolic case is a vector model with memory terms. An example is presented to illustrate the methodology.

Section: 22mentioning

confidence: 99%

“…The SC condition was introduced first by Khruslov [12] and was then revisited by many other authors (see, e.g. [14][15][16][17][18][19]). Examples of domains that satisfy this condition are classical periodic perforated domains (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

On the homogenization of some linear problems in domains weakly connected by a system of traps

Amaziane

Pankratov

2007

“…In this section, following Reference [17] (see also References [14,20,21]), we introduce the concept of convergence in domains (") …”

Section: Convergence In Domains Of Degenerating Measurementioning

confidence: 99%

“…Let us mention also that the asymptotic analysis for a non-Newtonian ow in a porous medium with a double porosity has been obtained in Reference [13]. Homogenization results for a single phase ow in a double porosity medium with thin ÿssures were obtained in References [14,15]. Di erent periodic homogenization problems in thin reticulated structures were studied in References [1,16].…”

Section: Introductionmentioning

confidence: 98%

Γ_D‐convergence for a class of quasilinear elliptic equations in thin structures

Amaziane

Гончаренко

Pankratov

2005

SUMMARYWe study the asymptotic behaviour of the solution of a stationary quasilinear elliptic problem posed in a domain (") of asymptotically degenerating measure, i.e. meas (") → 0 as " → 0, where " is the parameter that characterizes the scale of the microstructure. We obtain the convergence of the solution and the homogenized model of the problem is constructed using the notion of convergence in domains of degenerating measure. Proofs are given using the method of local characteristics of the medium (") associated with our problem in a variational form.

“…The effect of the appearance of memory terms in the homogenized equation was also investigated in many articles, see, in particular, [9][10][11][12][13][14]. Homogenization problems on manifolds with complicated microstructure were studied, except [2], in [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Positivity and time behavior of a linear reaction–diffusion system, non‐local in space and time

Khrabustovskyi

Stephan

2008

SUMMARYWe consider a general linear reaction-diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle and positivity of the solution and investigate its asymptotic behavior. Moreover, we give an explicit expression of the limit of the solution for large times. In order to obtain these results, we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution to a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction-diffusion system. Using this and the facts that the diffusion equation on manifolds satisfies the maximum principle and its solution converges to a easily calculated constant, we can obtain analogous properties for the original system.