Abstract
This paper is devoted to the homogenization of weakly coupled cooperativeparabolic systems in strong convection regime with purely periodiccoefficients. Our approach is to factor out oscillations from the solution viaprincipal eigenfunctions of an associated spectral problem and to cancel anyexponential decay in time of the solution using the principal eigenvalue of thesame spectral problem. We employ the notion of two-scale convergence with driftin the asymptotic analysis of the factorized model as the lengthscale of theoscillations tends to zero. This combination of the factorization method andthe method of two-scale convergence is applied to upscale an adsorption modelfor multicomponent flow in an heterogeneous porous medium.
Highlights
Upscaling reactive transport models in porous media is a problem of great practical importance and homogenization theory is a method of choice for achieving this goal
Denoting the unknown concentrations by uεα, for 1 ≤ α ≤ N, we study in the entire space Rd the following weakly coupled system of N parabolic equations with periodic bounded coefficients: (1.1) ρα x ε
The estimates in (3.1) are not uniform in ε. This renders the application of standard compactness theorems from homogenization theory useless for (2.1)
Summary
Upscaling reactive transport models in porous media is a problem of great practical importance and homogenization theory is a method of choice for achieving this goal (see [15] and references therein). The main result of [21] is that the solution to the Cauchy problem for the above system admits the asymptotic representation: uεα(t, x) ≈ φα x ε δ(x − b∗t) where φα is the first eigenfunction and there is no time exponential because λ = 0 happens to be the first eigenvalue for the specific choice of cooperative matrix Παβ made in [21]. This is a frequent case for adsorption or deposition of the chemical on the solid surface (cf. the discussion and references in [6])
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