Upscaling the interplay between diffusion and polynomial drifts through a composite thin strip with periodic microstruct
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ORIGINAL PAPERS
Upscaling the interplay between diffusion and polynomial drifts through a composite thin strip with periodic microstructure Emilio N. M. Cirillo . Ida de Bonis . Adrian Muntean
. Omar Richardson
Received: 3 December 2019 / Accepted: 26 September 2020 The Author(s) 2020
Abstract We study the upscaling of a system of many interacting particles through a heterogenous thin elongated obstacle as modeled via a two-dimensional diffusion problem with a one-directional nonlinear convective drift. Assuming that the obstacle can be described well by a thin composite strip with periodically placed microstructures, we aim at deriving the upscaled model equations as well as the effective transport coefficients for suitable scalings in terms of both the inherent thickness at the strip and the typical length scales of the microscopic heterogeneities. Aiming at computable scenarios, we consider that the heterogeneity of the strip is made of an array of periodically arranged impenetrable solid rectangles and identify two scaling regimes what concerns the small asymptotics parameter for the upscaling E. N. M. Cirillo Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Universita` di Roma, Roma, Italy e-mail: [email protected] I. de Bonis Universita` degli Studi ‘‘Giustino Fortunato’’, Benevento, Italy e-mail: [email protected] A. Muntean (&) O. Richardson Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden e-mail: [email protected]
procedure: the characteristic size of the microstructure is either significantly smaller than the thickness of the thin obstacle or it is of the same order of magnitude. We scale up the diffusion–polynomial drift model and list computable formulas for the effective diffusion and drift tensorial coefficients for both scaling regimes. Our upscaling procedure combines ideas of two-scale asymptotics homogenization with dimension reduction arguments. Consequences of these results for the construction of more general transmission boundary conditions are discussed. We illustrate numerically the concentration profile of the chemical species passing through the upscaled strip in the finite thickness regime and point out that trapping of concentration inside the strip is likely to occur in at least two conceptually different transport situations: (i) full diffusion/dispersion matrix and nonlinear horizontal drift, and (ii) diagonal diffusion matrix and oblique nonlinear drift. Keywords Diffusion Polynomial drifts Upscaling Dimension reduction Derivation of nonlinear transmission boundary conditions Concentration localization Mathematics Subject Classification 76M50 76M45
35B27
O. Richardson e-mail: [email protected]
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Meccanica
1 Introduction 1.1 Background. Motivation The study of the physics of interfaces has known a great impulse in the last decades; different point of views have been adopted and several related problem have been investigated,
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