[Homogénéisation d'un modèle de convection–diffusion avec réaction en milieu poreux]
On étudie l'homogénéisation d'un problème de convection–diffusion avec réaction en milieu poreux lorsque les nombres de Péclet et de Damkohler sont grands. Nous démontrons que, dans un repère dérivant à grande vitesse, l'équation homogénéisée est une équation de diffusion. Notre méthode est basée sur un principe de factorisation et sur la convergence à deux échelles. La conséquence pratique la plus importante est que nous obtenons ainsi une définition rigoureuse des coefficients homogénéisés qui justife des arguments heuristiques utilisés dans la méthode de la prise de moyenne. Nous avons effectué des calculs numériques en 2-d du coefficient homogénéisé de diffusion–dispersion qui donnent des valeurs très semblables à celles obtenues par prise de moyenne.
We study the homogenization of a convection–diffusion equation with reaction in a porous medium when both the Péclet and Damkohler numbers are large. We prove that, up to a large drift, the homogenized equation is a diffusion equation. Our method is based on a factorization principle and two-scale convergence. The main consequence is that we obtain rigorous definitions of homogenized coefficients which justify heuristic arguments in the method of volume averaging. We perform 2-d numerical computations of the diffusion–dispersion homogenized coefficient which are in very good agreement with previous results obtained by the method of volume averaging.
Accepté le :
Publié le :
Grégoire Allaire 1 ; Anne-Lise Raphael 1
@article{CRMATH_2007__344_8_523_0,
author = {Gr\'egoire Allaire and Anne-Lise Raphael},
title = {Homogenization of a convection{\textendash}diffusion model with reaction in a porous medium},
journal = {Comptes Rendus. Math\'ematique},
pages = {523--528},
publisher = {Elsevier},
volume = {344},
number = {8},
year = {2007},
doi = {10.1016/j.crma.2007.03.008},
language = {en},
}
TY - JOUR AU - Grégoire Allaire AU - Anne-Lise Raphael TI - Homogenization of a convection–diffusion model with reaction in a porous medium JO - Comptes Rendus. Mathématique PY - 2007 SP - 523 EP - 528 VL - 344 IS - 8 PB - Elsevier DO - 10.1016/j.crma.2007.03.008 LA - en ID - CRMATH_2007__344_8_523_0 ER -
Grégoire Allaire; Anne-Lise Raphael. Homogenization of a convection–diffusion model with reaction in a porous medium. Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 523-528. doi : 10.1016/j.crma.2007.03.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.008/
[1] Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23 (1992) no. 6, pp. 1482-1518
[2] Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg., Volume 187 (2000), pp. 91-117
[3] G. Allaire, A. Raphael, Homogénéisation d'un modèle de convection–diffusion avec chimie/absorption en milieu poreux, rapport interne n. 604, CMAP, Ecole Polytechnique, 2006
[4] Éléments de comparaison entre la méthode d'homogénéisation et la méthode de prise de moyenne avec fermeture, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, Volume 306 (1988) no. 7, pp. 463-466
[5] Dispersion resulting from flow through spatially periodic porous media. II. Surface and intraparticle transport, Philos. Trans. Roy. Soc. London Ser. A, Volume 307 (1982) no. 1498, pp. 149-200
[6] Homogenization of a neutronic multigroup evolution model, Asymptotic Anal., Volume 24 (2000) no. 2, pp. 143-165
[7] Homogenization of a diffusion with drift, C. R. Acad. Sci. Paris, Sér. I, Volume 327 (1998), pp. 807-812
[8] Homogenization of a neutronic critical diffusion problem with drift, Proc. Roy. Soc. Edinburgh Sect. A, Volume 132 (2002) no. 3, pp. 567-594
[9] F. Hecht, O. Pironneau, A. Le Hyaric, K. Ohtsuka, Freefem++, Version 2.0–0, http://www.freefem.org//ff++, Laboratoire Jacques Louis Lions, Université Pierre et Marie Curie, Paris
[10] Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, 1994
[11] Convection of microstructure and related problems, SIAM J. Appl. Math., Volume 45 (1985) no. 5, pp. 780-797
[12] Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena, Phys. Rep., Volume 314 (1999) no. 4–5, pp. 237-574
[13] Homogenization of a nonlinear convection–diffusion equation with rapidly oscillating coefficients and strong convection, J. London Math. Soc. (2), Volume 72 (2005) no. 2, pp. 391-409
[14] Dispersion, convection, and reaction in porous media, Phys. Fluids A, Volume 3 (May 1991) no. 5
[15] A. Mikelic, V. Devigne, C.J. van Duijn, Rigorous upscaling of the reactive flow through a pore, under dominant Peclet and Damkohler numbers, preprint
[16] A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., Volume 20 (1989) no. 3, pp. 608-623
[17] The Method of Volume Averaging, Theory Applications of Transport in Porous Media, vol. 13, Kluwer Academic Publishers, 1999
- Clarifications about upscaling diffusion with heterogeneous reaction in porous media, Acta Mechanica, Volume 236 (2025) no. 3, p. 1697 | DOI:10.1007/s00707-024-04214-4
- The Method of Finite Averages: A rigorous upscaling methodology for heterogeneous porous media, Advances in Water Resources, Volume 188 (2024), p. 104689 | DOI:10.1016/j.advwatres.2024.104689
- Accelerated computational micromechanics for solute transport in porous media, Computer Methods in Applied Mechanics and Engineering, Volume 426 (2024), p. 116976 | DOI:10.1016/j.cma.2024.116976
- A Learning‐Based Multiscale Model for Reactive Flow in Porous Media, Water Resources Research, Volume 60 (2024) no. 9 | DOI:10.1029/2023wr036303
- Homogenization based heating control for moist paperboard with evaporation on the pore surface, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 104 (2024) no. 3 | DOI:10.1002/zamm.202300673
- Upscaling coupled heterogeneous diffusion reaction equations in porous media, Acta Mechanica, Volume 234 (2023) no. 6, p. 2293 | DOI:10.1007/s00707-023-03501-w
- Upscaled model for the diffusion/heterogeneous reaction in porous media: Boundary layer problem, Advances in Water Resources, Volume 179 (2023), p. 104500 | DOI:10.1016/j.advwatres.2023.104500
- Prediction of local concentration fields in porous media with chemical reaction using a multi scale convolutional neural network, Chemical Engineering Journal, Volume 455 (2023), p. 140367 | DOI:10.1016/j.cej.2022.140367
- On the Periodic Homogenization of Elliptic Equations in Nondivergence Form with Large Drifts, Multiscale Modeling Simulation, Volume 21 (2023) no. 4, p. 1486 | DOI:10.1137/23m1550906
- Homogenization modelling of antibiotic diffusion and adsorption in viral liquid crystals, Royal Society Open Science, Volume 10 (2023) no. 1 | DOI:10.1098/rsos.221120
- Population Balance Models for Particulate Flows in Porous Media: Breakage and Shear-Induced Events, Transport in Porous Media, Volume 146 (2023) no. 1-2, p. 197 | DOI:10.1007/s11242-022-01793-5
- Automated Symbolic Upscaling: 1. Model Generation for Extended Applicability Regimes, Water Resources Research, Volume 59 (2023) no. 7 | DOI:10.1029/2022wr033600
- A spectral approach for homogenization of diffusion and heterogeneous reaction in porous media, Applied Mathematical Modelling, Volume 104 (2022), p. 666 | DOI:10.1016/j.apm.2021.12.017
- Homogenized model for diffusion and heterogeneous reaction in porous media: Numerical study and validation., Applied Mathematical Modelling, Volume 111 (2022), p. 486 | DOI:10.1016/j.apm.2022.07.001
- The Role of the Relative Fluid Velocity in an Objective Continuum Theory of Finite Strain Poroelasticity, Journal of Elasticity, Volume 150 (2022) no. 1, p. 151 | DOI:10.1007/s10659-022-09903-6
- Modelling of filamentous phage-induced antibiotic tolerance of P. aeruginosa, PLOS ONE, Volume 17 (2022) no. 4, p. e0261482 | DOI:10.1371/journal.pone.0261482
- Advection-dominated transport past isolated disordered sinks: stepping beyond homogenization, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 478 (2022) no. 2262 | DOI:10.1098/rspa.2022.0032
- Upscaling and Automation: Pushing the Boundaries of Multiscale Modeling through Symbolic Computing, Transport in Porous Media, Volume 140 (2021) no. 1, p. 313 | DOI:10.1007/s11242-021-01628-9
- Limits of Classical Homogenization Procedure for Coupled Diffusion-Heterogeneous Reaction Processes in Porous Media, Transport in Porous Media, Volume 140 (2021) no. 2, p. 437 | DOI:10.1007/s11242-021-01683-2
- Upscaling diffusion–reaction in porous media, Acta Mechanica, Volume 231 (2020) no. 5, p. 2011 | DOI:10.1007/s00707-020-02631-9
- Macroscopic models for filtration and heterogeneous reactions in porous media, Advances in Water Resources, Volume 141 (2020), p. 103605 | DOI:10.1016/j.advwatres.2020.103605
- Homogenization approach to the upscaling of a reactive flow through particulate filters with wall integrated catalyst, Advances in Water Resources, Volume 146 (2020), p. 103779 | DOI:10.1016/j.advwatres.2020.103779
- Finite element approximation of elliptic homogenization problems in nondivergence-form, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 54 (2020) no. 4, p. 1221 | DOI:10.1051/m2an/2019093
- Homogenization of biased convolution type operators, Asymptotic Analysis, Volume 115 (2019) no. 3-4, p. 241 | DOI:10.3233/asy-191533
- Upscaling Flow and Transport Processes, Flowing Matter (2019), p. 137 | DOI:10.1007/978-3-030-23370-9_5
- Multiscale Finite Element Methods for Advection-Dominated Problems in Perforated Domains, Multiscale Modeling Simulation, Volume 17 (2019) no. 2, p. 773 | DOI:10.1137/17m1152048
- Upscaling of Diffusion–Reaction Phenomena by Homogenisation Technique: Possible Appearance of Morphogenesis, Transport in Porous Media, Volume 127 (2019) no. 1, p. 191 | DOI:10.1007/s11242-018-1187-y
- On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption, Communications on Pure Applied Analysis, Volume 17 (2018) no. 2, p. 579 | DOI:10.3934/cpaa.2018031
- A robust upscaling of the effective particle deposition rate in porous media, Journal of Contaminant Hydrology, Volume 212 (2018), p. 3 | DOI:10.1016/j.jconhyd.2017.09.002
- An FFT method for the computation of thermal diffusivity of porous periodic media, Acta Mechanica, Volume 228 (2017) no. 9, p. 3019 | DOI:10.1007/s00707-017-1885-5
- Analytical and variational numerical methods for unstable miscible displacement flows in porous media, Journal of Computational Physics, Volume 335 (2017), p. 444 | DOI:10.1016/j.jcp.2017.01.021
- Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows, Numerische Mathematik, Volume 136 (2017) no. 3, p. 603 | DOI:10.1007/s00211-016-0854-6
- Convergence Along Mean Flows, SIAM Journal on Mathematical Analysis, Volume 49 (2017) no. 1, p. 222 | DOI:10.1137/16m1068657
- A survey on properties of Nernst–Planck–Poisson system. Application to ionic transport in porous media, Applied Mathematical Modelling, Volume 40 (2016) no. 2, p. 846 | DOI:10.1016/j.apm.2015.06.013
- A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift, Discrete and Continuous Dynamical Systems - Series S, Volume 9 (2016) no. 5, p. 1393 | DOI:10.3934/dcdss.2016056
- On the homogenization of multicomponent transport, Discrete and Continuous Dynamical Systems - Series B, Volume 20 (2015) no. 8, p. 2527 | DOI:10.3934/dcdsb.2015.20.2527
- A priori error estimate of a multiscale finite element method for transport modeling, SeMA Journal, Volume 67 (2015) no. 1, p. 1 | DOI:10.1007/s40324-014-0023-8
- Ground states of singularly perturbed convection-diffusion equation with oscillating coefficients, ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 4, p. 1059 | DOI:10.1051/cocv/2014007
- Discontinuous Galerkin finite element heterogeneous multiscale method for advection–diffusion problems with multiple scales, Numerische Mathematik, Volume 126 (2014) no. 4, p. 589 | DOI:10.1007/s00211-013-0578-9
- Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare?, Advances in Water Resources, Volume 62 (2013), p. 178 | DOI:10.1016/j.advwatres.2013.09.006
- Homogenization and concentration for a diffusion equation with large convection in a bounded domain, Journal of Functional Analysis, Volume 262 (2012) no. 1, p. 300 | DOI:10.1016/j.jfa.2011.09.014
- Homogenization of a nonstationary convection-diffusion equation in a thin rod and in a layer, SeMA Journal, Volume 58 (2012) no. 1, p. 53 | DOI:10.1007/bf03322605
- RIGOROUS DERIVATION OF A HYPERBOLIC MODEL FOR TAYLOR DISPERSION, Mathematical Models and Methods in Applied Sciences, Volume 21 (2011) no. 05, p. 1095 | DOI:10.1142/s0218202510005264
- Homogenization of convection-diffusion equation in infinite cylinder, Networks Heterogeneous Media, Volume 6 (2011) no. 1, p. 111 | DOI:10.3934/nhm.2011.6.111
- A homogenization approach to flashing ratchets, Nonlinear Differential Equations and Applications NoDEA, Volume 18 (2011) no. 1, p. 45 | DOI:10.1007/s00030-010-0083-0
- Transport in the placenta: homogenizing haemodynamics in a disordered medium, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 369 (2011) no. 1954, p. 4162 | DOI:10.1098/rsta.2011.0170
- Homogenization of nonlinear reaction-diffusion equation with a large reaction term, ANNALI DELL'UNIVERSITA' DI FERRARA, Volume 56 (2010) no. 1, p. 141 | DOI:10.1007/s11565-010-0095-z
- Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media, Chemical Engineering Science, Volume 65 (2010) no. 7, p. 2292 | DOI:10.1016/j.ces.2009.09.010
- The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift, Networks Heterogeneous Media, Volume 5 (2010) no. 4, p. 711 | DOI:10.3934/nhm.2010.5.711
- Homogenization Approach to the Dispersion Theory for Reactive Transport through Porous Media, SIAM Journal on Mathematical Analysis, Volume 42 (2010) no. 1, p. 125 | DOI:10.1137/090754935
- Asymmetric potentials and motor effect: a homogenization approach, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 26 (2009) no. 6, p. 2055 | DOI:10.1016/j.anihpc.2008.10.003
- Chapter 1 Effective Dispersion Equations for Reactive Flows with Dominant Péclet and Damkohler Numbers, Advances in Chemical Engineering - Mathematics in Chemical Kinetics and Engineering, Volume 34 (2008), p. 1 | DOI:10.1016/s0065-2377(08)00001-x
- Laplace transform approach to the rigorous upscaling of the infinite adsorption rate reactive flow under dominant Peclet number through a pore, Applicable Analysis, Volume 87 (2008) no. 12, p. 1373 | DOI:10.1080/00036810802140699
Cité par 53 documents. Sources : Crossref
Commentaires - Politique