Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift
Authors:
Vishnu Raveendran, Emilio N. M. Cirillo and Adrian Muntean
Journal:
Quart. Appl. Math. 80 (2022), 641-667
MSC (2020):
Primary 35B27, 35Q92, 35A01
DOI:
https://doi.org/10.1090/qam/1622
Published electronically:
May 18, 2022
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Abstract: We study a reaction-diffusion-convection problem with non-linear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this non-linearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model equations as well as the structure of the effective transport tensors. Finally, we use Schauder’s fixed point theorem as well as monotonicity arguments to study the weak solvability of the upscaled model posed in an unbounded domain. This study wants to contribute with theoretical understanding needed when designing thin composite materials that are resistant to high velocity impacts.
References
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References
- Grégoire Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences, vol. 146, Springer-Verlag, New York, 2002. MR 1859696, DOI 10.1007/978-1-4684-9286-6
- G. Allaire, R. Brizzi, A. Mikelić, and A. Piatnitski, Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media, Chemical Engineering Science 65 (2010), no. 7, 2292–2300.
- G. Allaire, S Desroziers, G. Enchéry, and F. Ouaki, A multiscale finite element method for transport modelling, CD-ROM Proceedings of the 6th European Congress on Computational Methods in Applied Sciences and Engineering, Vienna University of Technology, Austria, 2012.
- G. Allaire and H. Hutridurga, Homogenization of reactive flows in porous media and competition between bulk and surface diffusion, IMA J. Appl. Math. 77 (2012), no. 6, 788–815. MR 2999138, DOI 10.1093/imamat/hxs049
- Grégoire Allaire and Harsha Hutridurga, Upscaling nonlinear adsorption in periodic porous media—homogenization approach, Appl. Anal. 95 (2016), no. 10, 2126–2161. MR 3523144, DOI 10.1080/00036811.2015.1038254
- Grégoire Allaire, Andro Mikelić, and Andrey Piatnitski, Homogenization approach to the dispersion theory for reactive transport through porous media, SIAM J. Math. Anal. 42 (2010), no. 1, 125–144. MR 2596548, DOI 10.1137/090754935
- Grégoire Allaire and Rafael Orive, Homogenization of periodic non self-adjoint problems with large drift and potential, ESAIM Control Optim. Calc. Var. 13 (2007), no. 4, 735–749. MR 2351401, DOI 10.1051/cocv:2007030
- G. Allaire, I. Pankratova, and A. Piatnitski, Homogenization and concentration for a diffusion equation with large convection in a bounded domain, J. Funct. Anal. 262 (2012), no. 1, 300–330. MR 2852263, DOI 10.1016/j.jfa.2011.09.014
- Jean-Pierre Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris 256 (1963), 5042–5044 (French). MR 152860
- K. R. Bagnall, Y. S. Muzychka, and E. N. Wang, Application of the Kirchhoff transform to thermal spreading problems with convection boundary conditions, IEEE Transactions on Components, Packaging and Manufacturing Technology 4 (2013), no. 3, 408–420.
- J. Bear, Dynamics of Fluids in Porous Media, Dover Publications, 1988.
- Doina Cioranescu and Patrizia Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, vol. 17, The Clarendon Press, Oxford University Press, New York, 1999. MR 1765047
- Emilio N. M. Cirillo, Ida de Bonis, Adrian Muntean, and Omar Richardson, Upscaling the interplay between diffusion and polynomial drifts through a composite thin strip with periodic microstructure, Meccanica 55 (2020), no. 11, 2159–2178. MR 4173437, DOI 10.1007/s11012-020-01253-8
- Emilio N. M. Cirillo, Oleh Krehel, Adrian Muntean, Rutger van Santen, and Aditya Sengar, Residence time estimates for asymmetric simple exclusion dynamics on strips, Phys. A 442 (2016), 436–457. MR 3412980, DOI 10.1016/j.physa.2015.09.037
- Michael Eden, Christos Nikolopoulos, and Adrian Muntean, A multiscale quasilinear system for colloids deposition in porous media: weak solvability and numerical simulation of a near-clogging scenario, Nonlinear Anal. Real World Appl. 63 (2022), Paper No. 103408, 29. MR 4305337, DOI 10.1016/j.nonrwa.2021.103408
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- Patrick Henning and Mario Ohlberger, The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift, Netw. Heterog. Media 5 (2010), no. 4, 711–744. MR 2740530, DOI 10.3934/nhm.2010.5.711
- D. Hilhorst, J. R. King, and M. Röger, Mathematical analysis of a model describing the invasion of bacteria in burn wounds, Nonlinear Anal. 66 (2007), no. 5, 1118–1140. MR 2286623, DOI 10.1016/j.na.2006.01.009
- H. Hutridurga, Homogenization of complex flows in porous media and applications, Ph.D. thesis, École Polytechnique, Palaiseau, France, 2013.
- Ekeoma R. Ijioma and Adrian Muntean, Fast drift effects in the averaging of a filtration combustion system: a periodic homogenization approach, Quart. Appl. Math. 77 (2019), no. 1, 71–104. MR 3897920, DOI 10.1090/qam/1509
- O. Iliev, A. Mikelić, T. Prill, and A. Sherly, Homogenization approach to the upscaling of a reactive flow through particulate filters with wall integrated catalyst, Advances in Water Resources 146 (2020), 103779.
- Gautam Iyer, Tomasz Komorowski, Alexei Novikov, and Lenya Ryzhik, From homogenization to averaging in cellular flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), no. 5, 957–983. MR 3258362, DOI 10.1016/j.anihpc.2013.06.003
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
- S. Liu and J. H. Masliyah, Dispersion in porous media, Handbook of Porous Media (2nd ed.) (K. Vafai, ed.), CRC Press, 2005, pp. 81–141.
- Eduard Maru ić-Paloka and Andrey L. Piatnitski, Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection, J. London Math. Soc. (2) 72 (2005), no. 2, 391–409. MR 2156660, DOI 10.1112/S0024610705006824
- Franck Ouaki, Grégoire Allaire, Sylvain Desroziers, and Guillaume Enchéry, A priori error estimate of a multiscale finite element method for transport modeling, SeMA J. 67 (2015), 1–37. MR 3301085, DOI 10.1007/s40324-014-0023-8
- Iryna Pankratova and Andrey Piatnitski, Homogenization of convection-diffusion equation in infinite cylinder, Netw. Heterog. Media 6 (2011), no. 1, 111–126. MR 2777012, DOI 10.3934/nhm.2011.6.111
- Andrey Piatnitski and Mariya Ptashnyk, Homogenization of biomechanical models of plant tissues with randomly distributed cells, Nonlinearity 33 (2020), no. 10, 5510–5542. MR 4151416, DOI 10.1088/1361-6544/ab95ab
- Radu Precup, Linear and semilinear partial differential equations, De Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2013. An introduction. MR 2986215
- A. L. Pyatnitskiĭ, Averaging of a singularly perturbed equation with rapidly oscillating coefficients in a layer, Mat. Sb. (N.S.) 121(163) (1983), no. 1, 18–39 (Russian). MR 699735
- Vishnu Raveendran, Emilio N. M. Cirillo, Ida de Bonis, and Adrian Muntean, Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer, Quart. Appl. Math. 80 (2022), no. 1, 157–200. MR 4360553, DOI 10.1090/qam/1607
- Mohammad Abdulhadi Al-Mahmeed, The analysis of autoregressive processes: the identification and the prior, posterior, and predictive analysis, ProQuest LLC, Ann Arbor, MI, 1982. Thesis (Ph.D.)–Oklahoma State University. MR 2632736
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
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Additional Information
Vishnu Raveendran
Affiliation:
Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden
MR Author ID:
1483393
ORCID:
0000-0001-5168-0841
Email:
vishnu.raveendran@kau.se
Emilio N. M. Cirillo
Affiliation:
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, Rome, Italy
MR Author ID:
606246
ORCID:
0000-0003-3673-2054
Email:
emilio.cirillo@uniroma1.it
Adrian Muntean
Affiliation:
Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden
MR Author ID:
684703
ORCID:
0000-0002-1160-0007
Email:
adrian.muntean@kau.se
Keywords:
Two-scale periodic homogenization asymptotics with drift,
reaction-diffusion equations with non-linear drift,
effective dispersion tensors for reactive flow in porous media,
weak solvability of quasi-linear systems in unbounded domains
Received by editor(s):
April 2, 2022
Published electronically:
May 18, 2022
Additional Notes:
The work of the first and third authors is partly supported by the project “Homogenization and dimension reduction of thin heterogeneous layers”, grant nr. VR 2018-03648 of the Swedish Research Council.
Article copyright:
© Copyright 2022
Brown University
