Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers

Tasnim Fatima; Ekeoma Ijioma; Toshiyuki Ogawa; Adrian Muntean; Tasnim Fatima; Ekeoma Ijioma; Toshiyuki Ogawa; Adrian Muntean

doi:10.3934/nhm.2014.9.709

Networks and Heterogeneous Media

2014, Volume 9, Issue 4: 709-737. doi: 10.3934/nhm.2014.9.709

Previous Article Next Article

Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers

1.
Department of Mathematics and Computer Science, CASA - Center for Analysis, Scientific computing and Applications, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven
2.
Graduate School of Advanced Mathematical Science, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525
3.
CASA - Centre for Analysis, Scientific computing and Applications, Department of Mathematics and Computer Science, Institute of Complex Molecular Systems, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven

Received: 01 February 2014 Revised: 01 September 2014
Primary: 35B27; Secondary: 76S05, 74E30, 80A25, 35B25, 35B40.

We study the homogenization of a reaction-diffusion-convection system posed in an $\varepsilon$-periodic $\delta$-thin layer made of a two-component (solid-air) composite material. The microscopic system includes heat flow, diffusion and convection coupled with a nonlinear surface chemical reaction. We treat two distinct asymptotic scenarios: (1) For a fixed width $\delta>0$ of the thin layer, we homogenize the presence of the microstructures (the classical periodic homogenization limit $\varepsilon\to 0$); (2) In the homogenized problem, we pass to $\delta\to 0$ (the vanishing limit of the layer's width). In this way, we are preparing the stage for the simultaneous homogenization ($\varepsilon\to 0$) and dimension reduction limit ($\delta\to 0$) with $\delta=\delta(\epsilon)$. We recover the reduced macroscopic equations from [25] with precise formulas for the effective transport and reaction coefficients. We complement the analytical results with a few simulations of a case study in smoldering combustion. The chosen multiscale scenario is relevant for a large variety of practical applications ranging from the forecast of the response to fire of refractory concrete, the microstructure design of resistance-to-heat ceramic-based materials for engines, to the smoldering combustion of thin porous samples under microgravity conditions.
- Homogenization,
- dimension reduction,
- thin layers,
- filtration combustion,
- two-scale convergence,
- anisotropic singular perturbations
Citation: Tasnim Fatima, Ekeoma Ijioma, Toshiyuki Ogawa, Adrian Muntean. Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers[J]. Networks and Heterogeneous Media, 2014, 9(4): 709-737. doi: 10.3934/nhm.2014.9.709

Related Papers:

Abstract

We study the homogenization of a reaction-diffusion-convection system posed in an $\varepsilon$-periodic $\delta$-thin layer made of a two-component (solid-air) composite material. The microscopic system includes heat flow, diffusion and convection coupled with a nonlinear surface chemical reaction. We treat two distinct asymptotic scenarios: (1) For a fixed width $\delta>0$ of the thin layer, we homogenize the presence of the microstructures (the classical periodic homogenization limit $\varepsilon\to 0$); (2) In the homogenized problem, we pass to $\delta\to 0$ (the vanishing limit of the layer's width). In this way, we are preparing the stage for the simultaneous homogenization ($\varepsilon\to 0$) and dimension reduction limit ($\delta\to 0$) with $\delta=\delta(\epsilon)$. We recover the reduced macroscopic equations from [25] with precise formulas for the effective transport and reaction coefficients. We complement the analytical results with a few simulations of a case study in smoldering combustion. The chosen multiscale scenario is relevant for a large variety of practical applications ranging from the forecast of the response to fire of refractory concrete, the microstructure design of resistance-to-heat ceramic-based materials for engines, to the smoldering combustion of thin porous samples under microgravity conditions.

References

[1]	I. Aganovic, J. Tambaca and Z. Tutek, A note on reduction of dimension for linear elliptic equations, Glasnik Matematicki, 41 (2006), 77-88. doi: 10.3336/gm.41.1.08
[2]	B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. Watson, Molecular Biology of the Cell, Garland, NY, 2002.
[3]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084
[4]	G. Allaire and Z. Habibi, Homogenization of a conductive, convective, and radiative heat transfer problem in a heterogeneous domain, SIAM J. Math. Anal., 45 (2013), 1136-1178. doi: 10.1137/110849821
[5]	B. Amaziane, L. Pankratov and V. Pytula, Homogenization of one phase flow in a highly heterogeneous porous medium including a thin layer, Asymptotic Analysis, 70 (2010), 51-86.
[6]	L. Barbu and G. Morosanu, Singularly Perturbed Boundary Value Problems, vol. 156 of International Series of Numerical Mathematics, Birkhäuser, Basel, 2007.
[7]	M. Beneš and J. Zeman, Some properties of strong solutions to nonlinear heat and moisture transport in multi-layer porous structures, Nonlinear Anal. RWA, 13 (2012), 1562-1580. doi: 10.1016/j.nonrwa.2011.11.015
[8]	G. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization Methods and Applications, vol. 234 of Translations of Mathematical Monographs, AMS, Providence, Rhode Island USA, 2007.
[9]	R.-H. Chen, G. B. Mitchell and P. D. Ronney, Diffusive-thermal instability and flame extinction in nonpremixed combustion, in Symposium (International) on Combustion, Elsevier, 24 (1992), 213-221. doi: 10.1016/S0082-0784(06)80030-5
[10]	M. Chipot, $l$ goes to Infinity, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8173-9
[11]	M. Chipot and S. Guesmia, On some anisotropic, nonlocal, parabolic singular perturbations problems, Applicable Analysis, 90 (2011), 1775-1789. doi: 10.1080/00036811003627542
[12]	D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.
[13]	D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7
[14]	D. Ciorănescu and A. Oud Hammouda, Homogenization of elliptic problems in perforated domains with mixed boundary conditions, Rev. Roumaine Math. Pures Appl., 53 (2008), 389-406.
[15]	D. Ciorănescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer Verlag, Berlin, 1999. doi: 10.1007/978-1-4612-2158-6
[16]	P. Constantin, A. Kiselev, A. Oberman and L. Ryzhik, Bulk burning rate in passive-reactive diffusion, Arch. Ration. Mech. Anal., 154 (2000), 53-91. doi: 10.1007/s002050000090
[17]	B. Denet and P. Haldenwang, Numerical study of thermal-diffusive instability of premixed flames, Combustion Science and Technology, 86 (1992), 199-221. doi: 10.1080/00102209208947195
[18]	A. Fasano, M. Mimura and M. Primicerio, Modelling a slow smoldering combustion process, Math. Methods Appl. Sci., 33 (2010),1211-1220. doi: 10.1002/mma.1301
[19]	T. Fatima and A. Muntean, Sulfate attack in sewer pipes: Derivation of a concrete corrosion model via two-scale convergence, Nonlinear Analysis: Real World Applications, 15 (2014), 326-344. doi: 10.1016/j.nonrwa.2012.01.019
[20]	B. Gustafsson and J. Mossino, Non-periodic explicit homogenization and reduction of dimension: the linear case, IMA J. Appl. Math., 68 (2003), 269-298. doi: 10.1093/imamat/68.3.269
[21]	Z. Habibi, Homogéneisation et Convergence à Deux Échelles lors D'échanges Thermiques Stationnaires et Transitoires. Application Aux Coeurs des Réacteurs Nucléaires à Caloporteur gaz., PhD thesis, École Polytechnique, Paris, 2011.
[22]	U. Hornung, Homogenization and Porous Media, Springer-Verlag New York, 1997. doi: 10.1007/978-1-4612-1920-0
[23]	U. Hornung and W. Jäger, Diffusion, convection, absorption, and reaction of chemicals in porous media, J. Diff. Eqs., 92 (1991), 199-225. doi: 10.1016/0022-0396(91)90047-D
[24]	E. R. Ijioma, Homogenization approach to filtration combustion of reactive porous materials: Modeling, simulation and analysis, PhD thesis, Meiji University, Tokyo, Japan, 2014.
[25]	E. R. Ijioma, A. Muntean and T. Ogawa, Pattern formation in reverse smouldering combustion: A homogenisation approach, Combustion Theory and Modelling, 17 (2013), 185-223. doi: 10.1080/13647830.2012.734860
[26]	K. Ikeda and M. Mimura, Mathematical treatment of a model for smoldering combustion, Hiroshima Math. J., 38 (2008), 349-361.
[27]	L. Kagan and G. Sivashinsky, Pattern formation in flame spread over thin solid fuels, Combust. Theory Model., 12 (2008), 269-281. doi: 10.1080/13647830701639462
[28]	CASA Report.
[29]	V. Kurdyumov and E. Fernández-Tarrazo, Lewis number effect on the propagation of premixed laminar flames in narrow open ducts, Combustion and Flame, 128 (2002), 382-394, URL http://www.sciencedirect.com/science/article/pii/S00102180010 03583. doi: 10.1016/S0010-2180(01)00358-3
[30]	K. Kuwana, G. Kushida and Y. Uchida, Lewis number effect on smoldering combustion of a thin solid, Combustion Science and Technology, 186 (2014), 466-474. doi: 10.1080/00102202.2014.883220
[31]	J. L. Lions, Quelques Méthodes de Résolution des Problemes Aux Limites Nonlinéaires, Dunod, Paris, 1969.
[32]	Z. Lu and Y. Dong, Fingering instability in forward smolder combustion, Combustion Theory and Modelling, 15 (2011), 795-815. doi: 10.1080/13647830.2011.564658
[33]	S. Monsurro, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63.
[34]	S. Neukamm and I. Velcic, Derivation of a homogenized von-Karman plate theory from 3D nonlinear elasticity, Mathematical Models and Methods in Applied Sciences, 23 (2013), 2701-2748. doi: 10.1142/S0218202513500449
[35]	M. Neuss-Radu, Some extensions of two-scale convergence, C. R. Acad. Sci. Paris. Mathematique, 322 (1996), 899-904.
[36]	M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720. doi: 10.1137/060665452
[37]	G. Nguestseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal, 20 (1989), 608-623. doi: 10.1137/0520043
[38]	A. Oliveira and M. Kaviany, Nonequilibrium in the transport of heat and reactants in combustion in porous media, Progress in Energy and Combustion Science, 27 (2001), 523-545. doi: 10.1016/S0360-1285(00)00030-7
[39]	S. Olson, H. Baum and T. Kashiwagi, Finger-like smoldering over thin cellulose sheets in microgravity, Twenty-Seventh Symposium (International) on Combustion, 27 (1998), 2525-2533. doi: 10.1016/S0082-0784(98)80104-5
[40]	I. Ozdemir, W. A. M. Brekelmans and M. G. D. Geers, Computational homogenization for heat conduction in heterogeneous solids, International Journal for Numerical Methods in Engineering, 73 (2008), 185-204. doi: 10.1002/nme.2068
[41]	A. Bourgeat, G. A. Chechkin and A. L. Piatnitski, Singular double porosity model, Applicable Analysis, 82 (2003), 103-116. doi: 10.1080/0003681031000063739
[42]	P. Ronney, E. Roegner and J. Greenberg, Lewis number effects on flame spreading over thin solid fuels, Combust. Flame, 90 (1992), 71-83.
[43]	M. Sahraoui and M. Kaviany, Direct simulation vs volume-averaged treatment of adiabatic premixed flame in a porous medium, Int. J. Heat Mass Transf., 37 (1994), 2817-2834. doi: 10.1016/0017-9310(94)90338-7
[44]	H. F. W. Taylor, Cement Chemistry, London: Academic Press, 1990.
[45]	R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, 2005. doi: 10.1017/CBO9780511755422
[46]	S. Turns, An Introduction to Combustion: Concepts and Applications, McGraw-Hill Series in Mechanical Engineering, McGraw-Hill, 2000.
[47]	Advances in Chemical Engineering, 1-45.
[48]	C. J. van Duijn and I. S. Pop, Crystal dissolution and precipitation in porous media: Pore scale analysis, J. Reine Angew. Math, 577 (2004), 171-211. doi: 10.1515/crll.2004.2004.577.171
[49]	J.-P. Vassal, L. Orgéas, D. Favier and J.-L. Auriault, Upscaling the diffusion equations in particulate media made of highly conductive particles. I. Theoretical aspects, Physical Review E, 77 (2008), 011302, 10pp. doi: 10.1103/PhysRevE.77.011302
[50]	F. Yuan and Z. Lu, Structure and stability of non-adiabatic reverse smolder waves, Applied Mathematics and Mechanics, 34 (2013), 657-668. doi: 10.1007/s10483-013-1698-8
[51]	O. Zik, Z. Olami and E. Moses, Fingering instability in combustion, Phys. Rev. Lett., 81 (1998), 3868-3871. doi: 10.1103/PhysRevLett.81.3868

Reader Comments

Your name:*

Email:*
© 2014 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)