# American Institute of Mathematical Sciences

December  2014, 9(4): 709-737. doi: 10.3934/nhm.2014.9.709

## 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, Netherlands 2 Graduate School of Advanced Mathematical Science, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan, Japan 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  February 2014 Revised  September 2014 Published  December 2014

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.
Citation: Tasnim Fatima, Ekeoma Ijioma, Toshiyuki Ogawa, Adrian Muntean. Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers. Networks & Heterogeneous Media, 2014, 9 (4) : 709-737. doi: 10.3934/nhm.2014.9.709
##### References:
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Anal., 154 (2000), 53-91. doi: 10.1007/s002050000090.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [22] U. Hornung, Homogenization and Porous Media, Springer-Verlag New York, 1997. doi: 10.1007/978-1-4612-1920-0.  Google Scholar [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.  Google Scholar [24] E. R. Ijioma, Homogenization approach to filtration combustion of reactive porous materials: Modeling, simulation and analysis, PhD thesis, Meiji University, Tokyo, Japan, 2014. Google Scholar [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.  Google Scholar [26] K. 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Mathematique, 322 (1996), 899-904.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [41] A. Bourgeat, G. A. Chechkin and A. L. Piatnitski, Singular double porosity model, Applicable Analysis, 82 (2003), 103-116. doi: 10.1080/0003681031000063739.  Google Scholar [42] P. Ronney, E. Roegner and J. Greenberg, Lewis number effects on flame spreading over thin solid fuels, Combust. Flame, 90 (1992), 71-83. Google Scholar [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.  Google Scholar [44] H. F. W. Taylor, Cement Chemistry, London: Academic Press, 1990. Google Scholar [45] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, 2005. doi: 10.1017/CBO9780511755422.  Google Scholar [46] S. Turns, An Introduction to Combustion: Concepts and Applications, McGraw-Hill Series in Mechanical Engineering, McGraw-Hill, 2000. Google Scholar [47] C. van Duijn, A. Mikelic, I. S. Pop and C. Rosier, Mathematics in chemical kinetics and engineering, chapter on Effective dispersion equations for reactive flows with dominant Peclet and Damköhler numbers,, Advances in Chemical Engineering, (): 1.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [2] B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. Watson, Molecular Biology of the Cell, Garland, NY, 2002. Google Scholar [3] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] L. Barbu and G. Morosanu, Singularly Perturbed Boundary Value Problems, vol. 156 of International Series of Numerical Mathematics, Birkhäuser, Basel, 2007.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] M. Chipot, $l$ goes to Infinity, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8173-9.  Google Scholar [11] M. Chipot and S. Guesmia, On some anisotropic, nonlocal, parabolic singular perturbations problems, Applicable Analysis, 90 (2011), 1775-1789. doi: 10.1080/00036811003627542.  Google Scholar [12] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] D. Ciorănescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer Verlag, Berlin, 1999. doi: 10.1007/978-1-4612-2158-6.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [22] U. Hornung, Homogenization and Porous Media, Springer-Verlag New York, 1997. doi: 10.1007/978-1-4612-1920-0.  Google Scholar [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.  Google Scholar [24] E. R. Ijioma, Homogenization approach to filtration combustion of reactive porous materials: Modeling, simulation and analysis, PhD thesis, Meiji University, Tokyo, Japan, 2014. Google Scholar [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.  Google Scholar [26] K. Ikeda and M. Mimura, Mathematical treatment of a model for smoldering combustion, Hiroshima Math. J., 38 (2008), 349-361.  Google Scholar [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.  Google Scholar [28] K. Kumar, M. Neuss-Radu and I. S. Pop, Homogenization of a pore scale model for precipitation and dissolution in porous media,, CASA Report., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [31] J. L. Lions, Quelques Méthodes de Résolution des Problemes Aux Limites Nonlinéaires, Dunod, Paris, 1969.  Google Scholar [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.  Google Scholar [33] S. Monsurro, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63.  Google Scholar [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.  Google Scholar [35] M. Neuss-Radu, Some extensions of two-scale convergence, C. R. Acad. Sci. Paris. Mathematique, 322 (1996), 899-904.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [41] A. Bourgeat, G. A. Chechkin and A. L. Piatnitski, Singular double porosity model, Applicable Analysis, 82 (2003), 103-116. doi: 10.1080/0003681031000063739.  Google Scholar [42] P. Ronney, E. Roegner and J. Greenberg, Lewis number effects on flame spreading over thin solid fuels, Combust. Flame, 90 (1992), 71-83. Google Scholar [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.  Google Scholar [44] H. F. W. Taylor, Cement Chemistry, London: Academic Press, 1990. Google Scholar [45] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, 2005. doi: 10.1017/CBO9780511755422.  Google Scholar [46] S. Turns, An Introduction to Combustion: Concepts and Applications, McGraw-Hill Series in Mechanical Engineering, McGraw-Hill, 2000. Google Scholar [47] C. van Duijn, A. Mikelic, I. S. Pop and C. Rosier, Mathematics in chemical kinetics and engineering, chapter on Effective dispersion equations for reactive flows with dominant Peclet and Damköhler numbers,, Advances in Chemical Engineering, (): 1.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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