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 and Heterogeneous Media, 2014, 9 (4) : 709-737. doi: 10.3934/nhm.2014.9.709
References:
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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]

K. Kumar, M. Neuss-Radu and I. S. Pop, Homogenization of a pore scale model for precipitation and dissolution in porous media, 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]

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, 1-45.

[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.

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.

[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]

K. Kumar, M. Neuss-Radu and I. S. Pop, Homogenization of a pore scale model for precipitation and dissolution in porous media, 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]

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, 1-45.

[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.

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