Fast drift effects in the averaging of a filtration combustion system: A periodic homogenization approach
Authors:
Ekeoma R. Ijioma and Adrian Muntean
Journal:
Quart. Appl. Math. 77 (2019), 71-104
MSC (2010):
Primary 80A32, 76M50; Secondary 80A25, 35B27
DOI:
https://doi.org/10.1090/qam/1509
Published electronically:
June 19, 2018
MathSciNet review:
3897920
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Additional Information
Abstract: We target the periodic homogenization of a semi-linear reaction-diffusion-convection system describing filtration combustion, where fast drifts are triggered by the competition between heat and mass transfer processes in an asymptotic regime of dominant convection. In addition, we consider the interplay between surface nonlinear chemical reactions and transport processes. To handle the oscillations occurring due to the heterogeneity of the medium, we rely on the concept of two-scale convergence with drift to obtain, for suitably scaled model parameters, the upscaled system of equations together with effective transport parameters. The main difficulty is to treat the case of a coupled multi-physics problem. We proceed by extending the results reported by G. Allaire et al. and other related papers in this context to the case of a coupled system of evolution equations pertinent to filtration combustion.
References
- H. Yi and J. Moore, Self-propagating high-temperature (combustion) synthesis (SHS) of powder-compacted materials, J. Mater. Sci. 25 (2) (1990) 1159–1168.
- S. Wang and X. Zhang, Microgravity smoldering combustion of flexible polyurethane foam with central ignition, Microgravity Sci. Technol. 20 (2008) 99–105.
- S. Olson, H. Baum, and T. Kashiwagi, Finger-like smoldering over thin cellulose sheets in microgravity, Twenty-Seventh Symposium (International) on Combustion (1998) 2525–2533.
- A. P. Aldushin, A. Bayliss, and B. J. Matkowsky, On the transition from smoldering to flaming, Combust. Flame 145 (2006) 579–606.
- J. Thullie and A. Burghardt, Simplified procedure for estimating maximum cycling time of flow-reversal reactors, Chem.Eng. Sci. 50 (14) (1995) 2299–2309.
- A. Burghardt, M. Berezowski, and E. W. Jacobsen, Approximate characteristics of a moving temperature front in a fixed-bed catalytic reactor, Chem. Eng. and Process. 38 (1999) 19–34.
- G. Faeth and G. S. Samuelsen, Fast reaction nonpremixed combustion, Prog. Energy Combust. Sci. 12 (1986) 305–372.
- G. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, in: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 219, The Royal Society, 1953, pp. 186–203.
- J. Salles, J.-F. Thovert, R. Delannay, L. Prevors, J.-L. Auriault, and P. M. Adler, Taylor dispersion in porous media. Determination of the dispersion tensor, Phys. Fluids A 5 (1993), no. 10, 2348–2376. MR 1240905, DOI https://doi.org/10.1063/1.858751
- M. Fatehi and M. Kaviany, Adiabatic reverse combustion in a packed bed, Combust. Flame 99 (1994) 1–17.
- Catherine Choquet and Carole Rosier, Effective models for reactive flow under a dominant Péclet number and order one Damköhler number: numerical simulations, Nonlinear Anal. Real World Appl. 15 (2014), 345–360. MR 3110576, DOI https://doi.org/10.1016/j.nonrwa.2012.02.008
- M. H. Pedras and M. J. de Lemos, Thermal dispersion in porous media as a function of the solid-fluid conductivity ratio, Int. J. Heat Mass Tran. 51 (21-22) (2008) 5359–5367.
- C. Hsu and P. Cheng, Thermal dispersion in a porous medium, Int. J. Heat Mass Trans. 33 (8) (1990) 1587–1597.
- Y. Sano, F. Kuwahara, M. Mobedi, and A. Nakayama, Effects of thermal dispersion on heat transfer in cross-flow tubular heat exchangers, Heat Mass Tran. 48 (1) (2011) 183–189.
- C. Moyne, S. Didierjean, H. A. Souto, and O. da Silveira, Thermal dispersion in porous media: one-equation model, Int. J. Heat Mass Tran. 43 (20) (2000) 3853–3867.
- J. L. Auriault and P. Adler, Taylor dispersion in porous media: Analysis by multiple scale expansions, Adv. Water Resour. 18 (4) (1995) 217–226.
- 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 https://doi.org/10.1080/00036811.2015.1038254
- I. Brailovsky, P. V. Gordon, L. Kagan, G. Sivashinsky, Diffusive-thermal instabilities in premixed flames: Stepwise ignition-temperature kinetics, Combust. Flame 162 (5) (2015) 2077–2086.
- Tasnim Fatima and Adrian Muntean, Sulfate attack in sewer pipes: derivation of a concrete corrosion model via two-scale convergence, Nonlinear Anal. Real World Appl. 15 (2014), 326–344. MR 3110575, DOI https://doi.org/10.1016/j.nonrwa.2012.01.019
- Tasnim Fatima, Adrian Muntean, and Mariya Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl. Anal. 91 (2012), no. 6, 1129–1154. MR 2926708, DOI https://doi.org/10.1080/00036811.2011.625016
- Eduard Marušić-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 https://doi.org/10.1112/S0024610705006824
- 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 https://doi.org/10.1137/090754935
- H. Hutridurga, Homogenization of complex flows in porous media and applications, Ph.D. thesis, École Polytéchnique, Palaiseau, France (2013).
- Thomas Holding, Harsha Hutridurga, and Jeffrey Rauch, Convergence along mean flows, SIAM J. Math. Anal. 49 (2017), no. 1, 222–271. MR 3598785, DOI https://doi.org/10.1137/16M1068657
- Grégoire Allaire and Harsha Hutridurga, On the homogenization of multicomponent transport, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 8, 2527–2551. MR 3423245, DOI https://doi.org/10.3934/dcdsb.2015.20.2527
- 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 https://doi.org/10.1016/j.jfa.2011.09.014
- J. Bruining, A. V. Mailybaev, and D. Marchesin, Filtration combustion in wet porous medium, SIAM J. Appl. Math. 70 (2009), no. 4, 1157–1177. MR 2546357, DOI https://doi.org/10.1137/080741318
- John D. Buckmaster (ed.), The mathematics of combustion, Frontiers in Applied Mathematics, vol. 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. MR 806548
- A. Fasano, M. Mimura, and M. Primicerio, Modelling a slow smoldering combustion process, Math. Methods Appl. Sci. 33 (2010), no. 10, 1211–1220. MR 2675040, DOI https://doi.org/10.1002/mma.1301
- E. R. Ijioma, A. Muntean, and T. Ogawa, Pattern formation in reverse smouldering combustion: A homogenisation approach, Combust. Theor. and Model. 17 (2) (2013) 185–223.
- E. R. Ijioma, A. Muntean, and T. Ogawa, Effect of material anisotropy on the fingering instability in reverse smoldering combustion, Int. J. Heat Mass Tran. 81 (0) (2015) 924–938.
- P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, ZAMP 19 (4) (1968) 614–627.
- 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
- Doina Cioranescu, Patrizia Donato, and Rachad Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal. 53 (2007), no. 4, 209–235. MR 2350739
- J. L. Auriault, Heterogeneous medium. is an equivalent macroscopic description possible?, Int. J. Eng. Sci. 29 (7) (1991) 785–795.
- G. I. Sivashinsky, Instabilities, pattern formation, and turbulence in flames, Annu. Rev. Fluid Mech. 15 (1) (1983) 179–199.
- Ulrich Hornung and Willi Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media, J. Differential Equations 92 (1991), no. 2, 199–225. MR 1120903, DOI https://doi.org/10.1016/0022-0396%2891%2990047-D
- Sara Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl. 13 (2003), no. 1, 43–63. MR 2002395
- G. Allaire, R. Brizzi, A. Mikelic, and A. Piatnitski, Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media, Chem. Eng. Sci. 65 (7) (2010) 2292–2300.
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
References
- H. Yi and J. Moore, Self-propagating high-temperature (combustion) synthesis (SHS) of powder-compacted materials, J. Mater. Sci. 25 (2) (1990) 1159–1168.
- S. Wang and X. Zhang, Microgravity smoldering combustion of flexible polyurethane foam with central ignition, Microgravity Sci. Technol. 20 (2008) 99–105.
- S. Olson, H. Baum, and T. Kashiwagi, Finger-like smoldering over thin cellulose sheets in microgravity, Twenty-Seventh Symposium (International) on Combustion (1998) 2525–2533.
- A. P. Aldushin, A. Bayliss, and B. J. Matkowsky, On the transition from smoldering to flaming, Combust. Flame 145 (2006) 579–606.
- J. Thullie and A. Burghardt, Simplified procedure for estimating maximum cycling time of flow-reversal reactors, Chem.Eng. Sci. 50 (14) (1995) 2299–2309.
- A. Burghardt, M. Berezowski, and E. W. Jacobsen, Approximate characteristics of a moving temperature front in a fixed-bed catalytic reactor, Chem. Eng. and Process. 38 (1999) 19–34.
- G. Faeth and G. S. Samuelsen, Fast reaction nonpremixed combustion, Prog. Energy Combust. Sci. 12 (1986) 305–372.
- G. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, in: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 219, The Royal Society, 1953, pp. 186–203.
- J. Salles, J.-F. Thovert, R. Delannay, L. Prevors, J.-L. Auriault, and P. M. Adler, Taylor dispersion in porous media. Determination of the dispersion tensor, Phys. Fluids A 5 (1993), no. 10, 2348–2376. MR 1240905, DOI https://doi.org/10.1063/1.858751
- M. Fatehi and M. Kaviany, Adiabatic reverse combustion in a packed bed, Combust. Flame 99 (1994) 1–17.
- Catherine Choquet and Carole Rosier, Effective models for reactive flow under a dominant Péclet number and order one Damköhler number: numerical simulations, Nonlinear Anal. Real World Appl. 15 (2014), 345–360. MR 3110576, DOI https://doi.org/10.1016/j.nonrwa.2012.02.008
- M. H. Pedras and M. J. de Lemos, Thermal dispersion in porous media as a function of the solid-fluid conductivity ratio, Int. J. Heat Mass Tran. 51 (21-22) (2008) 5359–5367.
- C. Hsu and P. Cheng, Thermal dispersion in a porous medium, Int. J. Heat Mass Trans. 33 (8) (1990) 1587–1597.
- Y. Sano, F. Kuwahara, M. Mobedi, and A. Nakayama, Effects of thermal dispersion on heat transfer in cross-flow tubular heat exchangers, Heat Mass Tran. 48 (1) (2011) 183–189.
- C. Moyne, S. Didierjean, H. A. Souto, and O. da Silveira, Thermal dispersion in porous media: one-equation model, Int. J. Heat Mass Tran. 43 (20) (2000) 3853–3867.
- J. L. Auriault and P. Adler, Taylor dispersion in porous media: Analysis by multiple scale expansions, Adv. Water Resour. 18 (4) (1995) 217–226.
- 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 https://doi.org/10.1080/00036811.2015.1038254
- I. Brailovsky, P. V. Gordon, L. Kagan, G. Sivashinsky, Diffusive-thermal instabilities in premixed flames: Stepwise ignition-temperature kinetics, Combust. Flame 162 (5) (2015) 2077–2086.
- Tasnim Fatima and Adrian Muntean, Sulfate attack in sewer pipes: derivation of a concrete corrosion model via two-scale convergence, Nonlinear Anal. Real World Appl. 15 (2014), 326–344. MR 3110575, DOI https://doi.org/10.1016/j.nonrwa.2012.01.019
- Tasnim Fatima, Adrian Muntean, and Mariya Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl. Anal. 91 (2012), no. 6, 1129–1154. MR 2926708, DOI https://doi.org/10.1080/00036811.2011.625016
- Eduard Marušić-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 https://doi.org/10.1112/S0024610705006824
- 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 https://doi.org/10.1137/090754935
- H. Hutridurga, Homogenization of complex flows in porous media and applications, Ph.D. thesis, École Polytéchnique, Palaiseau, France (2013).
- Thomas Holding, Harsha Hutridurga, and Jeffrey Rauch, Convergence along mean flows, SIAM J. Math. Anal. 49 (2017), no. 1, 222–271. MR 3598785, DOI https://doi.org/10.1137/16M1068657
- Grégoire Allaire and Harsha Hutridurga, On the homogenization of multicomponent transport, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 8, 2527–2551. MR 3423245, DOI https://doi.org/10.3934/dcdsb.2015.20.2527
- 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 https://doi.org/10.1016/j.jfa.2011.09.014
- J. Bruining, A. V. Mailybaev, and D. Marchesin, Filtration combustion in wet porous medium, SIAM J. Appl. Math. 70 (2009), no. 4, 1157–1177. MR 2546357, DOI https://doi.org/10.1137/080741318
- John D. Buckmaster (ed.), The mathematics of combustion, Frontiers in Applied Mathematics, vol. 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. MR 806548
- A. Fasano, M. Mimura, and M. Primicerio, Modelling a slow smoldering combustion process, Math. Methods Appl. Sci. 33 (2010), no. 10, 1211–1220. MR 2675040, DOI https://doi.org/10.1002/mma.1301
- E. R. Ijioma, A. Muntean, and T. Ogawa, Pattern formation in reverse smouldering combustion: A homogenisation approach, Combust. Theor. and Model. 17 (2) (2013) 185–223.
- E. R. Ijioma, A. Muntean, and T. Ogawa, Effect of material anisotropy on the fingering instability in reverse smoldering combustion, Int. J. Heat Mass Tran. 81 (0) (2015) 924–938.
- P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, ZAMP 19 (4) (1968) 614–627.
- 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
- Doina Cioranescu, Patrizia Donato, and Rachad Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal. 53 (2007), no. 4, 209–235. MR 2350739
- J. L. Auriault, Heterogeneous medium. is an equivalent macroscopic description possible?, Int. J. Eng. Sci. 29 (7) (1991) 785–795.
- G. I. Sivashinsky, Instabilities, pattern formation, and turbulence in flames, Annu. Rev. Fluid Mech. 15 (1) (1983) 179–199.
- Ulrich Hornung and Willi Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media, J. Differential Equations 92 (1991), no. 2, 199–225. MR 1120903, DOI https://doi.org/10.1016/0022-0396%2891%2990047-D
- Sara Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl. 13 (2003), no. 1, 43–63. MR 2002395
- G. Allaire, R. Brizzi, A. Mikelic, and A. Piatnitski, Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media, Chem. Eng. Sci. 65 (7) (2010) 2292–2300.
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
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Additional Information
Ekeoma R. Ijioma
Affiliation:
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
Address at time of publication:
University of Limerick, Castletroy, Limerick, V94 T9PX.
MR Author ID:
1091502
Email:
e.r.ijioma@gmail.com
Adrian Muntean
Affiliation:
Department of Mathematics and Computer Science, Karlstad University, Sweden
MR Author ID:
684703
ORCID:
0000-0002-1160-0007
Email:
adrian.muntean@kau.se
Keywords:
Filtration combustion,
thermal dispersion,
periodic homogenization,
two-scale convergence with drift
Received by editor(s):
March 29, 2018
Published electronically:
June 19, 2018
Additional Notes:
The first author was supported by Science Foundation Ireland (SFI) under Grant Number 14/SP/2750.
The second author was supported by NWO MPE “Theoretical estimates of heat losses in geothermal wells” (Grant Number 657.014.004)
Article copyright:
© Copyright 2018
Brown University
