December  2014, 9(4): 739-762. doi: 10.3934/nhm.2014.9.739

Homogenization of a thermo-diffusion system with Smoluchowski interactions

1. 

Department of Mathematics and Computer Science, CASA - Center for Analysis, Scientific computing and Engineering, Eindhoven University of Technology, 5600 MB, PO Box 513, Eindhoven, Netherlands

2. 

Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women's University, Tokyo, 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  April 2014 Revised  September 2014 Published  December 2014

We study the solvability and homogenization of a thermal-diffusion reaction problem posed in a periodically perforated domain. The system describes the motion of populations of hot colloidal particles interacting together via Smoluchowski production terms. The upscaled system, obtained via two-scale convergence techniques, allows the investigation of deposition effects in porous materials in the presence of thermal gradients.
Citation: Oleh Krehel, Toyohiko Aiki, Adrian Muntean. Homogenization of a thermo-diffusion system with Smoluchowski interactions. Networks & Heterogeneous Media, 2014, 9 (4) : 739-762. doi: 10.3934/nhm.2014.9.739
References:
[1]

G. Allaire, Homogenization and two-scale convergence,, SIAM Journal on Mathematical Analysis, 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar

[2]

B. Andreianov, M. Bendahmane and R. Ruiz-Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics,, Mathematical Models and Methods in Applied Sciences, 21 (2011), 307. doi: 10.1142/S0218202511005064. Google Scholar

[3]

M. Beneš and R. Štefan, Global weak solutions for coupled transport processes in concrete walls at high temperatures,, ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, 93 (2013), 233. doi: 10.1002/zamm.201200018. Google Scholar

[4]

M. Beneš, R. Štefan and J. Zeman, Analysis of coupled transport phenomena in concrete at elevated temperatures,, Applied Mathematics and Computation, 219 (2013), 7262. doi: 10.1016/j.amc.2011.02.064. Google Scholar

[5]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 374,, American Mathematical Soc., (2011). Google Scholar

[6]

S. de Groot and P. Mazur, Non-equilibrium Thermodynamics,, Series in physics, (1962). Google Scholar

[7]

M. Elimelech, J. Gregory, X. Jia and R. Williams, Particle Deposition and Aggregation: Measurement, Modelling and Simulation,, Elsevier, (1998). Google Scholar

[8]

L. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics,, American Mathematical Society, (1998). Google Scholar

[9]

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. doi: 10.1016/j.nonrwa.2012.01.019. Google Scholar

[10]

T. Funaki, H. Izuhara, M. Mimura and C. Urabe, A link between microscopic and macroscopic models of self-organized aggregation,, Networks and Heterogeneous Media, 7 (2012), 705. doi: 10.3934/nhm.2012.7.705. Google Scholar

[11]

R. Golestanian, Collective behavior of thermally active colloids,, Physical Review Letters, 108 (2012). doi: 10.1103/PhysRevLett.108.038303. Google Scholar

[12]

Z.-X. Gong and A. S. Mujumdar, Development of drying schedules for one-side-heating drying of refractory concrete slabs based on a finite element model,, Journal of the American Ceramic Society, 79 (1996), 1649. doi: 10.1111/j.1151-2916.1996.tb08777.x. Google Scholar

[13]

U. Hornung and W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media,, Journal of Differential Equations, 92 (1991), 199. doi: 10.1016/0022-0396(91)90047-D. Google Scholar

[14]

O. Krehel, A. Muntean and P. Knabner, On modeling and simulation of flocculation in porous media,, In A.J. Valochi (Ed.), (2012), 1. Google Scholar

[15]

O. Krehel, A. Muntean and P. Knabner, Multiscale modeling of colloidal dynamics in porous media including aggregation and deposition,, Technical Report No. 14-12, (2014), 14. Google Scholar

[16]

J. Lions, Quelques méthodes de résolution des problèmes aux limites non linèaires},, Dunod, (1969). Google Scholar

[17]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques,, SIAM Journal on Mathematical Analysis, 40 (2008), 215. doi: 10.1137/050645269. Google Scholar

[18]

N. Masmoudi and M. L. Tayeb, Diffusion limit of a semiconductor boltzmann-poisson system,, SIAM Journal on Mathematical Analysis, 38 (2007), 1788. doi: 10.1137/050630763. Google Scholar

[19]

C. C. Mei and B. Vernescu, Homogenization Methods for Multiscale Mechanics.,, World Scientific, (2010). doi: 10.1142/7427. Google Scholar

[20]

M. Neuss-Radu, Some extensions of two-scale convergence,, Comptes Rendus de l'Académie des Sciences. Série 1, 322 (1996), 899. Google Scholar

[21]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM Journal on Mathematical Analysis, 20 (1989), 608. doi: 10.1137/0520043. Google Scholar

[22]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices of the AMS, 45 (1998), 9. Google Scholar

[23]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115. Google Scholar

[24]

N. Ray, A. Muntean and P. Knabner, Rigorous homogenization of a stokes-nernst-planck-poisson system,, Journal of Mathematical Analysis and Applications, 390 (2012), 374. doi: 10.1016/j.jmaa.2012.01.052. Google Scholar

[25]

S. Rothstein, W. Federspiel and S. Little, A unified mathematical model for the prediction of controlled release from surface and bulk eroding polymer matrices,, Biomaterials, 30 (2009), 1657. doi: 10.1016/j.biomaterials.2008.12.002. Google Scholar

[26]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, Journal of Theoretical Biology, 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

[27]

M. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen,, Z. Phys. Chem, 92 (1917), 129. Google Scholar

[28]

J. Soares and P. Zunino, A mixture model for water uptake, degradation, erosion and drug release from polydisperse polymeric networks,, Biomaterials, 31 (2010), 3032. doi: 10.1016/j.biomaterials.2010.01.008. Google Scholar

[29]

V. K. Vanag and I. R. Epstein, Cross-diffusion and pattern formation in reaction-diffusion systems,, Physical Chemistry Chemical Physics, 11 (2009), 897. doi: 10.1039/b813825g. Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence,, SIAM Journal on Mathematical Analysis, 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar

[2]

B. Andreianov, M. Bendahmane and R. Ruiz-Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics,, Mathematical Models and Methods in Applied Sciences, 21 (2011), 307. doi: 10.1142/S0218202511005064. Google Scholar

[3]

M. Beneš and R. Štefan, Global weak solutions for coupled transport processes in concrete walls at high temperatures,, ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, 93 (2013), 233. doi: 10.1002/zamm.201200018. Google Scholar

[4]

M. Beneš, R. Štefan and J. Zeman, Analysis of coupled transport phenomena in concrete at elevated temperatures,, Applied Mathematics and Computation, 219 (2013), 7262. doi: 10.1016/j.amc.2011.02.064. Google Scholar

[5]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 374,, American Mathematical Soc., (2011). Google Scholar

[6]

S. de Groot and P. Mazur, Non-equilibrium Thermodynamics,, Series in physics, (1962). Google Scholar

[7]

M. Elimelech, J. Gregory, X. Jia and R. Williams, Particle Deposition and Aggregation: Measurement, Modelling and Simulation,, Elsevier, (1998). Google Scholar

[8]

L. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics,, American Mathematical Society, (1998). Google Scholar

[9]

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. doi: 10.1016/j.nonrwa.2012.01.019. Google Scholar

[10]

T. Funaki, H. Izuhara, M. Mimura and C. Urabe, A link between microscopic and macroscopic models of self-organized aggregation,, Networks and Heterogeneous Media, 7 (2012), 705. doi: 10.3934/nhm.2012.7.705. Google Scholar

[11]

R. Golestanian, Collective behavior of thermally active colloids,, Physical Review Letters, 108 (2012). doi: 10.1103/PhysRevLett.108.038303. Google Scholar

[12]

Z.-X. Gong and A. S. Mujumdar, Development of drying schedules for one-side-heating drying of refractory concrete slabs based on a finite element model,, Journal of the American Ceramic Society, 79 (1996), 1649. doi: 10.1111/j.1151-2916.1996.tb08777.x. Google Scholar

[13]

U. Hornung and W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media,, Journal of Differential Equations, 92 (1991), 199. doi: 10.1016/0022-0396(91)90047-D. Google Scholar

[14]

O. Krehel, A. Muntean and P. Knabner, On modeling and simulation of flocculation in porous media,, In A.J. Valochi (Ed.), (2012), 1. Google Scholar

[15]

O. Krehel, A. Muntean and P. Knabner, Multiscale modeling of colloidal dynamics in porous media including aggregation and deposition,, Technical Report No. 14-12, (2014), 14. Google Scholar

[16]

J. Lions, Quelques méthodes de résolution des problèmes aux limites non linèaires},, Dunod, (1969). Google Scholar

[17]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques,, SIAM Journal on Mathematical Analysis, 40 (2008), 215. doi: 10.1137/050645269. Google Scholar

[18]

N. Masmoudi and M. L. Tayeb, Diffusion limit of a semiconductor boltzmann-poisson system,, SIAM Journal on Mathematical Analysis, 38 (2007), 1788. doi: 10.1137/050630763. Google Scholar

[19]

C. C. Mei and B. Vernescu, Homogenization Methods for Multiscale Mechanics.,, World Scientific, (2010). doi: 10.1142/7427. Google Scholar

[20]

M. Neuss-Radu, Some extensions of two-scale convergence,, Comptes Rendus de l'Académie des Sciences. Série 1, 322 (1996), 899. Google Scholar

[21]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM Journal on Mathematical Analysis, 20 (1989), 608. doi: 10.1137/0520043. Google Scholar

[22]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices of the AMS, 45 (1998), 9. Google Scholar

[23]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115. Google Scholar

[24]

N. Ray, A. Muntean and P. Knabner, Rigorous homogenization of a stokes-nernst-planck-poisson system,, Journal of Mathematical Analysis and Applications, 390 (2012), 374. doi: 10.1016/j.jmaa.2012.01.052. Google Scholar

[25]

S. Rothstein, W. Federspiel and S. Little, A unified mathematical model for the prediction of controlled release from surface and bulk eroding polymer matrices,, Biomaterials, 30 (2009), 1657. doi: 10.1016/j.biomaterials.2008.12.002. Google Scholar

[26]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, Journal of Theoretical Biology, 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

[27]

M. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen,, Z. Phys. Chem, 92 (1917), 129. Google Scholar

[28]

J. Soares and P. Zunino, A mixture model for water uptake, degradation, erosion and drug release from polydisperse polymeric networks,, Biomaterials, 31 (2010), 3032. doi: 10.1016/j.biomaterials.2010.01.008. Google Scholar

[29]

V. K. Vanag and I. R. Epstein, Cross-diffusion and pattern formation in reaction-diffusion systems,, Physical Chemistry Chemical Physics, 11 (2009), 897. doi: 10.1039/b813825g. Google Scholar

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