Search Advanced

Networks and Heterogeneous Media

2014, Volume 9, Issue 4: 739-762. doi: 10.3934/nhm.2014.9.739
Previous Article Next Article

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
  • 2.
    Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women's University, Tokyo
  • 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 April 2014 Revised: 01 September 2014

  • Primary: 35B27, 35K59; Secondary: 80M40, 80A25.

  • 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[J]. Networks and Heterogeneous Media, 2014, 9(4): 739-762. doi: 10.3934/nhm.2014.9.739

    Related Papers:

    [1] Oleh Krehel, Toyohiko Aiki, Adrian Muntean . Homogenization of a thermo-diffusion system with Smoluchowski interactions. Networks and Heterogeneous Media, 2014, 9(4): 739-762. doi: 10.3934/nhm.2014.9.739
    [2] 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
    [3] Alexander Mielke, Sina Reichelt, Marita Thomas . Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks and Heterogeneous Media, 2014, 9(2): 353-382. doi: 10.3934/nhm.2014.9.353
    [4] Feiyang Peng, Yanbin Tang . Inverse problem of determining diffusion matrix between different structures for time fractional diffusion equation. Networks and Heterogeneous Media, 2024, 19(1): 291-304. doi: 10.3934/nhm.2024013
    [5] Iryna Pankratova, Andrey Piatnitski . Homogenization of convection-diffusion equation in infinite cylinder. Networks and Heterogeneous Media, 2011, 6(1): 111-126. doi: 10.3934/nhm.2011.6.111
    [6] M. Berezhnyi, L. Berlyand, Evgen Khruslov . The homogenized model of small oscillations of complex fluids. Networks and Heterogeneous Media, 2008, 3(4): 831-862. doi: 10.3934/nhm.2008.3.831
    [7] Junlong Chen, Yanbin Tang . Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure. Networks and Heterogeneous Media, 2023, 18(3): 1118-1177. doi: 10.3934/nhm.2023049
    [8] Delio Mugnolo . Gaussian estimates for a heat equation on a network. Networks and Heterogeneous Media, 2007, 2(1): 55-79. doi: 10.3934/nhm.2007.2.55
    [9] Mostafa Bendahmane, Kenneth H. Karlsen . Martingale solutions of stochastic nonlocal cross-diffusion systems. Networks and Heterogeneous Media, 2022, 17(5): 719-752. doi: 10.3934/nhm.2022024
    [10] Thomas Geert de Jong, Georg Prokert, Alef Edou Sterk . Reaction–diffusion transport into core-shell geometry: Well-posedness and stability of stationary solutions. Networks and Heterogeneous Media, 2025, 20(1): 1-14. doi: 10.3934/nhm.2025001
  • 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.


    [1] G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518. doi: 10.1137/0523084
    [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), 307-344. doi: 10.1142/S0218202511005064
    [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), 233-251. doi: 10.1002/zamm.201200018
    [4] M. Beneš, R. Štefan and J. Zeman, Analysis of coupled transport phenomena in concrete at elevated temperatures, Applied Mathematics and Computation, 219 (2013), 7262-7274. doi: 10.1016/j.amc.2011.02.064
    [5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 374, American Mathematical Soc., 2011.
    [6] S. de Groot and P. Mazur, Non-equilibrium Thermodynamics, Series in physics, North-Holland Publishing Company - Amsterdam, 1962.
    [7] M. Elimelech, J. Gregory, X. Jia and R. Williams, Particle Deposition and Aggregation: Measurement, Modelling and Simulation, Elsevier, 1998.
    [8] L. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, 1998.
    [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), 326-344. doi: 10.1016/j.nonrwa.2012.01.019
    [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), 705-740. doi: 10.3934/nhm.2012.7.705
    [11] R. Golestanian, Collective behavior of thermally active colloids, Physical Review Letters, 108 (2012), 038303. doi: 10.1103/PhysRevLett.108.038303
    [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), 1649-1658. doi: 10.1111/j.1151-2916.1996.tb08777.x
    [13] U. Hornung and W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media, Journal of Differential Equations, 92 (1991), 199-225. doi: 10.1016/0022-0396(91)90047-D
    [14] O. Krehel, A. Muntean and P. Knabner, On modeling and simulation of flocculation in porous media, In A.J. Valochi (Ed.), Proceedings of XIX International Conference on Water Resources. (pp. 1-8) CMWR, University of Illinois at Urbana-Champaign, 2012.
    [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, CASA Report, Eindhoven, 2014.
    [16] J. Lions, Quelques méthodes de résolution des problèmes aux limites non linèaires}, Dunod, Paris, 1969.
    [17] A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques, SIAM Journal on Mathematical Analysis, 40 (2008), 215-237. doi: 10.1137/050645269
    [18] N. Masmoudi and M. L. Tayeb, Diffusion limit of a semiconductor boltzmann-poisson system, SIAM Journal on Mathematical Analysis, 38 (2007), 1788-1807. doi: 10.1137/050630763
    [19] C. C. Mei and B. Vernescu, Homogenization Methods for Multiscale Mechanics., World Scientific, 2010. doi: 10.1142/7427
    [20] M. Neuss-Radu, Some extensions of two-scale convergence, Comptes Rendus de l'Académie des Sciences. Série 1, Mathématique, 322 (1996), 899-904.
    [21] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623. doi: 10.1137/0520043
    [22] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices of the AMS, 45 (1998), 9-18.
    [23] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
    [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), 374-393. doi: 10.1016/j.jmaa.2012.01.052
    [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), 1657-1664. doi: 10.1016/j.biomaterials.2008.12.002
    [26] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, Journal of Theoretical Biology, 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3
    [27] M. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Z. Phys. Chem, 92 (1917), 129-168.
    [28] J. Soares and P. Zunino, A mixture model for water uptake, degradation, erosion and drug release from polydisperse polymeric networks, Biomaterials, 31 (2010), 3032-3042. doi: 10.1016/j.biomaterials.2010.01.008
    [29] V. K. Vanag and I. R. Epstein, Cross-diffusion and pattern formation in reaction-diffusion systems, Physical Chemistry Chemical Physics, 11 (2009), 897-912. doi: 10.1039/b813825g

  • This article has been cited by:

    1. Michal Beneš, Igor Pažanin, Homogenization of degenerate coupled transport processes in porous media with memory terms, 2019, 42, 0170-4214, 6227, 10.1002/mma.5718
    2. Pietro Artale Harris, Emilio N. M. Cirillo, Adrian Muntean, Weak solutions to Allen–Cahn-like equations modelling consolidation of porous media, 2017, 82, 0272-4960, 224, 10.1093/imamat/hxw013
    3. Ansgar Jüngel, Mariya Ptashnyk, Homogenization of degenerate cross-diffusion systems, 2019, 267, 00220396, 5543, 10.1016/j.jde.2019.05.036
    4. Oleh Krehel, Adrian Muntean, Peter Knabner, Multiscale modeling of colloidal dynamics in porous media including aggregation and deposition, 2015, 86, 03091708, 209, 10.1016/j.advwatres.2015.10.005
    5. Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma, On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit, 2021, 26, 1553-524X, 2451, 10.3934/dcdsb.2020190
    6. Michal Beneš, Lukáš Krupička, Weak solutions of coupled dual porosity flows in fractured rock mass and structured porous media, 2016, 433, 0022247X, 543, 10.1016/j.jmaa.2015.07.052
    7. Vo Anh Khoa, Adrian Muntean, Asymptotic analysis of a semi-linear elliptic system in perforated domains: Well-posedness and correctors for the homogenization limit, 2016, 439, 0022247X, 271, 10.1016/j.jmaa.2016.02.068
    8. Toyohiko Aiki, Adrian Muntean, Large-time behavior of solutions to a thermo-diffusion system with Smoluchowski interactions, 2017, 263, 00220396, 3009, 10.1016/j.jde.2017.04.024
    9. Michal Beneš, 2018, 1978, 0094-243X, 030009, 10.1063/1.5043659
    10. Laurent Desvillettes, Silvia Lorenzani, Homogenization of the discrete diffusive coagulation–fragmentation equations in perforated domains, 2018, 467, 0022247X, 1100, 10.1016/j.jmaa.2018.07.042
    11. Michal Beneš, Igor Pažanin, Homogenization of degenerate coupled fluid flows and heat transport through porous media, 2017, 446, 0022247X, 165, 10.1016/j.jmaa.2016.08.041
    12. Adrian Muntean, Sina Reichelt, Corrector Estimates for a Thermodiffusion Model with Weak Thermal Coupling, 2018, 16, 1540-3459, 807, 10.1137/16M109538X
    13. Bruno Franchi, Silvia Lorenzani, Homogenization of Smoluchowski-type equations with transmission boundary conditions, 2024, 24, 2169-0375, 952, 10.1515/ans-2023-0143
  • Reader Comments
  • © 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Networks and Heterogeneous Media
1.3 1.9

Metrics

Article views(4010) PDF downloads(99) Cited by(13)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog