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Multistability of Convective Flows in a Porous Enclosure

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Sixty Shades of Generalized Continua

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 170))

Abstract

We discuss multistability for the problem of convection in a porous medium. Nontrivial phenomena of extreme multistability arise in the mathematical model of the Darcy convection with a linear temperature profile on the boundary. It manifests in the appearance of one-parameter families of steady states. An illustrative example concerns the anisotropic convection problem. We review here some issues of transition from extreme to standard multistability. Then we describe numerical and algorithmic aspects of the extreme multistability and provide necessary references.

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References

  1. Huang P-W, Wellmann F (2021) An Explanation to the Nusselt–Rayleigh discrepancy in naturally convected porous media, Transport in Porous Media 137(1):195–214. https://doi.org/10.1007/s11242-021-01556-8

  2. Nield DA, Bejan A (2017) Convection in Porous Media, New York, Springer. https://doi.org/10.1007/978-3-319-49562-0

  3. Soboleva EB (2019) A method for numerical simulation of Haline convective flows in porous media as applied to geology, Computational Mathematics and Mathematical Physics 59(11):1893–1903. https://doi.org/10.1134/S0965542519110101

  4. Storesletten L, Rees DAS (2019) Onset of convection in an inclined anisotropic porous layer with internal heat generation, Fluids 4(2):75. https://doi.org/10.3390/fluids4020075

  5. Maryshev B, Lyubimova T, Lyubimov D (2013) Two-dimensional thermal convection in porous enclosure subjected to the horizontal seepage and gravity modulation, Physics of Fluids 25(8):084105. https://doi.org/10.1063/1.4817375

  6. Rionero S (2012) Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive-convective fluid mixtures, Physics of Fluids 24(10):104101. https://doi.org/10.1063/1.4757858

  7. Yudovich VI (1991) Cosymmetry, degeneration of solutions of operator equations, and onset of a filtration convection, Mathematical Notes of the Academy of Sciences of the USSR 49(5):540–545. https://doi.org/10.1007/BF01142654

  8. Yudovich VI (1995) Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it, Chaos 5(2):402–411. https://doi.org/10.1063/1.166110

  9. Yudovich VI (2004) Bifurcations under perturbations violating cosymmetry, Doklady Physics 49(9):522–526. https://doi.org/10.1134/1.1810578

  10. Bratsun DA, Lyubimov DV, Roux B (1995) Co-symmetry breakdown in problems of thermal convection in porous medium, Physica D: Nonlinear Phenomena 82(4):398–417. https://doi.org/10.1016/0167-2789(95)00045-6

  11. Glukhov AF, Putin GF (1999) Experimental study of convective structures in a liquid-saturated porous medium near the threshold of instability of mechanical equilibrium (in Russ.), Hydrodynamics 12:104–120.

    Google Scholar 

  12. Lyubimov DV (1975) Convective motions in a porous medium heated from below, Journal of Applied Mechanics and Technical Physics 16:257–261. https://doi.org/10.1007/BF00858924

  13. Govorukhin VN (1996) Computer experiments with cosymmetric models, ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik 76(Suppl. 4):559–562.

    Google Scholar 

  14. Govorukhin VN (2000) Calculation of one-parameter families of stationary regimes in a cosymmetric case and analysis of plane filtrational convection problem, In: Continuation methods in fluid dynamics (Eds Henry D, Bergeon A), Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 74, pp 133–144.

    Google Scholar 

  15. Abdelhafez MA, Tsybulin VG (2017) Numerical simulation of convective motion in an anisotropic porous medium and cosymmetry conservation, Computational Mathematics and Mathematical Physics 57(10):1706–1719. https://doi.org/10.1134/S0965542517100025

  16. Govorukhin VN (1998) Numerical simulation of the loss of stability for secondary steady regimes in the Darcy plane-convection problem, Doklady Physics 43(12):806–808.

    Google Scholar 

  17. Govorukhin VN (1999) Analysis of families of secondary steady-state regimes in the problem of plane flow through a porous medium in a rectangular vessel, Fluid Dynamics 34(5):652–659.

    Google Scholar 

  18. Govorukhin VN, Shevchenko IV (2003) Numerical investigation of the second transition in the problem of plane convective flow through a porous medium, Fluid Dynamics 38(5):760–771. https://doi.org/10.1023/B:FLUI.0000007838.46669.1a

  19. Govorukhin VN (2014) On the action of internal heat sources on convective motion in a porous medium heated from below, J. Applied Mechanics and Technical Phys. 55(2):225–233. https://doi.org/10.1134/S0021894414020059

  20. Kantur OY, Tsybulin VG (2002) A spectral-difference method for computing convective fluid motions in a porous medium and cosymmetry preservation, Comput. Mathematics and Mathematical Phys. 42(6):878–888.

    Google Scholar 

  21. Karasözen B, Tsybulin VG (1999) Finite-difference approximations and cosymmetry conservation in filtration convection problem, Physics Letters, Section A 262:321–329. https://doi.org/10.1016/S0375-9601(99)00599-X

  22. Karasözen B, Tsybulin VG (2005) Cosymmetry preserving finite-difference methods for convection equations in a porous medium, Applied Numerical Mathematics 55(1):69–82. https://doi.org/10.1016/j.apnum.2004.10.008

  23. Karasözen B, Tsybulin VG (2005) Mimetic discretization of two-dimensional Darcy convection, Computer Physics Communications 167(3):203–213. https://doi.org/10.1016/j.cpc.2004.12.012

  24. Karasözen B, Nemtsev AD, Tsybulin VG (2012) Staggered grids for three dimensional convection of a multicomponent fluid in a porous medium, Computers and Mathematics with Applications 64(6):1740–1751. https://doi.org/10.1016/j.camwa.2012.02.007

  25. Karasözen B, Trofimova AV, Tsybulin VG (2012) Natural convection in porous annular domains: Mimetic scheme and family of steady states, J. Comput. Physics 231(7):2995–3005. https://doi.org/10.1016/j.jcp.2012.01.004

  26. Abdelhafez MA, Tsybulin VG (2018) Anisotropic problem of Darcy convection: family of steady flows and its disintegration during the destruction of cosymmetry, Fluid Dynamics 53(6):738–749. https://doi.org/10.1134/S0015462818060125

  27. Tsybulin VG, Karasözen B (2008) Destruction of the family of steady states in the planar problem of Darcy convection, Physics Letters A 372(35):5639–5643. https://doi.org/10.1016/j.physleta.2008.07.006

  28. Govorukhin VN, Tsybulin VG (2021) Multistability, scattering and selection of equilibria in the mechanical system with constraint, Communications in Nonlinear Science and Numerical Simulation 95:105602. https://doi.org/10.1016/j.cnsns.2020.105602

  29. Govorukhin VN, Shevchenko IV (2013) Selection of steady regimes of a one parameter family in the problem of plane convective flow through a porous medium, Fluid Dynamics 48(4):523–532. https://doi.org/10.1134/S001546281304011X

  30. Karasözen B, Tsybulin VG (2004) Cosymmetric families of steady states in Darcy convection and their collision, Physics Letters, Section A: General, Atomic and Solid State Physics 323(1–2):67–76. https://doi.org/10.1016/j.physleta.2004.01.053

  31. Kantur OYu, Tsibulin VG (2004) Numerical investigation of the plane problem of convection of a multicomponent fluid in a porous medium, Fluid Dynamics 39(3):464–473. https://doi.org/10.1023/B:FLUI.0000038565.09347.ac

  32. Trofimova AV, Tsibulin BG (2011) Convective motions in a porous ring sector, J. Applied Mechanics and Technical Phys. 52(3):427–435. https://doi.org/10.1134/S0021894411030138

  33. Trofimova AV, Tsibulin VG (2014) Filtration convection in an annular domain and branching of a family of steady-state regimes, Fluid Dynamics 49(4):481–490. https://doi.org/10.1134/S0015462814040085

  34. Karasözen B, Nemtsev AD, Tsybulin VG (2008) Staggered grids discretization in three-dimensional Darcy convection, Computer Physics Commun. 178(12):885–893. https://doi.org/10.1016/j.cpc.2008.02.004

  35. Nemtsev AD, Tsibulin VG (2007) Numerical investigation of the first transition in the three-dimensional problem of convective flow in a porous medium, Fluid Dynamics 42(4):637–643. https://doi.org/10.1007/s10697-0070066-y

  36. Abdelhafez MA, Tsybulin VG (2017) Anisotropy effect on the convection of a heat-conducting fluid in a porous medium and cosymmetry of the Darcy problem, Fluid Dynamics 52(1):49–57. https://doi.org/10.1134/S0015462817010057

  37. Abdelhafez MA, Tsybulin VG (2018) Modeling of anisotropic convection for the binary fluid in porous medium, Computer Research and Modeling 10(6):801–816. https://doi.org/10.20537/2076-7633-2018-10-6-801-816

  38. Tsybulin VG, Karasözen B, Ergench T (2006) Selection of steady states in planar Darcy convection, Physics Letters, Section A 356(3):189–194. https://doi.org/10.1016/j.physleta.2006.03.043

  39. Eremeyev VA, Lebedev LP, Altenbach H (2013) Foundations of Micropolar Mechanics, Springer, Heidelberg.

    Google Scholar 

  40. Sumbatyan MA, Scalia A (2005) Equations of Mathematical Diffraction Theory, CRC Press, Boca Raton, Florida.

    Google Scholar 

  41. Govorukhin VN, Shevchenko IV (2006) Scenarios of the onset of unsteady regimes in the problem of plane convective flow through a porous medium, Fluid Dynamics 41(6):967–975. https://doi.org/10.1007/s10697-006-0111-2

  42. Govorukhin VN, Shevchenko IV (2017) Multiple equilibria, bifurcations and selection scenarios in cosymmetric problem of thermal convection in porous medium, Physica D: Nonlinear Phenomena, 361:42–58. https://doi.org/10.1016/j.physd.2017.08.012

  43. Yudovich VI (1996) Implicit function theorem for cosymmetric equations, Mathematical Notes 60(2):235–238. https://doi.org/10.1007/bf02305191

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Authors and Affiliations

  1. Vorovich Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Milchakova Street 8a, 344090, Rostov-on-Don, Russian Federation

    Vasily Govorukhin, Mezhlum Sumbatyan & Vyacheslav Tsybulin

Authors
  1. Vasily Govorukhin
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  2. Mezhlum Sumbatyan
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  3. Vyacheslav Tsybulin
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Corresponding author

Correspondence to Vasily Govorukhin .

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Editors and Affiliations

  1. Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

    Holm Altenbach

  2. Tallinn University of Technology, Tallinn, Estonia

    Arkadi Berezovski

  3. Università degli Studi dell’Aquila, L'Aquila, Italy

    Francesco dell'Isola

  4. Institute of Problems of Mechanical Engineering, Saint Petersburg, Russia

    Alexey Porubov

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Govorukhin, V., Sumbatyan, M., Tsybulin, V. (2023). Multistability of Convective Flows in a Porous Enclosure. In: Altenbach, H., Berezovski, A., dell'Isola, F., Porubov, A. (eds) Sixty Shades of Generalized Continua. Advanced Structured Materials, vol 170. Springer, Cham. https://doi.org/10.1007/978-3-031-26186-2_19

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