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A study on the influence of a local thermal non-equilibrium on the onset of Darcy–Bénard convection in a liquid-saturated anisotropic porous medium

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Abstract

The present study investigates the onset of Darcy–Bénard convection in a liquid-saturated anisotropic porous medium when phases are in local thermal non-equilibrium (LTNE) analytically. Flow is small-scale convective motion, and the Boussinesq approximation is valid. A thermodynamically more accurate two-phase energy model (LTNE model) is used in the study, and the results will agree with the system’s thermodynamical behavior. The solid and liquid phases thermal conductivity and permeability of the medium are heterogeneous along the principal axes with invariant along the vertical direction. Oscillatory convection is not possible due to the validation of the principle of exchange of stabilities. The weighted residue Galerkin technique is employed to obtain the expression of the thermal Rayleigh number. The heterogeneity in thermal conductivity of a solid phase significantly affects the onset of convection for higher values of the inter-phase heat transfer coefficient. The LTNE effect ceases at higher rates of inter-phase heat transfer coefficient, gives rise to the thermal equilibrium situations, and results coincide with that of locally thermal equilibrium (LTE) case. Quadratic heterogeneity significantly impacts the convective instability in the LTE case and stabilizes the system due to the rapid exchange of heat between the liquid and solid phases. The ratio of thermal conductivities destabilizes the system, whereas the opposite phenomenon is observed for the inter-phase heat transfer coefficient.

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References

  1. Horton CW, Rogers JF. Convection currents in a porous medium. J Appl Phys. 1945;16:367–70.

    Article  Google Scholar 

  2. Lapwood E. Convection of a fluid in a porous medium. In: Mathematical Proceedings of the Cambridge Philosophical Society. vol. 44. Cambridge University Press; 1948. p. 508–21.

  3. Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. Oxford, UK: Clarendon Press; 1961.

    Google Scholar 

  4. Lappa M. Thermal Convection: Patterns, Evolution and Stability. United Kingdom: John Wiley & Sons; 2009.

    Book  Google Scholar 

  5. Nield DA, Bejan A. Convection in Porous Media. New Zealand: Springer; 2006.

    Google Scholar 

  6. Vafai K. Handbook of Porous Media. New York: CRC Press; 2015.

    Book  Google Scholar 

  7. Quintard M, Kaviany M, Whitaker S. Two-medium treatment of heat transfer in porous media: numerical results for effective properties. Adv Water Resour. 1997;20(2–3):77–94.

    Article  Google Scholar 

  8. Banu N, Rees DAS. Onset of Darcy-Bénard convection using a thermal non-equilibrium model. Int J Heat Mass Transfer. 2002;45:2221–8.

    Article  Google Scholar 

  9. Barletta A, Rees DAS. Local thermal non-equilibrium effects in the Darcy-Bénard instability with isoflux boundary conditions. Int J Heat Mass Transfer. 2012;55(1):384–94.

    Article  Google Scholar 

  10. Postelnicu A, Rees DAS. The onset of Darcy-Brinkman convection in a porous layer using a thermal nonequlibrium model part I: stress-free boundaries. Int J Eng Res. 2003;27:961–73.

    Google Scholar 

  11. Postelnicu A. The onset of a Darcy-Brinkman convection using a thermal nonequilibrium model. Part II Int J Therm Sci. 2008;47:1587–94.

    Article  Google Scholar 

  12. Malashetty MS, Shivakumara IS, Kulkarni S. The onset of convection in an anisotropic porous layer using a thermal non-equilibrium model. Transp Porous Media. 2005;60:199–215.

    Article  Google Scholar 

  13. Malashetty MS, Shivakumara IS, Kulkarni S. The onset of Lapwood-Brinkman convection using a thermal non-equilibrium model. Int J Heat Mass Transfer. 2005;48:1155–63.

    Article  Google Scholar 

  14. Siddheshwar PG, Siddabasappa C. Linear and weakly nonlinear stability analyses of two-dimensional, steady Brinkman-Bénard convection using local thermal non-equilibrium model. Transp Porous Media. 2017;120(3):605–31.

    Article  CAS  Google Scholar 

  15. Bhadauria BS, Agarwal S. Convective transport in a nanofluid saturated porous layer with thermal non equilibrium model. Transp Porous Media. 2011;88(1):107–31.

    Article  CAS  Google Scholar 

  16. Siddheshwar PG, Sakshath TN. Steady finite-amplitude Rayleigh–Bénard convection of ethylene glycol–copper nanoliquid in a high-porosity medium made of 30% glass fiber-reinforced polycarbonate. J Therm Anal Calorim. 2020;1–18.

  17. Siddheshwar PG, Siddabasappa C. Unsteady natural convection in a liquid-saturated porous enclosure with local thermal non-equilibrium effect. Meccanica. 2020;55(9):1763–80.

    Article  Google Scholar 

  18. Siddabasappa C, Sakshath TN. Effect of thermal non-equilibrium and internal heat source on Brinkman-Bénard convection. Physica A. 2021;566:125617.

    Article  CAS  Google Scholar 

  19. Barnoon P, Toghraie D. Numerical investigation of laminar flow and heat transfer of non-Newtonian nanofluid within a porous medium. Powder Technol. 2018;325:78–91.

    Article  CAS  Google Scholar 

  20. Mirzaeyan M, Toghraie D. Numerical investigation of laminar heat transfer and nanofluid flow between two porous horizontal concentric cylinders. J Cent South Univ. 2019;26(7):1976–99.

    Article  CAS  Google Scholar 

  21. Moradi A, Toghraie D, Isfahani AHM, Hosseinian A. An experimental study on MWCNT-water nanofluids flow and heat transfer in double-pipe heat exchanger using porous media. J Therm Anal Calorim. 2019;137(5):1797–807.

    Article  CAS  Google Scholar 

  22. Nazari S, Toghraie D. Numerical simulation of heat transfer and fluid flow of Water-CuO Nanofluid in a sinusoidal channel with a porous medium. Physica E. 2017;87:134–40.

    Article  CAS  Google Scholar 

  23. Shahsavar A, Entezari S, Toghraie D, Barnoon P. Effects of the porous medium and water-silver biological nanofluid on the performance of a newly designed heat sink by using first and second laws of thermodynamics. Chin J Chem Eng. 2020;28(11):2928–37.

    Article  CAS  Google Scholar 

  24. Toghraie D, Mahmoudi M, Akbari OA, Pourfattah F, Heydari M. The effect of using water/CuO nanofluid and L-shaped porous ribs on the performance evaluation criterion of microchannels. J Therm Anal Calorim. 2019;135(1):145–59.

    Article  CAS  Google Scholar 

  25. Barnoon P, Toghraie D, Rostami S. Optimization of heating-cooling generators with porous components/cryogenic conductors on natural convection in a porous enclosure: using different two-phase models and single-phase model and using different designs. Int Commun Heat Mass. 2020;111:104472.

    Article  CAS  Google Scholar 

  26. Sheikholeslami M. New computational approach for exergy and entropy analysis of nanofluid under the impact of Lorentz force through a porous media. Comput Methods Appl Mech Eng. 2019;344:319–33.

    Article  Google Scholar 

  27. Shehzad S, Sheikholeslami M, Ambreen T, Shafee A. Convective MHD flow of hybrid-nanofluid within an elliptic porous enclosure. Phys Lett A. 2020;384(28): 126727.

    Article  CAS  Google Scholar 

  28. Castinel G, Combarnous M. Critere d’apparition de la convection naturelle dans une couche poreuse anisotrope horizontal. CR Acad Sci B. 1974;278:701–4.

    Google Scholar 

  29. Epherre J. Critère d’apparition de la convection naturelle dans une couche poreuse anisotrope. Rev Gen Thermique. 1975;168:949–50.

    Google Scholar 

  30. Aly AM, Ahmed SE. An incompressible smoothed particle hydrodynamics method for natural/mixed convection in a non-Darcy anisotropic porous medium. Int J Heat Mass Transfer. 2014;77:1155–68.

    Article  Google Scholar 

  31. Ennis-King J, Preston I, Paterson L. Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. Phys Fluids. 2005;17(8):084107.

    Article  Google Scholar 

  32. Kvernvold O, Tyvand PA. Nonlinear thermal convection in anisotropic porous media. J Fluid Mech. 1979;90(4):609–24.

    Article  Google Scholar 

  33. Mahajan A, Nandal R. Anisotropic porous penetrative convection for a local thermal non-equilibrium model with Brinkman effects. Int J Heat Mass Transfer. 2017;115:235–50.

    Article  Google Scholar 

  34. Malashetty MS, Basavaraja D. Rayleigh-Bénard convection subject to time dependent wall temperature/gravity in a fluid-saturated anisotropic porous medium. Heat Mass Transfer. 2002;38:551–63.

    Article  CAS  Google Scholar 

  35. Malashetty MS, Basavaraja D. The effect of thermal/gravity modulation on the onset of convection in a horizontal anisotropic porous layer. Int J Appl Mech Eng. 2003;8:425–39.

    Google Scholar 

  36. Malashetty MS, Basavaraja D. Effect of time-periodic boundary temperatures on the onset of double diffusive convection in a horizontal anisotropic porous layer. Int J Heat Mass Transfer. 2004;47:2317–27.

    Article  Google Scholar 

  37. Rees DAS, Postelnicu A. The onset of convection in an inclined anisotropic porous layer. Int J Heat Mass Transfer. 2001;44:4127–38.

    Article  Google Scholar 

  38. Shivakumara IS, Lee J, Mamatha AL, Ravisha M. Boundary and thermal non-equilibrium effects on convective instability in an anisotropic porous layer. J Mech Sci Technol. 2011;25:911–21.

    Article  Google Scholar 

  39. Yadav D. The density-driven nanofluid convection in an anisotropic porous medium layer with rotation and variable gravity field: a numerical investigation. J Appl Comput Mech. 2020;6(3):699–712.

    Google Scholar 

  40. Chandra H, Bera P, Sharma AK. Natural convection in a square cavity filled with an anisotropic porous medium due to sinusoidal heat flux on horizontal walls. Num Heat Transfer A. 2020;77(3):317–41.

    Article  CAS  Google Scholar 

  41. Storesletten L. Effects of anisotropy on convective flow through porous media. In: Transport Phenomena in Porous Media. Pergamon, Oxford; 1998. p. 261–83.

  42. Storesletten L. Effects of anisotropy on convection in horizontal and inclined porous layers. In: Emerging Technologies and Techniques in Porous Media. vol. 134. Springer; 2004. p. 285–306.

  43. Tyvand PA, Storesletten L. Onset of convection in an anisotropic porous layer with vertical principal axes. Transp Porous Media. 2015;108(3):581–93.

    Article  Google Scholar 

  44. Siddheshwar PG, Siddabasappa C, Laroze D. Küppers-Lortz instability in the rotating Brinkman-Bénard problem. Transp Porous Media. 2020;132:465–93.

    Article  Google Scholar 

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  1. Department of Mathematics and Statistics, M. S. Ramaiah University of Applied Sciences, Bangalore, 560 058, India

    C. Siddabasappa

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Siddabasappa, C. A study on the influence of a local thermal non-equilibrium on the onset of Darcy–Bénard convection in a liquid-saturated anisotropic porous medium. J Therm Anal Calorim 147, 5937–5947 (2022). https://doi.org/10.1007/s10973-021-10919-8

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