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Competitive porous double diffusion with Korteweg stress

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Abstract

A model is developed to describe thermal convection in a mixture of fluids in a porous medium, where the layer is heated from below while simultaneously the fluid density at the base of the porous layer is greater than that higher up. In addition to buoyancy forces which are essentially due to gravity the fluid mixture is subject to Korteweg stresses which arise because of density gradients in the mixture. A complete stability analysis is provided and the critical Rayleigh number for convective motion is derived for both stationary and oscillatory convection and this is complemented with a global energy stability analysis. The analogous problem in a bidisperse porous medium is also briefly discussed.

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References

  1. Truesdell, C., Noll, W.: The Nonlinear Field Theories of Mechanics, 2nd edn. Springer, Cham (1992)

    Google Scholar 

  2. Massoudi, M., Kirwan, A.D.: On the thermodynamics of a nonlinear heat conducting suspension. Fluids 1, 19 (2016)

    ADS  Google Scholar 

  3. Mehrabdi, M.M., Cowin, S.C., Massoudi, M.: Conservation laws and constitutive equations for density-gradient-dependent viscous fluids. Cont. Mech. Thermodyn. 17, 183–200 (2005)

    Google Scholar 

  4. Clingan, H., Rusk, D., Smith, K., Garcia, A.A.: Viscous fingering of miscible liquids in porous and swellable media for rapid diagnostic tests. Bioengineering 5, 94 (2018)

    CAS  PubMed  PubMed Central  Google Scholar 

  5. Galdi, G.P., Joseph, D.D., Preziosi, L., Rionero, S.: Mathematical problems for miscible incompressible fluids with Korteweg stresses. Eur. J. Mech. B/Fluids 10, 253–267 (1991)

    ADS  Google Scholar 

  6. Gentile, M., Rionero, S.: The uniqueness problem for a model of an incompressible fluid mixture. Le Matematiche 46, 159–167 (1991)

    MathSciNet  Google Scholar 

  7. Pojman, J.A., Chekanov, Y., Wyatt, V., Bessonov, N., Volpert, V.: Numerical simulations of convection induced by Korteweg stresses in a miscible polymer–monomer system: effects of variable transport coefficients, polymerization rate and volume changes. Microgravity Sci. Technol. 21, 225–237 (2009)

    CAS  ADS  Google Scholar 

  8. Volpert, V., Pojman, J. A., Picard, R. Texier. C. R.: Convection induced by composition gradients in miscible systems. Mecanique 330, 353–358 (2002)

  9. Bessonov, N., Pojman, J.A., Volpert, V.: Modelling of diffuse interfaces with temperature gradients. J. Eng. Math. 49, 321–338 (2004)

    MathSciNet  Google Scholar 

  10. Bessonov, N., Volpert, V., Pojman, J.A., Zoltowski, B.D.: Numerical simulations of convection induced by Korteweg stresses in miscible polymer–monomer systems. Microgravity Sci. Technol. 17, 8–12 (2005)

    CAS  ADS  Google Scholar 

  11. Pramanik, S., Mishra, M.: Linear stability analyses of Korteweg stresses effect on miscible viscous fingering in porous media. Phys. Fluids 25, 074104 (2013)

    ADS  Google Scholar 

  12. Pramanik, S., Mishra, M.: Comparison of Korteweg stresses effect on the fingering instability of higher or less viscous miscible slices. Chem. Eng. Sci 110, 144–152 (2014)

    CAS  Google Scholar 

  13. Pramanik, S., Mishra, M.: Nonlinear simulations of miscible viscous fingering with gradient stresses in porous media. Chem. Eng. Sci 122, 523–532 (2015)

    CAS  Google Scholar 

  14. Swernath, S., Malengier, B., Pushpavanam, S.: Effect of Korteweg stress on viscous fingering of solute plugs in a porous medium. Chem. Eng. Sci. 65, 2284–2291 (2010)

    CAS  Google Scholar 

  15. Badday, A.J., Harfash, A.J.: Thermosolutal convection in rotating porous media with general boundary conditions. Special Top. Rev. Porous Media 13, 29–48 (2022)

    Google Scholar 

  16. Barletta, A., Nield, D.A.: Thermosolutal convective instability and viscous dissipation effect in a fluid-saturated porous medium. Int. J. Heat Mass Transf. 54, 1641–1648 (2011)

    Google Scholar 

  17. Capone, F., De Luca, R., Massa, G.: The onset of double diffusive convection in a rotating bidisperse porous medium. Eur. Phys. J. Plus 137, 1034 (2022)

    Google Scholar 

  18. Hughes, D.W., Proctor, M.R.E., Eltayeb, I.A.: Maxwell–Cattaneo double diffusive convection: limiting cases. J. Fluid Mech. 927, A13 (2021)

    MathSciNet  CAS  ADS  Google Scholar 

  19. Kumar, G., Narayana, P.A.L., Sahn, K.C.: Linear and nonlinear thermosolutal instabilities in an inclined porous layer. Proc. R. Soc. Lond. A 476, 20190705 (2020)

    MathSciNet  ADS  Google Scholar 

  20. Kuznetsov, A.V., Nield, D.A., Barletta, A., Celli, M.: Local thermal non-equilibrium and heterogeneity effects on the onset of double diffusive convection in an internally heated and soluted porous medium. Trans. Porous Media 109, 393–409 (2015)

    MathSciNet  CAS  Google Scholar 

  21. Nield, D.A., Kuznetsov, A.V., Barletta, A., Celli, M.: The effects of double diffusion and local thermal non-equilibrium on the onset of convection in a layered porous medium: non - oscillatory instability. Trans. Porous Media 107, 261–279 (2015)

    MathSciNet  CAS  Google Scholar 

  22. Nield, D.A., Kuznetsov, A.V.: Do isoflux boundary conditions inhibit oscillatory double: diffusive convection. Transp. Porous Media 112, 609–618 (2016)

    MathSciNet  Google Scholar 

  23. Nield, D.A., Simmons, C.T.: A brief introduction to convection in porous media. Trans. Porous Media 130, 237–250 (2019)

    MathSciNet  Google Scholar 

  24. Noon, N.J., Haddad, S.A.: Stability analysis for rotating double: diffusive convection in the presence of variable gravity and reaction effects: Darcy model. Special Top. Rev. Porous Media 13, 1–22 (2022)

    Google Scholar 

  25. Papanicolaou, N.C., Christov, C.I., Jordan, P.M.: The influence of thermal relaxation on the oscillatory properties of two-gradient convection in a vertical slot. Euro. J. Mech. - B/Fluids 30, 68–75 (2011)

    ADS  Google Scholar 

  26. Rionero, S.: Triple diffusive convection in porous media. Acta Mech. 224, 447–458 (2013)

    MathSciNet  Google Scholar 

  27. Rionero, S.: Heat and mass transfer by convection in multicomponent Navier–Stokes mixtures: absence of subcritical instabilities and global nonlinear stability via the auxiliary system method. Rendiconti Lincei, Matematica e Applicazioni 25, 369–412 (2014)

    MathSciNet  Google Scholar 

  28. Rionero, S.: Cold convection in porous layers salted from above. Meccanica 49, 2061–2068 (2014)

    MathSciNet  Google Scholar 

  29. Rionero, S.: Upper and lower estimates of multicomponent convection instability threshold via auxiliary Bénard problems. Rendiconti Lincei, Matematica e Applicazioni 28, 229–253 (2017)

    MathSciNet  Google Scholar 

  30. Rionero, S.: Heat and mass transfer driven by stratified viscosity and thermal diffusion. Ricerche di Matematica 68, 253–264 (2019)

    MathSciNet  Google Scholar 

  31. Rionero, S.: Hopf bifurcations in dynamical systems. Ricerche di Matematica 68, 811–840 (2019)

    MathSciNet  Google Scholar 

  32. Rionero, S.: Hopf bifurcations in quarternary dynamical systems of rotating thermofluid mixtures, driven by spectrum characteristic coefficients. Ricerche di Matematica 70, 331–346 (2021)

    Google Scholar 

  33. Shankar, B.M., Naveen, S.B., Shivakumara, I.S.: Stability of double-diffusive natural convection in a vertical porous layer. Transp. Porous Media 141, 87–105 (2022)

    MathSciNet  CAS  Google Scholar 

  34. Shivakumara, I.S., Raghunatha, A.: Changes in the onset of double-diffusive local thermal nonequilibrium convection due to the introduction of a of third component. Transp. Porous Media 143, 225–242 (2022)

    CAS  Google Scholar 

  35. Straughan, B.: Effect of temperature upon double diffusive convection in Navier–Stokes–Voigt models with Kazkhikov–Smagulov and Korteweg terms. Appl. Math. Optim. 87, 54 (2023)

    Google Scholar 

  36. Joseph, D.D.: Fluid dynamics of two miscible liquids with slow diffusion and Korteweg stresses. Eur. J. Mech. B/Fluids 9, 565–596 (1990)

    Google Scholar 

  37. Barletta, A.: The Boussinesq approximation for buoyant flows. Mech. Res. Commun. 124, 103939 (2022)

    Google Scholar 

  38. Straughan, B.: The Energy Method, Stability, and Nonlinear Convection. vol. 91, 2nd edn. Springer, New York (2004)

  39. Capone, F., De Luca, R., Gentile, M.: Coriolis effect on thermal convection in a rotating bidispersive porous layer. Proc. R. Soc. Lond. A 476, 20190875 (2020)

    MathSciNet  ADS  Google Scholar 

  40. Capone, F., De Luca, R., Gentile, M.: Penetrative convection in rotating anisotropic bidispersive porous layers. Mech. Res. Commun. 110, 103601 (2020)

    Google Scholar 

  41. Capone, F., Massa, G.: The effects of Vadasz term, anisotropy and rotation on bidisperse convection. Int. J. Nonlinear Mech. 135, 103749 (2021)

    ADS  Google Scholar 

  42. Capone, F., Massa, G., Gentile, M.: The onset of thermal convection in anisotropic and rotating bidisperse porous media. ZAMP 72, 169 (2021)

    MathSciNet  CAS  ADS  Google Scholar 

  43. Gentile, M., Straughan, B.: Bidispersive thermal convection with relatively large macropores. J. Fluid Mech. 898, A14 (2020)

    MathSciNet  ADS  Google Scholar 

  44. Ramchandraiah, C., Kishnan, N., Reddy, G.S.K., Paidpati, K.K., Chesneau, C.: Double-diffusive convection in bidisperse porous media with Coriolis effect. Math. Comput. Appl. 27, 56 (2022)

    Google Scholar 

  45. Straughan, B.: Convection with Local Thermal Non-equilibrium and Microfluidic Effects. Advances in Mechanics and Mathematics Series, vol. 32. Springer, Cham, Switzerland (2015)

  46. Zhou, T., Ioannidou, K., Masoero, E., Mirzadeh, M., Pellenq, R.J.M., Bazant, M.Z.: Capillary stresses and structural relaxation in moist granular materials. Langmuir 35, 4397–4402 (2019)

    CAS  PubMed  Google Scholar 

  47. Zhou, T., Ioannidou, K., Ulm, F.J., Bazant, M.Z., Pellenq, R.J.M.: Multiscale poromechanics of wet cement paste. Proc. Natl. Acad. Sci. 116, 10652–10657 (2019)

    CAS  PubMed  PubMed Central  ADS  Google Scholar 

  48. Falsaperla, P., Mulone, G., Straughan, B.: Bidispersive-inclined convection. Proc. R. Soc. Lond. A 472, 20160480 (2016)

    MathSciNet  ADS  Google Scholar 

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Acknowledgements

This work was supported by an Emeritus Fellowship of the Leverhulme Trust, EM-2019-022/9. I am indebted to an anonymous referee whose pointed observations led to improvements in an earlier manuscript.

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  1. Department of Mathematics, University of Durham, Durham, DH1 3LE, UK

    Brian Straughan

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Straughan, B. Competitive porous double diffusion with Korteweg stress. Ricerche mat 73 (Suppl 1), 293–307 (2024). https://doi.org/10.1007/s11587-023-00790-0

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