Abstract
In this article, the onset of double-diffusive convection with the effect of viscous dissipation in a horizontal fluid-saturated porous layer is examined. Two impermeable isothermal and isosolutal walls bound the porous layer, and Darcy’s law models the flow. The onset of convective instability is studied by two approaches: the linear stability analysis and the nonlinear stability analysis. The nonlinear stability analysis is performed by utilizing the energy method. The literature on the nonlinear stability analysis of onset of convective instability with the viscous dissipation effect is limited. The present article aims to fill this gap. It is observed that, when the fluid is at rest, the effect of viscous dissipation does not influence the critical thermal Rayleigh number corresponding to both the linear and nonlinear stability analyses. Moreover, sub-critical instabilities do not occur when \(Ra_{S}>0\). But in contrast, the region of sub-critical instabilities increases along the negative \(Ra_{S}\) direction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barletta, A., Celli, M., Rees, D.A.S.: Darcy–Forchheimer flow with viscous dissipation in a horizontal porous layer: onset of convective instabilities. J. Heat Trans. 131(7), (2009)
Barletta, A.: On the thermal instability induced by viscous dissipation. Int. J. Therm. Sci. 88, 238–247 (2015)
Barletta, A., Mulone, G.: The energy method analysis of the Darcy-Bénard problem with viscous dissipation. Continuum Mech. Thermodyn. 33(1), 25–33 (2021)
Barletta, A., Nield, D.A.: Thermosolutal convective instability and viscous dissipation effect in a fluid-saturated porous medium. Int. J. Heat Mass Transf. 54(7–8), 1641–1648 (2011)
Barletta, A., Storesletten, L.: Viscous dissipation and thermoconvective instabilities in a horizontal porous channel heated from below. Int. J. Therm. Sci. 49(4), 621–630 (2010)
Barletta, A., Celli, M., Rees, D.A.S.: The onset of convection in a porous layer induced by viscous dissipation: a linear stability analysis. Int. J. Heat Mass Transf. 52(1–2), 337–344 (2009)
Barletta, A., Rossi di Schio, E., Storesletten, L.: Convective roll instabilities of vertical throughflow with viscous dissipation in a horizontal porous layer. Transp. Porous Media 81, 461–477 (2010)
Deepika, N.: Linear and nonlinear stability of double-diffusive convection with the Soret effect. Transp. Porous Media 121, 93–108 (2018)
Deepika, N., Narayana, P.A.L.: Linear stability of double diffusive convection of Hadley-prats flow with viscous dissipation in a porous media. Procedia Eng. 127, 193–200 (2015)
Deepika, N., Murthy, P.V.S.N., Narayana, P.A.L.: The effect of magnetic field on the stability of double-diffusive convection in a porous layer with horizontal mass throughflow. Transp. Porous Media 134, 435–452 (2020)
Gebhart, B.: Effects of viscous dissipation in natural convection. J. Fluid Mech. 14(2), 225–232 (1962)
Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice-Hall Inc, Englewood Cliffs (1963)
Guo, J., Kaloni, P.N.: Double-diffusive convection in a porous medium, nonlinear stability, and the Brinkman effect. Stud. Appl. Math. 94(3), 341–358 (1995)
Guo, J., Kaloni, P.N.: Nonlinear stability of convection induced by inclined thermal and solutal gradients. Zeitschrift für angewandte Mathematik und Physik ZAMP 46(5), 645–654 (1995)
Joseph, D.D.: Nonlinear stability of the Boussinesq equations by the method of energy. Arch. Ration. Mech. Anal. 22, 163–184 (1966)
Kumar, G., Narayana, P.A.L., Sahu, K.C.: Linear and nonlinear thermosolutal instabilities in an inclined porous layer. Proceed. R. Soc. A 476(2233), 20190705 (2020)
Love, A. J., Simmons, C. T., Nield, D.A.: Double-diffusive convection in groundwater wells. Water Resour. Res. 43(8), (2007)
Nield, D.A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 4(3), 553–560 (1968)
Nield, D.A., Bejan, A.: Convection in porous media, vol. 3. Springer, New York (2006)
Nield, D.A., Kuznetsov, A.V.: Do isoflux boundary conditions inhibit oscillatory double-diffusive convection? Transp. Porous Media 112(3), 609–618 (2016)
Nield, D.A., Barletta, A., Celli, M.: The effect of viscous dissipation on the onset of convection in an inclined porous layer. J. Fluid Mech. 679, 544–558 (2011)
Rionero, S., Mulone, G.: On the non-linear stability of a thermo-diffusive fluid mixture in a mixed problem. J. Math. Anal. Appl. 124(1), 165–188 (1987)
Straughan, B.: Stability and wave motion in porous media (Vol. 165). Springer, Berlin (2008)
Straughan, B.: The energy method, stability, and nonlinear convection, vol. 91. Springer, Berlin (2013)
Straughan, B.: Nonlinear stability for thermal convection in a Brinkman porous material with viscous dissipation. Transp. Porous Media 134(2), 303–314 (2020)
Trevisan, O.V., Bejan, A.: Natural convection with combined heat and mass transfer buoyancy effects in a porous medium. Int. J. Heat Mass Trans. 28(8), 1597–1611 (1985)
Trevisan, O.V., Bejan, A.: Combined heat and mass transfer by natural convection in a porous medium. Adv. Heat Trans. 20, 315–352 (1990)
Wooding, R.A.: Rayleigh instability of a thermal boundary layer in flow through a porous medium. J. Fluid Mech. 9(2), 183–192 (1960)
Wooding, R.A.: Convection in a saturated porous medium at large Rayleigh number or Peclet number. J. Fluid Mech. 15(4), 527–544 (1963)
Acknowledgements
The authors thank the reviewers for their constructive comments for the improvement of the paper to a great extent.
Funding
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Deepika, N., Narayana, P.A.L. & Hill, A.A. The Nonlinear Stability Analysis of Double-Diffusive Convection with Viscous Dissipation Effect. Transp Porous Med 150, 215–227 (2023). https://doi.org/10.1007/s11242-023-02006-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-023-02006-3