Skip to main content
SpringerLink
Log in

Lack of Thermal Equilibrium Effects on Convection in a Porous Medium saturated with an Ellis Fluid

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

The onset of thermal convection in an Ellis fluid-saturated porous medium in the presence of a horizontal pressure gradient yielding a basic throughflow and subject to a lack of thermal equilibrium between the fluid and solid phases is investigated. The eigenvalue problem is solved analytically and the condition for the onset of convection is obtained. It is found that the instability sets in as oscillatory mode contrary to the manifestation of stationary convection in a Newtonian fluid-saturated porous layer. The effects of local thermal nonequilibrium and Ellis fluid parameters are discussed on the onset of convection. It is found that an increase in the Ellis power-law index and the scaled interphase heat transfer coefficient is to delay the onset of convection, while an opposite behaviour is observed with an increase in the Darcy–Ellis number and the porosity-modified conductivities ratio. The pressure lines and isotherms are presented at the critical state. The results delineated under the limiting cases are shown to be in excellent agreement with those published previously.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Alazmi, B., Vafai, K.: Analysis of variants within the porous media transport models. J. Heat Transfer 122(2), 303–326 (2000)

    Article  Google Scholar 

  • Alazmi, B., Vafai, K.: Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer. Int. J. Heat Mass Transf. 44(9), 1735–1749 (2001)

    Article  Google Scholar 

  • Alloui, Z., Khelifa, N.B., Beji, H., Vasseur, P.: Onset of convection in a horizontal porous layer saturated by a power-law fluid. J. Heat Transf. 134(9), 092502–092510 (2012)

    Article  Google Scholar 

  • Alves, L.S.D.B., Barletta, A.: Convective instability of the Darcy-Bénard problem with through flow in a porous layer saturated by a power-law fluid. Int. J. Heat Mass Transf. 62, 495–506 (2013)

    Article  Google Scholar 

  • Alves, L.S.D.B., Barletta, A.: Convective to absolute instability transition in the Prats flow of a power-law fluid. Int. J. Therm. Sci. 94, 270–282 (2015)

    Article  Google Scholar 

  • Amiri, A., Vafai, K.: Analysis of dispersion effects and non-thermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media. Int. J. Heat Mass Transf. 37, 939–954 (1994)

    Article  Google Scholar 

  • Amiri, A., Vafai, K.: Transient analysis of incompressible flow through a packed bed. Int. J. Heat Mass Transf. 41, 4259–4279 (1998)

    Article  Google Scholar 

  • Andreozzi, A., Brunese, L., Iasiello, M., Tucci, C., Vanoli, G.P.: Bioheat transfer in a spherical biological tissue: a comparison among various models. J. Phys.: Conf. Ser. 1224, 012001 (2019)

    Google Scholar 

  • Banu, N., Rees, D.A.S.: Onset of Darcy-Benard convection using a thermal non-equilibrium model. Int. J. Heat Mass Transf. 45(11), 2221–2228 (2002)

    Article  Google Scholar 

  • Barletta, A., Nield, D.A.: Linear instability of the horizontal throughflow in a plane porous layer saturated by a power-law fluid. Phys. Fluids 23(1), 013102 (2011)

    Article  Google Scholar 

  • Barletta, A., Storesletten, L.: Linear instability of the vertical throughflow in a horizontal porous layer saturated by a power-law fluid. Int. J. Heat Mass Transf. 99, 293–302 (2016)

    Article  Google Scholar 

  • Bird, R.B.: Experimental tests of generalized Newtonian models containing a zero-shear viscosity and a characteristic time. Can J Chem Eng. 43(4), 161–168 (1965)

    Article  Google Scholar 

  • Bird, R.B., Armstrong, R.C., Hassager, O.: Dynamics of polymeric liquids. Vol. 1: Fluid mechanics (1987)

  • Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport phenomena, Revised 2nd edition (2006)

  • Celli, M., Barletta, A., Brandão, P.V.: Rayleigh-Bénard Instability of an Ellis fluid saturating a porous Medium. Transp. Porous Med. 138(3), 679–692 (2021)

    Article  Google Scholar 

  • Christopher, R.H., Middleman, S.: Power-law flow through a packed tube. Ind. Eng. Chem. Fundam. 4(4), 422–426 (1965)

    Article  Google Scholar 

  • Di Federico, V., Pinelli, M., Ugarelli, R.: Estimates of effective permeability for non-Newtonian fluid flow in randomly heterogeneous porous media. Stoch. Environ. Res. Risk Assess. 24(7), 1067–1076 (2010)

    Article  Google Scholar 

  • Dutta, J., Kundu, B., Yook, S.J.: Three-dimensional thermal assessment in cancerous tumors based on local thermal non-equilibrium approach for hyperthermia treatment. Int. J. Therm. Sci. 159, 106591 (2021)

    Article  Google Scholar 

  • Horton, C.W., Rogers, F.T.: Convection currents in a porous medium. J. Appl. Phys. 16, 367–370 (1945)

    Article  Google Scholar 

  • Kang, J., Fu, C., Tan, W.: Thermal convective instability of viscoelastic fluids in a rotating porous layer heated from below. J. Nonnewton Fluid Mech. 166(1–2), 93–101 (2011)

    Article  Google Scholar 

  • Kim, M.C., Lee, S.B., Kim, S., Chung, B.J.: Thermal instability of viscoelastic fluids in porous media. Int. J. Heat Mass Transf. 46(26), 5065–5072 (2003)

    Article  Google Scholar 

  • Kumari, S., Murthy, P.V.S.N.: Convective stability of vertical throughflow of a non-Newtonian fluid in a porous channel with Soret effect. Transp. Porous Media 122(1), 125–143 (2018)

    Article  Google Scholar 

  • Kumari, S., Murthy, P.V.S.N.: Stability of the Horizontal throughflow in a power-law fluid saturated porous layer. Transp. Porous Media 129(3), 653–672 (2019a)

    Article  Google Scholar 

  • Kumari, S., Murthy, P.V.S.N.: Stability of the horizontal throughflow of a power-law fluid in a double-diffusive porous layer under convective boundary conditions. Int. J. Therm. Sci. 146, 106098 (2019b)

    Article  Google Scholar 

  • Lapwood, E.R.: Convection of a fluid in a porous medium. Proc. Camb. Philos. Soc. 44, 508–521 (1948)

    Article  Google Scholar 

  • Malashetty, M.S.: Effect of thermal/gravity modulation on the onset of Rayleigh-Bénard convection in a couple stress fluid. Int. J. Trans. Phenom. 7, 31–44 (2005)

    Google Scholar 

  • Minkowycz, W.J., Haji-Sheikh, A., Vafai, K.F.: On departure from local thermal equilibrium in porous media due to a rapidly changing heat source: the sparrow number. Int. J. Heat Mass Transf. 42, 3373–3385 (1999)

    Article  Google Scholar 

  • Nakayama, A., Kuwahara, F.: A general bioheat transfer model based on the theory of porous media. Int. J. Heat Mass Transf. 51, 3190–3199 (2008)

    Article  Google Scholar 

  • Nield, D.A.: Effects of local thermal nonequilibrium in steady convective processes in a saturated porous medium: forced convection in a channel. J. Porous Media 1, 181–186 (1998)

    Google Scholar 

  • Nield, D.A.: A further note on the onset of convection in a layer of a porous medium saturated by a non-Newtonian fluid of power-law type. Transp. Porous Media 88(2), 187–191 (2011a)

    Article  Google Scholar 

  • Nield, D.A.: A note on the onset of convection in a layer of a porous medium saturated by a non-Newtonian nanofluid of power-law type. Transp. Porous Media 87(1), 121–123 (2011b)

    Article  Google Scholar 

  • Nield, D.A., Bejan, A.: Heat transfer through a porous medium. In: Convection in porous media, pp. 01–55. Springer, Cham (2017)

    Chapter  Google Scholar 

  • Postelnicu, A.: The effect of a horizontal pressure gradient on the onset of a Darcy-Bénard convection in thermal non-equilibrium conditions. Int. J. Heat Mass Transf. 53(1–3), 68–75 (2010)

    Article  Google Scholar 

  • Rees, D.A.S.: The effect of inertia on the onset of mixed convection in a porous layer heated from below. Int. Commun. Heat Mass Transf. 24(2), 277–283 (1997)

    Article  Google Scholar 

  • Rudraiah, N., Kaloni, P.N., Radhadevi, P.V.: Oscillatory convection in a viscoelastic fluid through a porous layer heated from below. Rheol. Acta 28(1), 48–53 (1989)

    Article  Google Scholar 

  • Sharma, R.C.: Effect of suspended particles on couple-stress fluid heated from below in the presence of rotation and magnetic field. Ind. J. Pure Appl. Math. 35, 973–989 (2004)

    Google Scholar 

  • Sharma, R.C., Pal, M.: On couple-stress fluid heated from below in porous medium in the presence of rotation. Appl. Mech. Eng. 5(4), 883–896 (2000)

    Google Scholar 

  • Shenoy, A.V.: Non-Newtonian fluid heat transfer in porous media. In: Advances in Heat transfer, vol. 24, pp. 101–190. Elsevier, Netherlands (1994)

    Google Scholar 

  • Shivakumara, I.S.: Onset of convection in a couple-stress fluid-saturated porous medium: effects of non-uniform temperature gradients. Arch. Appl. Mech. 80(8), 949–957 (2010)

    Article  Google Scholar 

  • Shivakumara, I.S., Sureshkumar, S.: Convective instabilities in a viscoelastic-fluid-saturated porous medium with throughflow. J. Geophys. Eng. 4(1), 104 (2007)

    Article  Google Scholar 

  • Shivakumara, I.S., Lee, J., Kumar, S.S., Devaraju, N.: Linear and nonlinear stability of double diffusive convection in a couple stress fluid–saturated porous layer. Arch. Appl. Mech. 81(11), 1697–1715 (2011a)

    Article  Google Scholar 

  • Shivakumara, I.S., Sureshkumar, S., Devaraju, N.: Coriolis effect on thermal convection in a couple-stress fluid-saturated rotating rigid porous layer. Arch. Appl. Mech. 81(4), 513–530 (2011b)

    Article  Google Scholar 

  • Shivakumara, I.S., Dhananjaya, M., Ng, C.O.: Thermal convective instability in an Oldroyd-B nanofluid saturated porous layer. Int. J. Heat Mass Transf. 84, 167–177 (2015)

    Article  Google Scholar 

  • Silva, R.A., Assato, M., de Lemos, M.J.: Mathematical modeling and numerical results of power-law fluid flow over a finite porous medium. Int. J. Therm. Sci. 100, 126–137 (2016)

    Article  Google Scholar 

  • Straughan, B.: Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid. Proc. R. Soc. London Ser. A 469, 20130187 (2013)

    Google Scholar 

  • Straughan, B.: Convection with local thermal non-equilibrium and microfluidic effects, vol. 32. Springer, Cham (2015)

    Google Scholar 

  • Vafai, K., Sozen, M.: Analysis of energy and momentum transport for fluid flow through a porous bed. J. Heat Transfer 112, 690–699 (1990)

    Article  Google Scholar 

  • Virto, L., Carbonell, M., Castilla, R., Gamez-Montero, P.J.: Heating of saturated porous media in practice: several causes of local thermal non-equilibrium. Int. J. Heat Mass Transf. 52, 5412–5422 (2009)

    Article  Google Scholar 

  • Zhang, Y.: Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues. Int. J. Heat Mass Transf. 52(21–22), 4829–4834 (2009)

    Article  Google Scholar 

  • Zhang, Z., Fu, C., Tan, W.: Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below. Phys. Fluids 20(8), 084103 (2008)

    Article  Google Scholar 

  • Zhuang, Y.J., Yu, H.Z., Zhu, Q.Y.: A thermal non-equilibrium model for 3D double diffusive convection of power-law fluids with chemical reaction in the porous medium. Int. J. Heat Mass Transf. 115, 670–694 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the reviewers profusely for their constructive comments which helped in improving the paper considerably.

Funding

 There are no funders to report for this submission.

Author information

Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bangalore, 560 056, India

    I. S. Shivakumara

  2. Department of Mathematics, Davangere University, Davangere, 577 002, India

    K. R. Raghunatha & Y. Vinod

  3. Department of Mathematics, Bangalore Institute of Technology, Bangalore, 560 004, India

    M. Dhananjaya

Authors
  1. I. S. Shivakumara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. R. Raghunatha
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Dhananjaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Y. Vinod
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. R. Raghunatha.

Ethics declarations

Conflict of interest

 No conflict of interest exists. We wish to confrm that there are no known conficts of interest associated with this publication.

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shivakumara, I.S., Raghunatha, K.R., Dhananjaya, M. et al. Lack of Thermal Equilibrium Effects on Convection in a Porous Medium saturated with an Ellis Fluid. Transp Porous Med 146, 703–720 (2023). https://doi.org/10.1007/s11242-022-01881-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-022-01881-6

Keywords

Search

Navigation