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The energy method analysis of the Darcy–Bénard problem with viscous dissipation

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Abstract

A nonlinear analysis of the effect of viscous dissipation on the Rayleigh–Bénard instability in a fluid saturated porous layer is performed. The saturated medium is modelled through Darcy’s law, with the layer bounded by two parallel impermeable walls kept at different uniform temperatures, so that heating from below is supplied. While it is well known that viscous dissipation does not influence the linear threshold to instability, a rigorous nonlinear analysis of the instability when viscous dissipation is taken into account is still lacking. This paper aims to fill this gap. The energy method is employed to prove the nonlinear conditional stability of the basic conduction state. In other words, it is shown that a finite initial perturbation exponentially decays in time provided that its initial amplitude is smaller than a given finite value.

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References

  1. Gray, D.D., Giorgini, A.: The validity of the Boussinesq approximation for liquids and gases. Int. J. Heat Mass Transf. 19, 545–551 (1976)

    Article  Google Scholar 

  2. Barletta, A.: On the thermal instability induced by viscous dissipation. Int. J. Therm. Sci. 88, 238–247 (2015)

    Article  Google Scholar 

  3. Straughan, B.: The Energy Method, Stability, and Nonlinear Convection. Springer, Berlin (2004)

    Book  Google Scholar 

  4. Straughan, B.: Stability and Wave Motion in Porous Media. Springer, Berlin (2008)

    MATH  Google Scholar 

  5. Nield, D.A., Bejan, A.: Convection in Porous Media, 5th edn. Springer, New York (2017)

    Book  Google Scholar 

  6. Barletta, A.: Thermal instabilities in a fluid saturated porous medium. In: Öchsner, A., Murch, G.E. (eds.) Heat Transfer in Multi-phase Materials, pp. 381–414. Springer, New York (2011)

    Google Scholar 

  7. Barletta, A.: Routes to Absolute Instability in Porous Media. Springer, New York (2019)

    Book  Google Scholar 

  8. Celli, M., Alves, L.S.B., Barletta, A.: Nonlinear stability analysis of Darcy’s flow with viscous heating. Proc. R. Soc. A Math. Phys. Eng. Sci. 472, 20160036 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  9. Rees, D.A.S., Magyari, E.: Hexagonal cell formation in Darcy–Bénard convection with viscous dissipation and form drag. Fluids 2(2), 27 (2017)

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  12. Rees, D.A.S.: The stability of Darcy–Bénard convection. In: Vafai, K., Hadim, H.A. (eds.) Handbook of Porous Media, pp. 521–558. CRC Press, New York (2000)

    Google Scholar 

  13. Tyvand, P.A.: Onset of Rayleigh–Bénard convection in porous bodies. In: Ingham, D.B., Pop, I. (eds.) Transport Phenomena in Porous Media II, pp. 82–112. Pergamon, New York (2002)

    Chapter  Google Scholar 

  14. Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)

    Article  Google Scholar 

  15. Bebendorf, M.: A note on the Poincaré inequality for convex domains. Z. Anal. Anwend. 22, 751–756 (2003)

    Article  MathSciNet  Google Scholar 

  16. Acosta, G., Durán, R.G.: An optimal Poincaré inequality in \(L^1\) for convex domains. Proc. Am. Math. Soc. 132, 195–202 (2004)

    Article  Google Scholar 

  17. Mikhlin, S.G.: The Problem of the Minimum of a Quadratic Functional. Holden-Day, San Francisco (1965)

    MATH  Google Scholar 

  18. Mulone, G., Salemi, F.: On the existence of hydrodynamic motion in a domain with free boundary type conditions. Meccanica 18, 136–144 (1983)

    Article  ADS  Google Scholar 

  19. Mulone, G., Rionero, S.: Necessary and sufficient conditions for nonlinear stability in the magnetic Bénard problem. Arch. Ration. Mech. Anal. 166, 197–218 (2003)

    Article  MathSciNet  Google Scholar 

  20. 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, 337–344 (2009a)

    Article  Google Scholar 

  21. 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 Transf. 131, 072602 (2009b)

    Article  Google Scholar 

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Acknowledgements

The research that led to the present paper was partially supported by the following Grants: 2017YBKNCE national project PRIN of Italian Ministry for University and Research; 2017F7KZWS national project PRIN of Italian Ministry for University and Research; PTRDMI-53722122113 “Analisi qualitativa per sistemi dinamici finito e infinito dimensionali con applicazioni a biomatematica, meccanica dei continui e termodinamica estesa classica e quantistica” of University of Catania. G. M. also acknowledges the group GNFM of INdAM, and the support from the project PON SCN 00451 CLARA - CLoud plAtform and smart underground imaging for natural Risk Assessment, Smart Cities and Communities and Social Innovation.

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Authors and Affiliations

  1. Dipartimento di Ingegneria Industriale, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136, Bologna, Italy

    A. Barletta

  2. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125, Catania, Italy

    G. Mulone

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  1. A. Barletta
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  2. G. Mulone
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Correspondence to A. Barletta.

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Communicated by Andreas Öchsner.

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Barletta, A., Mulone, G. The energy method analysis of the Darcy–Bénard problem with viscous dissipation. Continuum Mech. Thermodyn. 33, 25–33 (2021). https://doi.org/10.1007/s00161-020-00883-3

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  • DOI: https://doi.org/10.1007/s00161-020-00883-3

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