Abstract
The linear and weakly nonlinear stability analyses are carried out to study instabilities in Darcy–Bénard convection for non-Newtonian inelastic fluids. The rheological model considered here is the Darcy–Carreau model, which is an extension to porous media of Carreau rheological model usually used in clear fluid media. The linear stability approach showed that the critical Rayleigh number and wave number corresponding to the onset of convection are the same as for Newtonian fluids. By employing weakly nonlinear theory, we derived a cubic Landau equation that describes the temporal evolution of the amplitude of convection rolls in the unstable regime. It is found that the bifurcation from the conduction state to convection rolls is always supercritical for dilatant fluids. For pseudoplastic fluids, however, the interplay between the macroscale properties of the porous media and the rheological characteristics of the fluid determines the supercritical or subcritical nature of the bifurcation. In the parameter range where the bifurcation is supercritical, we determined and discussed the combined effects of the fluid properties and the porous medium characteristics on the amplitude of convection rolls and the corresponding average heat transfer for both pseudoplastic and dilatant fluids. Remarkably, we found that the curves describing these effects collapse onto the universal curve for Newtonian fluids, provided the average apparent viscosity is used to define Rayleigh number.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balmforth, N.J., Rust, A.C.: Weakly nonlinear viscoplastic convection. J. Non-Newton. Fluid Mech. 158, 36–45 (2009)
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)
Benouared, O., Mamou, M., Messaoudene, N.A.: Numerical nonlinear analysis of subcritical Rayleigh–Bénard convection in a horizontal confined enclosure filled with non-Newtonian fluids. Phys. Fluids 26(7), 073101 (2014)
Bouteraa, M., Nouar, C., Plaut, E., Métivier, C., Kalck, A.: Weakly nonlinear analysis of Rayleigh–Bénard convection in shear-thinning fluids: nature of the bifurcation and pattern selection. J. Fluid Mech. 767, 696–734 (2015)
Carreau, P.J.: Rheological equations from molecular network theories. Trans. Soc. Rheol. 16(1), 99–127 (1972)
Darbouli, M., Métivier, C., Leclerc, S., Nouar, C., Bouteera, M., Stemmelen, D.: Natural convection in shear-thinning fluids: experimental investigations by MRI. Int. J. Heat Mass Transf. 95, 742–754 (2016)
Hirata, S.C., Eny, G.E., Ouarzazi, M.N.: Nonlinear pattern selection and heat transfer in thermal convection of a viscoelastic fluid saturating a porous medium. Int. J. Therm. Sci. 95, 136–146 (2015)
Jenny, M., Plaut, E., Briard, A.: Numerical study of subcritical Rayleigh–Bénard convection rolls in strongly shear-thinning Carreau fluids. J. Non-Newton. Fluid Mech. 219, 19–34 (2015)
Joseph, D.D.: Stability of Fluid Motions I, vol. 27. Springer, Berlin (2013)
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)
Kolodner, P.: Oscillatory convection in viscoelastic DNA suspensions. J. non-Newton. Fluid Mech. 75(2–3), 167–192 (1998)
Liang, S.F., Acrivos, A.: Experiments on buoyancy driven convection in non-Newtonian fluid. Rheol. Acta 9(3), 447–455 (1970)
Longo, S., Di Federico, V., Chiapponi, L., Archetti, R.: Experimental verification of power-law non-Newtonian axisymmetric porous gravity currents. J. Fluid Mech. 731, R2 (2013)
Newell, A.C., Whitehead, J.A.: Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38(2), 279–303 (1969)
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. Trans. Porous Media 88(2), 187–191 (2011a)
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. Trans. Porous Media 87(1), 121–123 (2011b)
Nield, D.A., Bejan, A.: Convection in Porous Media, 5th edn. Springer, Berlin (2017)
Parmentier, E.M.: A study of thermal convection in non-Newtonian fluids. J. Fluid Mech. 84(1), 1–11 (1978)
Pascal, J.P., Pascal, H.: Nonlinear effects on some unsteady non-Darcian flows through porous media. Int. J. Non-Linear Mech. 32(2), 361–376 (1997)
Petrolo, D., Chiapponi, L., Longo, S., Celli, M., Barletta, A., Di Federico, V.: Onset of Darcy–Bénard convection under throughflow of a shear-thinning fluid. J. Fluid Mech. 889, R2 (2020)
Rees, D.A.S.: Darcy–Bénard–Bingham convection. Phys. Fluids 32, 084107 (2020)
Requilé, Y., Hirata, S.C., Ouarzazi, M.N., Barletta, A.: Weakly nonlinear analysis of viscous dissipation thermal instability in plane Poiseuille and plane Couette flows. J. Fluid Mech. 886, A26 (2020)
Savins, J.G.: Non-Newtonian flow through porous media. Ind. Eng. Chem. 61(10), 18–47 (1969)
Taleb, A., BenHamed, H., Ouarzazi, M.N., Beji, H.: Analytical and numerical analysis of bifurcations in thermal convection of viscoelastic fluids saturating a porous square box. Phys. Fluids 28(5), 053106 (2016)
Tien, C., Sheng, H., Sun, Z.: Thermal instability of a horizontal layer of non-Newtonian fluid heated from below. Int. J. Heat Mass Transf. 12(9), 1173–1178 (1969)
Yoon, D., Kim, M.C., Choi, C.K.: The onset of oscillatory convection in a horizontal porous layer saturated with viscoelastic liquid. Trans. Porous Media 55(3), 275–284 (2004)
Acknowledgements
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Grant \(\hbox {n}^{\circ }\) 88881.174085/2018-01.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Brandão, P.V., Ouarzazi, M.N. Darcy–Carreau Model and Nonlinear Natural Convection for Pseudoplastic and Dilatant Fluids in Porous Media. Transp Porous Med 136, 521–539 (2021). https://doi.org/10.1007/s11242-020-01523-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-020-01523-9