The onset of convection in a porous layer saturated by a power-law fluid is here investigated. The walls are considered to be isothermal, isobaric and permeable in such a way that a vertical throughflow is described. The threshold for a buoyancy-driven cellular flow is investigated by means of a linear stability analysis. This study consists in introducing disturbances with small amplitude. The disturbances are plane waves, i.e. a normal modes stability analysis of the basic stationary solution is performed. The resulting problem is an ordinary differential equation eigenvalue problem which is solved numerically by coupling the Runge–Kutta method with the shooting method. Results are presented in the form of marginal stability curves and their critical points representing the values of the control parameters such that the growth rate of the disturbances is zero. It is found that, among roll disturbances, the most unstable modes are stationary and uniform with infinite wavelength. For this reason, an asymptotic analysis for vanishing wave numbers is carried out. The results of this asymptotic analysis are obtained analytically displaying a very good agreement with the numerical solution. It is found that vertical throughflow plays a destabilising role for pseudoplastic fluids and a stabilising role for dilatant fluids.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Alves, L.S.B., Hirata, S.C., Ouarzazi, M.N.: Linear onset of convective instability for Rayleigh-Bénard–Couette flows of viscoelastic fluids. J. Non-Newton. Fluid Mech. 231, 79–90 (2016)
Alves, L.S.B., Hirata, S.C., Nunes, M.S.S., Barletta, A.: Identifying linear absolute instabilities from differential eigenvalue problems using sensitivity analysis. J. Fluid Mech. 870, 941–969 (2019)
Barletta, A.: Routes to Absolute Instability in Porous Media. Springer, Berlin (2019)
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)
Barletta, A., di Schio, E.R., Storesletten, L.: Convective roll instabilities of vertical throughflow with viscous dissipation in a horizontal porous layer. Transp. Porous Media 81, 461–477 (2010)
Christopher, R.H., Middleman, S.: Power-law flow through a packed tube. Ind. Eng. Chem. Fundam. 4, 422–426 (1965)
Homsy, G.M., Sherwood, A.E.: Convective instabilities in porous media with through flow. AIChE J. 22, 168–174 (1976)
Horton, C.W., Rogers Jr., F.T.: Convection currents in a porous medium. J. Appl. Phys. 16, 367–370 (1945)
Jones, M.C., Persichetti, J.M.: Convective instability in packed beds with throughflow. AIChE J. 32, 1555–1557 (1986)
Lapwood, E.R.: Convection of a fluid in a porous medium. Math. Proc. Camb. Philos. Soc. 44, 508–521 (1948)
Nield, D.A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 4, 553–560 (1968)
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, 121–123 (2011a)
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, 187–191 (2011b)
Nield, D.A., Bejan, A., et al.: Convection in Porous Media, 5th edn. Springer, Berlin (2017)
Prats, M.: The effect of horizontal fluid flow on thermally induced convection currents in porous mediums. J. Geophys. Res. 71, 4835–4838 (1966)
Rees, D.A.S., Bassom, A.: The onset of Darcy-Bénard convection in an inclined layer heated from below. Acta Mech. 144, 103–118 (2000)
Shenoy, A.V.: Non-Newtonian fluid heat transfer in porous media. Adv. Heat Transf. 24, 102–191 (1994)
Sutton, F.M.: Onset of convection in a porous channel with net through flow. Phys. Fluids 13, 1931–1934 (1970)
Wolfram, S., et al.: Mathematica. Cambridge University Press, Cambridge (1996)
Zhao, C., Hobbs, B.E., Ord, A.: Convective and Advective Heat Transfer in Geological Systems. Springer, Berlin (2008)
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Grant No 88881.174085/2018-01.
Conflict of interest
The authors declare that they have no conflict of interest.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Brandão, P.V., Celli, M., Barletta, A. et al. Convection in a Horizontal Porous Layer with Vertical Pressure Gradient Saturated by a Power-Law Fluid. Transp Porous Med 130, 613–625 (2019). https://doi.org/10.1007/s11242-019-01328-5
- Non-Newtonian fluids
- Thermal convection
- Modal stability analysis
- Asymptotic analysis