Abstract
We show that for many classes of convection problem involving a porous layer, or layers, interleaved with finite but non-deformable solid layers, the global nonlinear stability threshold is exactly the same as the linear instability one. The layer(s) of porous material may be of Darcy type, Brinkman type, possess an anisotropic permeability, or even be such that they are of local thermal non-equilibrium type where the fluid and solid matrix constituting the porous material may have different temperatures. The key to the global stability result lies in proving the linear operator attached to the convection problem is a symmetric operator while the nonlinear terms must satisfy appropriate conditions.
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Altawallbeh A.A., Bhadauria B.S., Hashim I.: Linear and nonlinear double—diffusive convection in a saturated anisotropic porous layer with Soret effect and internal heat source. Int. J. Heat Mass Transf. 59, 103–111 (2013)
Banu N., Rees D.A.S.: Onset of Darcy—Bénard convection using a thermal non-equilibrium model. Int. J. Heat Mass Transf. 45, 2221–2228 (2002)
Böttger P.H.M., Gusarov A.V., Shklover V., Patscheider J., Sobiech M.: Anisotropic layered media with microintrusions; thermal properties of arc—evaporation multilayer metal nitrides. Int. J. Thermal Sci. 77, 75–83 (2014)
Capone F., Gentile M., Hill A.A.: Penetrative convection in anisotropic porous media with variable permeability. Acta Mechanica 216, 49–58 (2011)
Capone F., Gentile M., Hill A.A.: Double diffusive penetrative convection simulated via internal heating in an anisotropic porous layer with throughflow. Int. J. Heat Mass Transf. 54, 1622–1626 (2011)
Capone F., de Cataldis V., de Luca R., Torcicollo I.: On the stability of vertical constant throughflows for binary mixtures in porous layers. Int. J. Non-Linear Mech. 59, 1–8 (2014)
Capone F., Rionero S.: Inertia effect on the onset of convection in rotating porous layers via the “auxilliary system method”. Int. J. Non-Linear Mech. 57, 192–200 (2013)
Dongarra J.J., Straughan B., Walker D.W.: Chebyshev tau—QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Appl. Numer. Math. 22, 399–435 (1996)
Egorov S.D.: Thermal convection in sections of multilayer cryogenic heat insulation. J. Eng. Phys. Thermophys. 61, 989–992 (1991)
Falsaperla P., Giacobbe A., Mulone G.: Does symmetry of the operator of a dynamical system help stability?. Acta Appl. Math. 122, 239–253 (2012)
Galdi G.P., Padula M.: A new approach to energy theory in the stability of fluid motion. Arch. Ration. Mech. Anal. 110, 187–286 (1990)
Galdi, G.P., Rionero, S.: Weighted energy methods in fluid dynamics and elasticity. In: Springer Lecture Notes in Mathematics, vol. 1134 (1985)
Galdi G.P., Straughan B.: Exchange of stabilities, symmetry and nonlinear stability. Arch. Ration. Mech. Anal. 89, 211–228 (1985)
Georgescu A., Palese L.: Extension of Joseph’s criterion to the nonlinear stability of mechanical equilibria in the presence of thermodiffusive conductivity. Theor. Comput. Fluid Mech. 8, 403–413 (1996)
Hill A.A., Malashetty M.S.: An operative method to obtain sharp nonlinear stability for systems with spatially dependent coefficients. Proc. R. Soc. Lond. A 468, 323–336 (2012)
Jordan P.M., Puri P.: Thermal stresses in a spherical shell under three thermoelastic models. J. Thermal Stresses 24, 47–70 (2001)
Kumar A., Bhadauria B.S.: Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid. Int. J. Non-Linear Mech. 46, 47–56 (2011)
Lombardo S., Mulone G., Trovato M.: Nonlinear stability in reaction-diffusion systems via optimal Lyapunov functions. J. Math. Anal. Appl. 342, 461–476 (2008)
Malashetty M.S., Shivakumara I.S., Kulkarni S.: The onset of Lapwood–Brinkman convection using a thermal non-equilibrium model. Int. J. Heat Mass Transf. 48, 1155–1163 (2005)
Malashetty M.S., Hill A.A., Swamy M.: Double diffusive convection in a viscoelastic fluid—saturated porous layer using a thermal non-equilibrium model. Acta Mechanica 223, 967–983 (2012)
Nield D.A.: Comment on the effect of anisotropy on the onset of convection in a porous medium. Adv. Water Resour. 30, 696–697 (2007)
Nield D.A.: A note on local thermal non-equilibrium in porous media near boundaries and interfaces. Trans. Porous Media 95, 581–584 (2012)
Nield D.A., Kuznetsov A.V.: Local thermal non-equilibrium effects in forced convection in a porous medium channel: a conjugate problem. Int. J. Heat Mass Transf. 42, 3245–3252 (1999)
Nield D.A., Kuznetsov A.V.: The effect of local thermal non-equilibrium on the onset of convection in an nanofluid. J. Heat Transf. 132, 052405 (2010)
Patil P.M., Rees D.A.S.: The onset of convection in a porous layer with multiple horizontal solid partitions. Int. J. Heat Mass Transf. 68, 234–246 (2014)
Rees D.A.S.: Microscopic modelling of the two-temperature model for conduction in porous media. J. Porous Media 13, 125–143 (2010)
Rees D.A.S., Bassom A.P., Siddheshwar P.G.: Local thermal non-equilibrium effects arising from the injection of a hot fluid into a porous medium. J. Fluid Mech. 594, 379–398 (2008)
Rees D.A.S., Genc G.: The onset of convection in porous layers with multiple horizontal partitions. Int. J. Heat Mass Transf. 54, 3081–3089 (2011)
Rionero S.: Onset of convection in porous materials with vertically stratfied porosity. Acta Mechanica 222, 261–272 (2011)
Rionero, S.: Multicomponent diffusive—convective fluid motions in porous layers: Ultimately boundedness, absence of subcritical instabilities, and global stability for any number of salts. Phys. Fluids 25 (2013), Article Number 054104
Saravan S., Brindha D.: Linear and nonlinear stability limits for centrifugal convection in an anisotropic layer. Int. J. Non-Linear Mech. 46, 65–72 (2011)
Saravan S., Brindha D.: Onset of centrifugal filtration convection: departure from thermal equilibrium. Proc. R. Soc. Lond. A 469, 20120655 (2013)
Saravan S., Sivakumar T.: Onset of thermovibrational filtration convection: departure from thermal equilibrium. Phys. Rev. E 84, 026307 (2011)
Scott N.L., Straughan B.: A nonlinear stability analysis of convection in a porous vertical channel including local thermal nonequilibrium. J. Math. Fluid Mech. 15, 171–178 (2013)
Shiina Y., Hishida M.: Critical Rayleigh number of natural convection in high porosity anisotropic horizontal porous layers. Int. J. Heat Mass Transf. 53, 1507–1513 (2010)
Shivakumara I.S., Lee J., Chavaraddi K.B.: Onset of surface tension driven convection in a fluid overlying a layer of an anisotropic porous medium. Int. J. Heat Mass Transf. 54, 994–1001 (2011)
Straughan, B.: The energy method, stability, and nonlinear convection. Ser. Appl. Math. Sci., vol. 91, 2nd edn. Springer, New York (2004)
Straughan B.: Global nonlinear stability in porous convection with a thermal non-equilibrium model. Proc. R. Soc. Lond. A 462, 409–418 (2006)
Straughan, B.: Stability and wave motion in porous media. Ser. Appl. Math. Sci., vol. 165. Springer, New York (2008)
Straughan B.: Green-Naghdi fluid with non-thermal equilibrium effects. Proc. R. Soc. Lond. A 466, 2021–2032 (2010)
Straughan B.: Porous convection with local thermal non-equilibrium effects and Cattaneo effects in the solid. Proc. R. Soc. Lond. A 469, 20130187 (2013)
Straughan B.: Anisotropic inertia effect in microfluidic porous thermosolutal convection. Microfluidics Nanofluidics 16, 361–368 (2014)
Straughan B., Walker D.W.: Anisotropic porous penetrative convection. Proc. R. Soc. Lond. A 452, 97–115 (1996)
Tiwari A.K., Singh A.K., Chandran P., Sacheti N.C.: Natural convection in a cavity with a sloping upper surface filled with an anisotropic porous material. Acta Mechanica 223, 95–108 (2012)
Vadasz P.: Small Nield number convection in a porous layer heated from below via a constant heat flux and subject to lack of local thermal equilibrium. J. Porous Media 15, 249–258 (2012)
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Communicated by G.P. Galdi
Research supported by a “Tipping Points, Mathematics, Metaphors and Meanings” grant of the Leverhulme Trust. I am indebted to an anonymous referee for helpful criticism.
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Straughan, B. Nonlinear Stability of Convection in a Porous Layer with Solid Partitions. J. Math. Fluid Mech. 16, 727–736 (2014). https://doi.org/10.1007/s00021-014-0183-4
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DOI: https://doi.org/10.1007/s00021-014-0183-4