年間 12 号発行
ISSN 印刷: 1091-028X
ISSN オンライン: 1934-0508
Indexed in
CONVECTIVE INSTABILITIES OF A MAXWELL-CATTANEO POROUS LAYER
要約
Motivated by the need to better understand the influence of the Maxwell-Cattaneo effect (or hyperbolic heat flow) on the dynamics of porous media in local thermal non-equilibrium, the stability of a porous Darcy-Brinkman layer is studied when the Maxwell-Cattaneo (MC) relation of temperature and heat flux is introduced to a fluid and solid. We first prove that, in the absence of the MC effect, the porous layer cannot support oscillatory motions. When the MC effect is present in the fluid only, propagation of oscillatory motions is possible, provided that the MC effect parameter exceeds a certain threshold. The oscillatory motions are then preferred only if the thermal interphase interaction parameter H is small. On the other hand, when the MC effect is present in the solid only, the oscillatory instability is enhanced when H is large. The contrasting influences of the MC effect on the fluid and solid lead to some novel features when the MC effect is present simultaneously in both fluid and solid. Here, oscillatory motions can be preferred for intermediate values of H, depending on the two MC parameters measuring the influences in the solid and fluid. Although the presence of the MC effect introduces new modes so that the frequency equation changes from linear in the frequency squared to cubic, the unstable mode is always provided by the mode which is stable in the absence of the MC effect made unstable by the presence of the MC effect. The new modes are never preferred, but they can possess Taken-Bogdanov's bifurcations in addition to the Hopf bifurcations present in all the cases. When the analysis is applied to crude oil in sandstone and water in sandstone, we find that they possess contrasting stability properties.
-
Ashwin, T.R., Narasimham, G.S.V.L., and Jacob, S., CFD Analysis of High Frequency Miniature Pulse Tube Refrigerators for Space Applications with Thermal Non-Equilibrium Model, Appl. Therm.. Eng., vol. 30, pp. 152-166, 2010.
-
Bissell, J.J., On Oscillatory Convection with the Cattaneo-Christov Hyperbolic Heat-Flow Model, Proc. R. Soc. Lond. Ser. A, vol. 471, p. 20140845,2015.
-
Bissell, J.J., Thermal Convection in a Magnetized Conducting Fluid with the Cattaneo-Christov Heat-Flow Model, Proc. R. Soc. Lond. Ser. A, vol. 472, p. 20160649, 2016.
-
Carrassi, M. and Morro, A., A Modified Navier-Stokes Equation, and Its Consequences on Sound Dispersion, Nuovo Cimento B Serie, vol. 9, pp. 321-343, 1972.
-
Cattaneo, C., Sulla Conduzione del Calore, Atti Mat. Fis. Univ. Modena, vol. 3, pp. 83-101, 1948.
-
Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford, UK: Clarendon Press, 1961.
-
Christov, C.I., On Frame Indifferent Formulation of the Maxwell-Cattaneo Model of Finite Speed Heat Conduction, Mech. Res. Commun., vol. 36, pp. 481-486, 2009.
-
Dai, W., Wang, H., Jordan, P.M., Mickens, R.E., and Bejan, A., A Mathematical Model for Skin Burn Injury Induced by Radiation Heating, Int. J. Heat Mass Transf., vol. 51, pp. 5497-5510, 2008.
-
Donnelly, R.J., The Two-Fluid Theory and Second Sound in Liquid Helium, Phys. Today, vol. 62, pp. 34-39,2009.
-
Eltayeb, I.A., Propagation and Stability of Wave Motions in Rotating Magnetic Systems, Phys. Earth Planet Inters., vol. 24, pp. 259-271, 1981.
-
Eltayeb, I.A., Stability of a Porous Benard-Brinkman Layer in Local Thermal Non-Equilibrium with Cattaneo Effect in Solid, Int. J Therm. Sci., vol. 98, pp. 208-218, 2015.
-
Eltayeb, I.A., The Stability of Second Sound Waves in a Rotating Darcy-Brinkman Porous Layer in Local Thermal Non-Equilibrium, FluidDyn. Res, vol. 49, p. 045512, 2017a.
-
Eltayeb, I.A., Convective Instabilities of Maxwell-Cattaneo Fluids, Proc. R. Soc. Lond. Ser. A, vol. 473, p. 20160712, 2017b.
-
Eltayeb, I.A., Hughes, D.W., and Proctor, M.R.E., The Convective Instability of a Maxwell-Cattaneo Fluid in the Presence of a Vertical Magnetic Field, Proc. R. Soc. Lond. Ser. A, vol. 476, p. 20200494,2020.
-
Fox, N., Low Temperature Effects and Generalized Thermoelasticity, IMA J. Appl. Math, vol. 5, pp. 373-386, 1969.
-
Gnedin, O.Y., Yakovlev, D.G., and Potekhin, A.Y., Thermal Relaxation in Young Neutron Stars, Mon. Not. R. Astron. Soc, vol. 324, pp. 725-736,2001.
-
Haddad, S.A.M. and Straughan, B., Porous Convection and Thermal Oscillations, Ricerche Mat., vol. 61, pp. 307-320,2012.
-
Haddad, S.A.M., Thermal Instability in Brinkman Porous Media with Cattaneo-Christov Heat Flux, Int. J. Heat Mass Transf., vol. 68, pp. 659-668, 2014.
-
Hayes, A.M., Khan, J.A., Shaaban, A.H., and Spearing, I.G., The Thermal Modelling of a Matrix Heat Exchanger Using a Porous Medium and the Thermal Non-Equilibrium Model, Int. J. Therm.. Sci., vol. 47, pp. 1306-1315, 2008.
-
Herrera, L. and Falcon, N., Heat Waves and Thermohaline Instability in a Fluid, Phys. Lett. A, vol. 201, pp. 33-37, 1995.
-
Hill, A.A. and Morad, M.R., Convective Stability of Carbon Sequestration in Anisotropic Media, Proc. R. Soc. Lond. A, vol. 470, p. 373,2014.
-
Horton, C.W. and Rogers, R.T., Convection Currents in a Porous Medium, J. Appl. Phys., vol. 16, pp. 367-370, 1945.
-
Hughes, D.W., Proctor, M.R.E., and Eltayeb, I.A., Maxwell-Cattaneo Double Diffusive Convection: Limiting Cases, J. Fluid Mech., vol. 927,2021. DOI: 10.1017/jfm.2021.721. DOI: 10.1017/jfm.2021.721
-
Jou, D., Sellitto, A., and Alvarez, F.X., Heat Waves and Phonon-Wall Collisions in Nanowires, Proc. R. Soc. Lond. Ser. A, vol. 467, pp. 2520-2533,2011.
-
Kuznetsov, A.V. and Nield, D.A., Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: Brinkman Model, Transp. Porous Media, vol. 81, pp. 409-422, 2010.
-
Lapwood, E.R., Convection of a Fluid in a Porous Medium, Proc. Cambridge Philos. Soc., vol. 44, pp. 508-521, 1948.
-
Lebon, G. and Cloot, A., Benard-Marangoni Instability in a Maxwell-Cattaneo Fluid, Phys. Lett. A, vol. 105, pp. 361-364,1984.
-
Lebon, G., Machrafi, H., Grmela, M., and Dubois, C., An Extended Thermodynamic Model of Transient Heat Conduction at Sub-Continuum Scales, Proc. R. Soc. Lond. Ser. A, vol. 467, pp. 3241-3256, 2011.
-
Lefebvre, L.P., Banhart, J., and Dunand, D.C., Porous Metals and Metallic Foams: Current Status and Recent Developments, Adv. Eng. Mater, vol. 10, pp. 775-787,2008.
-
Liepmann, H.W. and Laguna, G.A., Nonlinear Interactions in the Fluid Mechanics of Helium II, Ann. Rev. Fluid Mech., vol. 16, pp. 139-177,2008.
-
Maxwell, J.C., On the Dynamical Theory of Gases, Phil. Trans. R. Soc. Lond. A, vol. 157, pp. 49-88, 1867.
-
Miranville, A. and Quintanilla, R., A Generalization of the Caginalp Phase-Field System Based on the Cattaneo Law, Nonlinear Anal., 2009. DOI: 10.1016/j.na.2009.01.061. DOI: 10.1016/j.na.2009.01.061
-
Nield, D.A. and Bejan, A., Convection in Porous Media, Second ed., Berlin: Springer, 1998.
-
Nield, D.A. and Kuznetsov, A.V, The Effect of Local Thermal Non-Equilibrium on the Onset of Convection in a Nanofluid, J. Heat Transf., vol. 132, p. 052405, 2010.
-
Oldroyd, J.G., On the Formulation of Rheological Equations of State, Proc. R. Soc. Lond. Ser. A, vol. 200, pp. 523-541,1950.
-
Saidane, S., Aliouat, M., Benzohra, M., and Ketata, A., Transmission Line Matrix (TLM) Study of Hyperbolic Heat Conduction in Biological Materials, J. Food Eng., vol. 68, pp. 491-496, 2005.
-
Stranges, D.F., Khayat, R.E., and Albaalbaki, B., Thermal Convection of Non-Fourier Fluids-Linear Stability, Int. J. Therm. Sci., vol. 74, pp. 14-23,2013.
-
Stranges, D.F., Khayat, R.E., and de Bruyn, J., Finite Thermal Convection of Non-Fourier Fluids, Int. J. Therm. Sci., vol. 104, pp. 437-447, 2016.
-
Straughan, B., Heat Waves, New York: Springer, 2011.
-
Straughan, B., Oscillatory Convection and the Cattaneo Law of Heat Conduction, Ricerche di Matematica, vol. 58, p. 57, 2009.
-
Straughan, B., Tipping Points in Cattaneo-Christov Thermohaline Convection, Proc. R. Soc., vol. 467, 2010. DOI: 10.1098/rspa.2010.0104. DOI: 10.1098/rspa.2010.0104
-
Straughan, B., Porous Convection with Local Thermal Non-Equilibrium Temperatures and with Cattaneo Effects in the Solid, Proc. R Soc. Lond. Ser. A, vol. 469, p. 20130187,2013.
-
Straughan, B. and Franchi, F., Benard Convection and the Cattaneo Law of Heat Conduction, Proc. R. Soc. Edin., vol. 96, pp. 175-178, 1984.
-
Tung, M.M., Trujillo, M., Lopez Molina, J.A., Rivera, M.J., and Berjano, E.J., Modeling the Heating of Biological Tissue Based on the Hyperbolic Heat Transfer Equation, Math. Comput. Model., vol. 50, pp. 665-672, 2009.
-
Vafai, K., Handbook of Porous Media, Boca Raton, FL: CRC Press, 2005.
-
Vidal, F. and Lhuillier, D., Second-Sound Velocity in Rotating Superfluid Helium, Phys. Rev. B, vol. 13, p. 148, 1976.