Abstract
The problem of thermal convection is investigated for a layer of fluid when the heat flux law of Cattaneo is adopted. The boundary conditions are those appropriate to two fixed surfaces. It is shown that for small Cattaneo number the critical Rayleigh number initially increases from its classical value of 1707.765 until a critical value of the Cattaneo number is reached. For Cattaneo numbers greater than this critical value a notable Hopf bifurcation is observed with convection occurring at lower Rayleigh numbers and by oscillatory rather than stationary convection. The aspect ratio of the convection cells likewise changes.
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Capone F., Gentile M., Hill A.A.: Penetrative convection in a fluid layer with throughflow. Ricerche di Matematica 57, 251–260 (2008)
Cattaneo C.: Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3, 83–101 (1948)
Chandrasekhar S.: Hydrodynamic and Hydromagnetic Stability. Dover, New York (1981)
Dai W., Wang H., Jordan P.M., Mickens R.E., Bejan A.: A mathematical model for skin burn injury induced by radiation heating. Int. J. Heat Mass Transf. 51, 5497–5510 (2008)
Dauby P.C., Nélis M., Lebon G.: Generalized Fourier equations and thermoconvective instabilities. Revista Mexicana de Fisica 48, 57–62 (2002)
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)
Flavin J.N., Rionero S.: Qualitative Estimates for Partial Differential Equations. CRC Press, Boca Raton (1995)
Franchi F., Straughan B.: Thermal convection at low temperature. J. Non-Equilibrium Thermodyn. 19, 368–374 (1994)
Lebon G., Cloot A.: Bénard–Marangoni instabiity in a Maxwell–Cattaneo fluid. Phys. Lett. A 105, 361–364 (1984)
Liu H., Bussmann M., Mostaghimi J.: A comparison of hyperbolic and parabolic models of phase change of a pure metal. Int. J. Heat Mass Transf. 52, 1177–1184 (2009)
Miranville, A., 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
Mulone G., Rionero S.: Necessary and sufficient conditions for nonlinear stability in the magnetic Bénard problem. Arch. Rational. Mech. Anal. 166, 197–218 (2003)
Puri P., Jordan P.M.: Wave structure in Stokes’ second problem for a dipolar fluid with nonclassical heat conduction. Acta Mech. 133, 145–160 (1999)
Puri P., Jordan P.M.: Stokes’s first problem for a dipolar fluid with nonclassical heat conduction. J. Eng. Math. 36, 219–240 (1999)
Puri P., Kythe P.K.: Discontinuities in velocity gradients and temperature in the Stokes’ first problem with nonclassical heat conduction. Q. Appl. Math. 55, 167–176 (1997)
Puri P., Kythe P.K.: Stokes’ first and second problems for Rivlin–Ericksen fluids with nonclassical heat conduction. J. Heat Transf. ASME 120, 44–50 (1998)
Rionero S.: Metodi variazionali per la stabilità asintotica in media in magnetoidrodinamica. Ann. Matem. Pura Appl. 78, 339–364 (1968)
Rionero S.: A new approach to nonlinear L 2-stability of double diffusive convection in porous media: necessary and sufficient conditions for global stability via a linearization principle. J. Math. Anal. Appl. 333, 1036–1057 (2007)
Rionero S., Mulone G.: Non-linear stability analysis of the magnetic Bénard problem through the Lyapunov direct method. Arch. Rational. Mech. Anal. 103, 347–368 (1988)
Saidane A., Aliouat S., Benzohra M., Ketata M.: A transmission line matrix (TLM) study of hyperbolic heat conduction in biological materials. J. Food Eng. 68, 491–496 (2005)
Straughan, B.: The energy method, stability, and nonlinear convection, 2nd edn, vol. 91, Appl. Math. Sci. Ser., Springer, Heidelberg (2004)
Straughan B.: A note on convection with nonlinear heat flux. Ricerche di Matematica 56, 229–239 (2007)
Straughan, B. Stability and Wave Motion in Porous Media. Springer, Ser. Appl. Math. Sci. vol. 165, New York (2008)
Straughan B., Franchi F.: Bénard convection and the Cattaneo law of heat conduction. Proc. R. Soc. Edinb. A 96, 175–178 (1984)
Vadasz P.: Lack of oscillations in dual-phase-lagging heat conduction for a porous slab subject to heat flux and temperature. Int. J. Heat Mass Transf. 48, 2822–2828 (2005)
Vadasz J.J., Govender S., Vadasz P.: Heat transfer enhancement in nano-fluids suspensions: possible mechanisms and explanations. Int. J. Heat Mass Transf. 48, 2673–2683 (2005)
Xu L., Yang S.: Stability analysis of thermosolutal second-order fluid in porous Bénard layer. Ricerche di Matematica 56, 149–160 (2007)
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Straughan, B. Oscillatory convection and the Cattaneo law of heat conduction. Ricerche mat. 58, 157–162 (2009). https://doi.org/10.1007/s11587-009-0055-z
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DOI: https://doi.org/10.1007/s11587-009-0055-z