A note on convection with nonlinear heat flux

The problem of thermal convection is investigated when the heat flux is a nonlinear function of the temperature gradient. A complete analysis of the linear instability problem is given. The nonlinear stability problem is studied in a case which is believed to be physically relevant and the stability threshold is compared directly to that found by linear instability theory.

[1]  J. J. Vadasz,et al.  Heat transfer enhancement in nano-fluids suspensions: Possible mechanisms and explanations , 2005 .

[2]  F. Capone,et al.  Nonlinear stability analysis of convection for fluids with exponentially temperature-dependent viscosity , 1994 .

[3]  K. Ragsdell,et al.  The energy method , 1975 .

[4]  Salvatore Rionero,et al.  A rigorous reduction of the L2-stability of the solutions to a nonlinear binary reaction–diffusion system of PDE's to the stability of the solutions to a linear binary system of ODE's , 2006 .

[5]  Karen A. Ames,et al.  Non-Standard and Improperly Posed Problems , 1997 .

[6]  P. Kaloni,et al.  Double diffusive penetrative convection in porous media , 1995 .

[7]  M. Gentile,et al.  Nonlinear stability analysis of the Bénard problem for fluids with a convex nonincreasing temperature depending viscosity , 1995 .

[8]  J. Rodrigues,et al.  On the stationary Boussinesq-Stefan problem with constitutive power-laws , 1998 .

[9]  S. Rionero.  On the stability of binary reaction-diffusion systems , 2004 .

[10]  Dongsik Kim,et al.  Thermal Conductivity of Metal-Oxide Nanofluids: Particle Size Dependence and Effect of Laser Irradiation , 2007 .

[11]  Brian Straughan,et al.  Sharp global nonlinear stability for temperature-dependent viscosity convection , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  Lawrence E. Payne,et al.  Lower bounds for the blow-up time in a temperature dependent Navier-Stokes flow , 2007 .

[13]  B. Straughan,et al.  Continuous Dependence on the Body Force for Solutions to the Navier-Stokes Equations and on the Heat Supply in a Model for Double Diffusive Porous Convection , 1993 .

[14]  N. Galanis,et al.  Heat transfer enhancement by using nanofluids in forced convection flows , 2005 .

[15]  J. N. Flavin,et al.  Nonlinear stability for a thermofluid in a vertical porous slab , 1999 .

[16]  F. Capone.  On the onset of convection in porous media: temperature depending viscosity , 2001 .

[17]  Mehrdad Massoudi,et al.  On the heat flux vector for flowing granular materials—part II: derivation and special cases , 2006 .

[18]  Lawrence E. Payne,et al.  ON STABILIZING AGAINST MODELING ERRORS IN A PENETRATIVE CONVECTION PROBLEM FOR A POROUS MEDIUM , 1994 .

[19]  D. P. Wall,et al.  The linear stability of channel flow of fluid with temperature-dependent viscosity , 1996, Journal of Fluid Mechanics.

[20]  Mehrdad Massoudi,et al.  On the heat flux vector for flowing granular materials—Part I: effective thermal conductivity and background , 2006 .

[21]  S. Rionero.  A new approach to nonlinear L2-stability of double diffusive convection in porous media , 2007 .

[22]  B. Straughan,et al.  Global stability for convection when the viscosity has a maximum , 2004 .

[23]  L. Payne,et al.  On the possibility of global solutions for variant models of viscous flow on unbounded domains , 1989 .

[24]  Peter Wriggers,et al.  On the effects of microstress on macroscopic diffusion processes , 1999 .

[25]  Non‐linear stability for convection with quadratic temperature dependent viscosity , 2006 .

[26]  L. Payne,et al.  Continuous dependence results for solutions of the Navier-Stokes equations backward in time , 1994 .

[27]  Brian Straughan,et al.  Stability and Wave Motion in Porous Media , 2008 .

[28]  A. Mottana.  Storia Della Mineralogia Antica. I. La Mineralogia A Bisanzio Nel Xi Secolo D.C.: I Poteri Insiti Nelle Pietre Secondo Michele Psello , 2005 .

[29]  Continuous dependence results for a problem in penetrative convection , 1997 .

[30]  Brian Straughan,et al.  Explosive Instabilities in Mechanics , 1998 .