A note on convection with nonlinear heat flux

Abstract

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.

Heat transfer in granular materials: effects of nonlinear heat conduction and viscous dissipation

“…This shows that in its simplest form, as given by Equation , the heat flux vector should depend on the temperature gradient and the density (or volume fraction) gradient. Finally, it is important to mention that although the second term on the right‐hand side of the above equation, namely the gradient of the volume fraction, resembles the diffusion term in multicomponent situations , we prefer to refer to this term as the heat flux because of the gradient of the density (or the volume fraction). In reality, if we were dealing with multicomponent systems where the moisture content is important, then indeed there could be heat conduction because of diffusion.…”

Heat transfer and Couette flow of a chemically reacting non-linear fluid

“…will make the problem highly non-linear. Interestingly, Straughan [19], based on the work of Rodrigues and Urbano [34], studied and provided a complete analysis of the linear instability problem of fluid between two horizontal plates, where a power-law type model was used for the heat flux.…”

“…Equations (16)- (18) along with the boundary conditions (19) and (20) are to be made dimensionless and solved numerically. Let us define the dimensionless distanceȳ, the velocityū, the concentrationc, and the temperature¯ by the following equations:…”

Heat Transfer in Complex Fluids

“…In recent years, chemically reacting flows have been studied in a variety of applications. A class of problems where for example heat and salinity concentration compete with each other, and as a result the density distribution can be affected, has received much attention (Straughan, 2002(Straughan, , 2007. The phenomenon of doublediffusive convection in a fluid layer presents a challenging problem not only to engineers, but also to mathematicians since the stability of the flow is also of concern and interests.…”