Finite thermal convection of non-Fourier fluids

The thermal convection is examined theoretically for fluids possessing significant thermal relaxation time, characterizing the response of the heat flux and the temperature gradient to changes in one another. Fourier's law breaks down for such fluids. Non-Fourier heat transport occurs in a wide range of applications, including superfluid helium, fluids subjected to rapid heating, and strongly confined fluids. The parallels between non-Fourier fluids and viscoelastic polymeric solutions are established. For viscoelastic fluids, the constitutive equation for stress must be frame invariant, a condition that must also hold for the constitutive equation for heat flux. The stability of conduction state is first reviewed, and the convection state is obtained using a low-order dynamical system approach. The stability of steady convection is analysed, and the Nusselt number is obtained as function of the Rayleigh and Cattaneo numbers.