Natural convection of a Cattaneo–Christov fluid bounded by Cattaneo thick walls with finite thermal conductivity

Abstract

In order to compare with experimental conditions, it is important to use realistic boundary conditions. In this paper the linear natural convection of a Cattaneo–Christov fluid bounded by two Cattaneo thick horizontal walls with finite thermal conductivity is investigated. In this sense, fluid and walls present heat flux relaxation times. Of particular interest is to find the codimension two points where stationary and oscillatory convection compete to be the first to appear in convection. The specific case where the fluid heat flux satisfies a Cattaneo–Christov constitutive equation and the walls satisfy the usual Fourier constitutive equations is also investigated for the first time. The reason is that the more realistic wall to fluid thicknesses ratio d and thermal conductivities ratio X have not been used before in this problem as part of the main parameters. The critical Rayleigh number, wavenumber and frequency of oscillation are plotted against logX for fixed d and different magnitudes of the relaxation times of the fluid and the walls. The curves of criticality of oscillatory convection have a maximum (stabilizes) in the middle range of logX, close to the codimension-two point, that is, to the intersection with the curve of criticality of stationary convection. In contrast, the curves of criticality of oscillatory convection decrease (destabilize) with an increase of the relaxation time of the fluid. It is revealed that the wall relaxation time stabilizes oscillatory convection even more in the middle range of logX.