The influence of thermal relaxation on the oscillatory properties of two-gradient convection in a vertical slot

We study the effects of the Maxwell–Cattaneo (MC) law of heat conduction on the flow of a Newtonian fluid in a vertical slot subject to both vertical and horizontal temperature gradients. Working in one spatial dimension (1D), we employ a spectral expansion involving Rayleigh’s beam functions as the basis set, which are especially well-suited to the fourth order boundary value problem (b.v.p.) considered here, and the stability of the resulting dynamical system for the Galerkin coefficients is investigated. It is shown that the absolute value of the (negative) real parts of the eigenvalues are reduced, while the absolute values of the imaginary parts are somewhat increased, under the MC law. This means that the presence of the time derivative of the heat flux increases the order of the system, thus leading to more oscillatory regimes in comparison with the usual Fourier case. Moreover, no eigenvalues with positive real parts were found, which means that in this particular situation, the inclusion of thermal relaxation does not lead to destabilization of the motion.