Two-gradient convection in a vertical slot with Maxwell-Cattaneo heat conduction

We study the effect of the Maxwell-Cattaneo law of heat conduction (MCHC) on the 1D flow in a vertical slot subject to both vertical and horizontal temperature gradients. The gravitational acceleration is allowed to oscillate, which provides an opportunity to investigate the quantitative contribution of thermal inertia as epitomized by MCHC. The addition of the time derivative in MCHC increases the order of the system. We use a spectral expansion with Rayleigh's beam functions as the basis set, which is especially suited to fourth order boundary value problems (BVP). We show that the time derivative (relaxation of the thermal flux) has a dissipative nature and leads to the appearance of purely real negative eigenvalues. Yet it also increases the absolute value of the imaginary part and decreases the absolute value of the real part of the complex eigenvalues. Thus, the system has a somewhat more oscillatory behavior than the one based on Fourier's heat conduction law (FHC).