On oscillatory convection with the Cattaneo–Christov hyperbolic heat-flow model

Abstract

Adoption of the hyperbolic Cattaneo–Christov heat-flow model in place of the more usual parabolic Fourier law is shown to raise the possibility of oscillatory convection in the classic Bénard problem of a Boussinesq fluid heated from below. By comparing the critical Rayleigh numbers for stationary and oscillatory convection, R c and R S respectively, oscillatory convection is found to represent the preferred form of instability whenever the Cattaneo number C exceeds a threshold value C T ≥8/27 π 2 ≈0.03. In the case of free boundaries, analytical approaches permit direct treatment of the role played by the Prandtl number P 1 , which—in contrast to the classical stationary scenario—can impact on oscillatory modes significantly owing to the non-zero frequency of convection. Numerical investigation indicates that the behaviour found analytically for free boundaries applies in a qualitatively similar fashion for fixed boundaries, while the threshold Cattaneo number C T is computed as a function of P 1 ∈ [ 10 − 2 , 10 + 2 ] for both boundary regimes.