Thermal convection in a magnetized conducting fluid with the Cattaneo–Christov heat-flow model

Abstract

By substituting the Cattaneo–Christov heat-flow model for the more usual parabolic Fourier law, we consider the impact of hyperbolic heat-flow effects on thermal convection in the classic problem of a magnetized conducting fluid layer heated from below. For stationary convection, the system is equivalent to that studied by Chandrasekhar ( Hydrodynamic and Hydromagnetic Stability, 1961), and with free boundary conditions we recover the classical critical Rayleigh number R c ( c ) ( Q ) which exhibits inhibition of convection by the field according to R c ( c ) → π 2 Q as Q → ∞ , where Q is the Chandrasekhar number. However, for oscillatory convection we find that the critical Rayleigh number R c ( o ) ( Q , P 1 , P 2 , C ) is given by a more complicated function of the thermal Prandtl number P 1 , magnetic Prandtl number P 2 and Cattaneo number C . To elucidate features of this dependence, we neglect P 2 (in which case overstability would be classically forbidden), and thereby obtain an expression for the Rayleigh number that is far less strongly inhibited by the field, with limiting behaviour R c ( o ) → π Q / C , as Q → ∞ . One consequence of this weaker dependence is that onset of instability occurs as overstability provided C exceeds a threshold value C T ( Q ); indeed, crucially we show that when Q is large, C T ∝ 1 / Q , meaning that oscillatory modes are preferred even when C itself is small. Similar behaviour is demonstrated in the case of fixed boundaries by means of a novel numerical solution.