The convective instability of a Maxwell-Cattaneo fluid in the presence of a vertical magnetic field

Abstract

We study the instability of a B\'enard layer subject to a vertical uniform magnetic field, in which the fluid obeys the Maxwell-Cattaneo (MC) heat flux-temperature relation. We extend the work of Bissell (Proc. R. Soc. A, 472: 20160649, 2016) to non-zero values of the magnetic Prandtl number $p_m$. With non-zero $p_m$, the order of the dispersion relation is increased, leading to considerably richer behaviour. An asymptotic analysis at large values of the Chandrasekhar number $Q$ confirms that the MC effect becomes important when $C Q^{1/2}$ is $O(1)$, where $C$ is the Maxwell-Cattaneo number. In this regime, we derive a scaled system that is independent of $Q$. When $CQ^{1/2}$ is large, the results are consistent with those derived from the governing equations in the limit of Prandtl number $p\to \infty$ with $p_m$ finite; here we identify a new mode of instability, which is due neither to inertial nor induction effects. In the large $p_m$ regime, we show how a transition can occur between oscillatory modes of different horizontal scale. For $Q \gg 1$ and small values of $p$, we show that the critical Rayleigh number is non-monotonic in $p$ provided that $C>1/6$. While the analysis of this paper is performed for stress-free boundaries, it can be shown that other types of mechanical boundary conditions give the same leading order results.