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

TL;DR

This paper investigates the convective instability of a Maxwell-Cattaneo fluid under a vertical magnetic field, revealing new instability modes and complex behaviors influenced by magnetic Prandtl number and Maxwell-Cattaneo effects.

Contribution

It extends previous work to include non-zero magnetic Prandtl number, deriving new asymptotic results and identifying novel instability modes in Maxwell-Cattaneo fluids with magnetic fields.

Findings

Identification of a new instability mode independent of inertial and induction effects.

Non-monotonic critical Rayleigh number dependence on Prandtl number for certain parameters.

Transition between oscillatory modes with different horizontal scales at large magnetic Prandtl number.

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.