# The effect of finite-conductvity Hartmann walls on the linear stability of Hunt's flow

## TL;DR

This study numerically investigates how finite electrical conductivity of Hartmann walls influences the linear stability of Hunt's flow in a square duct under strong magnetic fields, revealing a critical effective conductance ratio that governs destabilization.

## Contribution

It provides new insights into the stability behavior of Hunt's flow with finite-conductivity walls, extending previous ideal wall models to more realistic conditions.

## Key findings

- Flow destabilization peaks at Ha≈30/c for small c.
- For strong magnetic fields, the flow's stability depends on the effective conductance ratio cHa.
- As cHa increases beyond 30, the flow stability approaches that of ideal conducting Hartmann walls.

## Abstract

We analyse numerically the linear stability of the fully developed liquid metal flow in a square duct with insulating side walls and thin electrically conducting horizontal walls with the wall conductance ratio $c=0.01\cdots 1$ subject to a vertical magnetic field with the Hartmann numbers up to $Ha=10^{4}.$ In a sufficiently strong magnetic field, the flow consists of two jets at the side walls walls and a near-stagnant core with the relative velocity $\sim(cHa)^{-1}.$ We find that for $Ha\gtrsim300,$ the effect of wall conductivity on the stability of the flow is mainly determined by the effective Hartmann wall conductance ratio $cHa.$ For $c\ll 1,$ the increase of the magnetic field or that of the wall conductivity has a destabilizing effect on the flow. Maximal destabilization of the flow occurs at $Ha\approx30/c.$ In a stronger magnetic field with $cHa\gtrsim 30,$ the destabilizing effect vanishes and the asymptotic results of Priede et al. [J. Fluid Mech. 649, 115, 2010] for the ideal Hunt's flow with perfectly conducting Hartmann walls are recovered.