# The evolution of non-linear disturbances in magnetohydrodynamic flows

## TL;DR

This paper investigates the stability loss of Hartmann flow in magnetohydrodynamics using disturbance equations, comparing eigenfunction and fluid injection initial perturbations, and finds injection is a cost-effective alternative for stability analysis.

## Contribution

It demonstrates that fluid injection can replace eigenfunction methods for stability analysis in MHD flows, reducing computational costs.

## Key findings

- Injection method yields similar stability results as eigenfunction approach.
- Injection technique is less computationally expensive.
- Both methods confirm stability loss characteristics in Hartmann flow.

## Abstract

In this article the stability loss of the Hartmann flow are investigated by applying the equations for disturbances. The velocity and electric potential quasi-static MHD model is used. The equations allow us to calculate time-dependent disturbance fields using a base flow and an initial disturbance. Two type of initial perturbations are considered: the eigenfunction of the linearized MHD equations and a fluid injection into the flow. These two approaches lead to identical stability results. However, we found a significant difference in the practical implementation of the two approaches. Dealing with the eigenproblem of the linearized MHD system is a laborious task. In terms of calculation costs it is equal to a series of nonlinear perturbation simulations, and if the Hartmann number is increased, the proportion becomes worse. The non-linear stability analysis produced by these two methods shows that the injection technique can also be used in numerical analysis, and that this method is less expensive in terms of calculation costs.