Analysis on Tollmien-Schlichting wave in the Prandtl-Hartmann Regime

TL;DR

This paper rigorously analyzes the Tollmien-Schlichting wave instability in the MHD system within the Prandtl-Hartmann regime, revealing that the transverse magnetic field does not stabilize the wave, aligning with physical intuition.

Contribution

It introduces a new decomposition of the Orr-Sommerfeld operator and determines the critical Gevrey index for instability, matching that of the Navier-Stokes equations.

Findings

The transverse magnetic field does not stabilize Tollmien-Schlichting waves.

A new decomposition of the Orr-Sommerfeld operator is proposed.

The critical Gevrey index for instability is established.

Abstract

In this paper, we study the instability induced by the Tollmien-Schlichting wave governed by the MHD system in the Prandtl-Hartmann regime. The interaction of the inviscid mode and viscous mode that leads to the instability is analyzed by the introduction of a new decomposition of the Orr-Sommerfeld operator on the velocity and magnetic fields. The critical Gevrey index for the instability is justified by constructing the growing mode in the essential frequency and it is shown to be the same as the incompressible Navier-Stokes equations in the Prandtl regime. This result justifies rigorously the physical understanding that the transverse magnetic field to the boundary in the Prandtl-Hartmann regime has no extra stabilizing effect on the Tollmien-Schlichting wave.