Analysis of the Tollmien-Schlichting wave in the Prandtl-Hartmann regime

Cheng-Jie Liu; Tong Yang; Zhu Zhang

doi:10.1016/j.matpur.2022.07.004

Author(s)

Cheng-Jie Liu
Tong Yang
Zhu Zhang

Related Research Unit(s)

Department of Mathematics

Detail(s)

Original language	English
Pages (from-to)	58-105
Journal / Publication	Journal des Mathematiques Pures et Appliquees
Volume	165
Online published	2 Aug 2022
Publication status	Published - Sept 2022

Link(s)

DOI	DOI https://doi.org/10.1016/j.matpur.2022.07.004 Final Published version
	Check@CityULib
Link to Scopus	https://www.scopus.com/record/display.uri?eid=2-s2.0-85136193041&origin=recordpage
Permanent Link	https://scholars.cityu.edu.hk/en/publications/publication(b43a4e14-1331-4f26-b59f-5b49a67e5485).html

Abstract

In this paper, we study the instability induced by the Tollmien-Schlichting wave governed by the incompressible 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.