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

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Original languageEnglish
Pages (from-to)58-105
Journal / PublicationJournal des Mathematiques Pures et Appliquees
Online published2 Aug 2022
Publication statusPublished - Sept 2022


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.

Research Area(s)

  • Hartmann layer, Incompressible MHD, Orr-Sommerfeld equation, Tollmien-Schlichting wave