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

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

Cheng-Jie Liu, Tong Yang^*, Zhu Zhang

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

3 Citations (Scopus)

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.

Original language	English
Pages (from-to)	58-105
Journal	Journal des Mathematiques Pures et Appliquees
Volume	165
Online published	2 Aug 2022
DOIs	https://doi.org/10.1016/j.matpur.2022.07.004
Publication status	Published - Sept 2022

Funding

The research of C.-J. Liu was supported by the National Key R&D Program of China (No. 2020YFA0712000) and National Natural Science Foundation of China (No. 11801364). The research of T. Yang was supported by the General Research Fund of Hong Kong CityU No. 11303521.

Research Keywords

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

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title = "Analysis of the Tollmien-Schlichting wave in the Prandtl-Hartmann regime",

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.",

keywords = "Hartmann layer, Incompressible MHD, Orr-Sommerfeld equation, Tollmien-Schlichting wave",

author = "Cheng-Jie Liu and Tong Yang and Zhu Zhang",

year = "2022",

month = sep,

doi = "10.1016/j.matpur.2022.07.004",

language = "English",

volume = "165",

pages = "58--105",

journal = "Journal des Mathematiques Pures et Appliquees",

issn = "0021-7824",

publisher = "Elsevier Masson SAS",

}

TY - JOUR

T1 - Analysis of the Tollmien-Schlichting wave in the Prandtl-Hartmann regime

AU - Liu, Cheng-Jie

AU - Yang, Tong

AU - Zhang, Zhu

PY - 2022/9

Y1 - 2022/9

N2 - 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.

AB - 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.

KW - Hartmann layer

KW - Incompressible MHD

KW - Orr-Sommerfeld equation

KW - Tollmien-Schlichting wave

UR - http://www.scopus.com/inward/record.url?scp=85136193041&partnerID=8YFLogxK

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85136193041&origin=recordpage

U2 - 10.1016/j.matpur.2022.07.004

DO - 10.1016/j.matpur.2022.07.004

M3 - RGC 21 - Publication in refereed journal

SN - 0021-7824

VL - 165

SP - 58

EP - 105

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

ER -

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

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Research Keywords

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GRF: Analysis on the Orr-Sommerfeld Equations for the Incompressible MHD System

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

Abstract

Funding

Research Keywords

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GRF: Analysis on the Orr-Sommerfeld Equations for the Incompressible MHD System

Cite this