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<p>It is well-known that shear flows in a strip or in the half plane are unstable for the Navier-Stokes equations with Dirichlet boundary conditions if the viscosity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\nu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation> is small enough, provided the horizontal wave number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> lies in a small interval, between the so called lower and upper marginal stability curves. The corresponding instabilities are called Tollmien-Schlichting waves. In this article, we give a simple presentation of the dispersion relation of these waves and study its mathematical properties.</p>

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The Dispersion Relation of Tollmien-Schlichting Waves

  • 期刊论文

  • 发表日期：

    2026年2月2日

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  • Dongfen Bian,
  • Shouyi Dai,
  • Emmanuel Grenier

摘要

It is well-known that shear flows in a strip or in the half plane are unstable for the Navier-Stokes equations with Dirichlet boundary conditions if the viscosity \(\nu \) ν is small enough, provided the horizontal wave number \(\alpha \) α lies in a small interval, between the so called lower and upper marginal stability curves. The corresponding instabilities are called Tollmien-Schlichting waves. In this article, we give a simple presentation of the dispersion relation of these waves and study its mathematical properties.