Abstract

An asymptotic analysis of the eigenvalues and eigenfunctions in the Orr–Sommerfeld problem is carried out in the case when the velocity of the main plane-parallel shear flow in a layer of a Newtonian viscous fluid is low in a certain measure. The eigenvalues and corresponding eigenfunctions in the layer at rest are used as a zero approximation. For their perturbations, explicit analytical expressions are obtained in the linear approximation. It is shown that, FOR low velocities of the main shear flow, the perturbations of eigenvalues corresponding to monotonic decay near the rest in a viscous layer are such that, regardless of the velocity profile, the decay decrement remains the same, but an oscillatory component appears that is smaller in order by one than this decrement.

中文翻译：

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低速无扰流的Orr-Sommerfeld问题特征值的渐近性

摘要

当牛顿粘性流体层中主平面平行剪切流的速度在一定程度上较低时，对Orr-Sommerfeld问题中的特征值和特征函数进行渐近分析。静止层中的特征值和相应的特征函数用作零近似值。对于它们的扰动，以线性逼近获得了明确的解析表达式。结果表明，对于主剪切流的低速度，与粘性层中其余部分附近的单调衰减相对应的特征值的扰动使得，无论速度分布如何，衰减减量均保持不变，但具有振荡分量似乎比此减量小一个顺序。