Abstract

From the standpoint of the linearized stability theory, two eigenvalue problems for the Orr–Sommerfeld equation with two groups of boundary conditions having a certain mechanical meaning are considered. The stability parameter, which is a real part of the spectral parameter, is estimated on the basis of the integral relations method operating with quadratic functionals. The technique of the method involves the application of the Friedrichs inequality for various classes of complex-valued functions. Using the minimizing property of the first positive eigenvalues in the corresponding problems, the values of the constants in some Friedrichs inequalities are increased, which entails the strengthening of the stability sufficient integral estimates for plane-parallel shear flows in a plane layer.

中文翻译：

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平面平行流的弗里德里希不等式和锐化充分稳定条件

摘要

从线性化稳定性理论的角度，考虑了具有一定力学意义的两组边界条件的Orr-Sommerfeld方程的两个特征值问题。稳定性参数是谱参数的实部，它是基于使用二次泛函的积分关系方法来估计的。该方法的技术涉及将弗里德里希不等式应用于各种类型的复值函数。利用相应问题中第一个正特征值的最小化性质，增加了一些Friedrichs不等式中的常数值，这需要加强平面层中平面平行剪切流的稳定性充分积分估计。