We study how the pattern of perturbations superimposed on a plane-parallel time-periodic flow of a Newtonian viscous fluid evolves in a layer in which one of the boundaries performs longitudinal harmonic vibrations along itself, with the zero-friction slip of material allowed on the other boundary. We pose a generalized Orr–Sommerfeld problem as a linearized problem of hydrodynamic stability of unsteady-state viscous incompressible flows. Using the integral relation method, based on variational inequalities for quadratic functionals and developed as applied to unsteady-state flows, we derive integral estimates sufficient for the exponential decay of the initial perturbations. For each wave number, these estimates are inequalities relating three constant dimensionless quantities, viz., period-average depth-maximum shear velocity in the layer, boundary vibration amplitude, and the Reynolds number. We compare the established stability estimates for the planar and three-dimensional perturbation patterns.

中文翻译：

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叠加在粘性层纵向谐波振动上的扰动指数衰减估计

我们研究了叠加在牛顿粘性流体的平面平行时间周期流上的扰动模式如何在其中一个边界沿其自身执行纵向谐波振动的层中演化，并且允许材料的零摩擦滑动其他边界。我们将广义 Orr-Sommerfeld 问题作为非稳态粘性不可压缩流动的流体动力学稳定性的线性化问题。使用积分关系方法，基于二次函数的变分不等式，并在应用于非稳态流动时得到发展，我们推导出足以满足初始扰动指数衰减的积分估计。对于每个波数，这些估计是与三个恒定的无量纲量相关的不等式，即，层中的周期平均深度-最大剪切速度，边界振动幅值和雷诺数。我们比较了平面和三维扰动模式的既定稳定性估计。