We study unsteady shear flows realized in a half-plane with viscous incompressible fluid, where the law of motion of the boundary oscillating along itself is given. Either the longitudinal velocity of the boundary or the shear stress on it can be specified. The statement of the linearized problem with respect to small initial perturbations imposed on the kinematics in the entire half-plane is presented. For a flat picture of perturbations, the statement consists of a single biparabolic equation with variable coefficients with respect to the complex-valued stream function that generalizes the Orr-Sommerfeld equation to the nonstationary case and of four homogeneous boundary conditions. Using the method of integral relations, we derive exponential estimates for the decay of perturbations. The result is compared with the three-dimensional picture of variations.

中文翻译：

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具边界振荡的粘性半平面中一维非定常剪切上摄动的演变

我们研究了在具有粘性不可压缩流体的半平面中实现的非恒定剪切流，其中给出了沿其自身振荡的边界运动定律。可以指定边界的纵向速度或边界上的剪应力。提出了关于在整个半平面上施加给运动学的小的初始扰动的线性化问题的陈述。对于扰动的平面图，该语句由一个具有可变系数（相对于复值流函数的系数可变）的双抛物线方程组成，该方程将Orr-Sommerfeld方程推广到非平稳情况和四个齐次边界条件。使用积分关系方法，我们得出了扰动衰减的指数估计。