The study presents symmetry classifications of the linearized Navier–Stokes equations, governing the three-dimensional incompressible plane shear flows. The linearization is done with respect to small perturbations. In the case of a two-dimensional shear flow with a linear profile, Nold and Oberlack (PoF, 2013) showed the existence of three different kinds of linear instability modes using the framework of Lie symmetry classification. Those perturbation modes are normal mode, kelvin mode, and a new type invariant mode. We have extended their analysis for a three-dimensional plane shear flow with linear as well as non-linear base profiles. The invariant ansatz functions are systematically derived employing the full set of symmetries. The analysis is done for both viscous and inviscid flows by considering the linear, exponential, and fractional shear flow profiles. In the derivation process, the set of infinitesimal generators for the generalized system is first obtained using the classical Lie symmetry analysis, and then, some additional symmetries are searched out for each sub-case. Further, the governing system of partial differential equations is converted into ordinary differential equations by using symmetries and invariant conditions. The most popular three-dimensional normal modes and the Orr–Sommerfeld equation are acquired by taking the general symmetry. Moreover, for each of the sub-cases, we have derived the possible exact solutions of the associated system, and the behaviors of the solutions are explored for different parameter ranges.

中文翻译：

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三维扰动剪切流的 On the Lie 对称分析


该研究提出了线性 Navier-Stokes 方程的对称分类，控制了三维不可压缩平面剪切流。线性化是针对小扰动完成的。在具有线性剖面的二维剪切流的情况下，Nold 和 Oberlack （PoF， 2013） 使用 Lie 对称分类框架展示了三种不同类型的线性不稳定模式的存在。这些扰动模式是法则模式、开尔文模式和一种新型的不变模式。我们扩展了他们对具有线性和非线性基础剖面的三维平面剪切流的分析。不变的 ansatz 函数是利用全套对称性系统地推导的。通过考虑线性、指数和分数剪切流剖面，对粘性流和无粘性流进行分析。在推导过程中，首先使用经典的 Lie 对称分析获得广义系统的无穷小生成器集，然后为每个子情况寻找一些额外的对称性。此外，通过使用对称性和不变条件，将偏微分方程的控制系统转换为常微分方程。最流行的三维正则模态和 Orr-Sommerfeld 方程是通过取一般对称性获得的。此外，对于每个子情况，我们推导出了相关系统可能的精确解，并针对不同的参数范围探索了解的行为。