磁流体方程平面Couette流的稳定性分析

层流扰动问题的稳定性是流体动力学稳定性理论中最重要的组成部分，最核心的问题就是找到临界的 Reynolds 数 用于判断所考察的运动是否稳定。然而在实际情况中总是不可避免地存在着干扰因素，所以不稳定的定常流在现实中是不可能存在的。我们之所以要研究层流中最简单的平面 Couette 流的稳定性，是因为对它理解的越充分，越会对理解更为复杂的流动提供更多有用的信息。本课题主要研究磁流体方程(MHD)平面Couette流的稳定性和不稳定性。首先，考虑三维可压缩MHD平面Couette流的稳定性，并且考察雷诺数和磁雷诺数对Couette流稳定性的影响。其次，考虑当磁场给定无滑移边界条件，三维不可压缩MHD平面Couette流的稳定性。最后，研究当磁场给定具有物理意义的完美磁传导边界条件，三维不可压缩MHD平面Couette流的稳定性和不稳定性，并分析不同边界条件带来的影响。

The most important part of the hydrodynamic stability theory is the investigation of how laminar flow becomes unstable, and it is of particular interest to determine that value of the Reynolds number at which the onset of instability occurs. This is an important issue since a stationary unstable flow in general can’t exist in reality. A better understanding of the stability of the simple examples of laminar flows just like the plane Couette flow could provide information useful for more complicated flows. This project is dedicated to the research of the stability or instability of the plane Couette flow for Magnetohydrodynamic equations (MHD). First, we consider the stability of three-dimensional compressible MHD around the plane Couette flow and want to show the influence of Reynolds and magnetic Reynolds number on the plane Couette flow. Then, we study the stability of three-dimensional incompressible MHD around the plane Couette flow with non-slip boundary condition imposed on the magnetic fields. Finally, we are also interested in stability or instability of three-dimensional incompressible MHD around the plane Couette flow with perfect conduction boundary condition imposed on the magnetic fields and want to show the influence result due to the different choices of boundary condition.