Analysis on the Orr-Sommerfeld Equations for the Incompressible MHD System

Description

It is a classical problem in fluid dynamics about the stability and instability of different hydrodynamic patterns in various physical settings, in particular the inviscid limit of laminar flow and boundary layer. The study can be traced back to the early work by Lord Rayleigh and Heisenberg among many others. In fact, for inviscid flow, the classical criterion for stability was given by Rayleigh saying that a necessary condition for instability of shear flow profile is that the shear flow must have an inflection point, and it was later refined by Fjortoft. On the other hand, for viscous flow, except the case of the linear Couette flow, which is proved to be linearly stable for all Reynolds numbers by Romanov, all other profiles (including those which are inviscid stable) are shown to be linearly unstable for large Reynolds numbers.One of the powerful analytic tools initiated by Sommerfeld and Orr is the spectral analysis by studying the Fourier normal mode behavior through the famous OrrSommerfeld equation (denoted by OS equation) derived from the linearization of the incompressible Navier-Stokes equations. As for the stability and instability investigation, tremendous progress has been made since the pioneer work by Heisenberg, C.C. Lin, Tollmien, and Schlichting, motivated by the study of the boundary layer around a solid body. The related theories influenced by the wing design for airplanes have crucial impact on the developement of aerodynamics because of the importance in understanding on the transition from the laminar flow to turbulence that is also related to the separation of boundary layer. A large number of literatures have been devoted to the estimation of the critical Reynolds number for stability based on the Navier-Stokes equations for various shear flow pattens, such as Poiseuille flow, Blasius profile, exponential suction profile, etc. Without viscosity, the OS equation is the Rayleigh equation for inviscid flow. Hence, OS equation can be viewed as a singular perturbation of Rayleigh equation in high Reynolds number.This project aims to study the OS system of equations derived from the linearization of the MHD system around a shear flow profile. With our previous understanding on the stability mechanism of strong tangential magnetic field on the boundary layer system, we will investigate the instability mechanism of the weak tangential magnetic field even when velocity profile has the corresponding stable structure for the incompressible Navier-Stokes equations.