Separation of variables in the hydrodynamic stability equations

The problem of variable separation in the linear stability equations, which govern the disturbance behaviour in viscous incompressible fluid flows, is discussed. The so-called direct approach, in which a form of the 'ansatz' for a solution with separated variables as well as a form of reduced ODEs are postulated from the beginning, is applied. The results of application of the method are the new coordinate systems and the most general forms of basic flows, which permit the postulated form of separation of variables. Then the basic flows are specified by the requirement that they themselves satisfy the Navier-Stokes equations. Calculations are made for the (1+3)-dimensional disturbance equations written in Cartesian and cylindrical coordinates. The fluid dynamics interpretation and stability properties of some classes of the exact solutions of the Navier-Stokes equations, defined as flows for which the stability analysis can be reduced via separation of variables to the eigenvalue problems of ordinary differential equations, are discussed. The eigenvalue problems are solved numerically with the help of the spectral collocation method based on Chebyshev polynomials. For some classes of perturbations, the eigenvalue problems can be solved analytically. Those unique examples of exact (explicit) solution of the nonparallel unsteady flow stability problems provide a very useful test for numerical methods of solution of eigenvalue problems, and for methods used in the hydrodynamic stability theory, in general.