Stability analysis of a class of unsteady nonparallel incompressible flows via separation of variables

Stability of some unsteady three-dimensional flows (exact solutions of the viscous incompressible Navier-Stokes equations in cylindrical coordinates) is studied via separation of variables in the linearized equations for the flow perturbations. The flows in an expanding rotating porous cylinder and in a gap between two coaxial rotating cylinders are considered. Converting the stability equations to the new variables allows perturbation forms (counterparts of normal modes of the steady state parallel flow stability problem) such that the linear stability problems are exactly reduced to eigenvalue problems of ordinary differential equations. The eigenvalue problems are solved numerically with the help of the spectral collocation method based on Chebyshev polynomials. The results showing dependence of the stability threshold on the parameters of the problems and a spatial structure of the unstable perturbation modes are presented. For some classes of perturbations, exact analytical solutions of the eigenvalue problems are available. A combination of analytical and numerical solutions can provide useful testing for numerical methods used in the hydrodynamic stability studies. It may also provide a basis for a well-grounded discussion of some problematic points of (numerical) stability analysis. In particular, in the present paper, a problem of formulation of the boundary conditions for perturbations at the axis r=0 is discussed on the basis of the solutions obtained.