Stability analysis of a class of unsteady nonparallel incompressible flows via separation of variables
We present the stability analysis of some viscous incompressible unsteady nonparallel flows (exact solutions of the continuity and Navier-Stokes equations in cylindrical coordinates) that is based on separation of variables in the linearized equations for the flow perturbations. The basic flow and perturbation solutions have been identified by applying (in some restricted form) the so-called direct approach to separation of variables in linear partial differential equations (Zhdanov and Zhalij 1999). We study stability properties of the unsteady nonparallel flows in an expanding rotating cylinder and in a gap between two concentric expanding rotating cylinders by solving the corresponding eigenvalue problems of ordinary differential equations. The eigenvalue problems were 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) solutions of the linear stability equations for 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.
Joint work with Alexander Zhalij (Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine) and Georgy Burde (Jacob Blaustein Institute for Desert Research, Ben-Gurion University, Sede-Boker Campus, Israel).