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

**Abstract:**

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).