  The so-called 'direct' approach to separation of variables in linear PDEs is
applied to the hydrodynamic stability problem. Calculations are made for the
complete linear stability equations in cylindrical coordinates. Several classes
of the exact solutions of the Navier-Stokes equations describing spatially
developing and unsteady flows, for which the linear stability problems can be
rigorously reduced to eigenvalue problems of ordinary differential equations,
are defined. Those exactly solvable nonparallel and unsteady flow stability
problems can be used for testing approximate approaches and the methods based
on direct numerical simulations of the (linearized) Navier-Stokes equations.
The exact solutions of the viscous incompressible Navier-Stokes equations
determined as the basic states, for which the linear stability problem is
exactly separable, may be themselves of interest from theoretical and
engineering points of view.
