Separable Non-Parallel and Unsteady Flow Stability Problems

Abstract

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.