INVESTIGATION of linear instability mechanisms is essential for understanding the process of transition from laminar to turbulent flow. Many studies over several decades have reported results in simple one-dimensional steady laminar basic flows, such as the boundary and shear layers. However, most flows of practical engineering significance remain unexplored. The main reason is that the geometry and underlying basic state in most applications depend on an inhomogeneous manner on more than one spatial direction, which does not permit use of simplified equations. The consideration of two or three inhomogeneous directions in the stability problem formulation results in formidable computational costs for completing parametric studies, which are, on the other hand, mandatory from a physical point of view.

Once a laminar, steady, or unsteady basic flow has been established, the Navier–Stokes equations can be written in terms of disturbance variables and linearized to study small-amplitude disturbance development in this basic flow. If the latter depends in an inhomogeneous manner on two spatial directions, the disturbance can be considered periodic on the third, homogeneous spatial direction, along which Fourier modes can be introduced. An exponential dependence on time is assumed for the disturbances, and therefore the problem becomes a two-dimensional partial derivative-