A case study in boundary layer flows
A two-dimensional perturbation in an incompressible boundary layer flow is determined using linear stability theory. The linearized Navier Stokes equations are rewritten in matrix operator form. This linear PDE with boundary conditions is reduced to an ordinary differential equations after a Fourier transform in space and Laplace transform in time. Knowing the singularities allows us to use the Residue Theorem to invert the Laplace transform and obtain a formal solution of the temporal evolution. The solution is a linear combination of the discrete and continuous eigenfunctions and describes the temporal evolution of flow perturbations inside and outside the boundary respectively. This solution determines the temporal stability behavior of the perturbations.