Instability of plane shear flows

This paper obtains the existence of unstable modes of the Rayleigh equation with smooth or piecewise smooth steady states. In the reasoning, we employ a crucial tool that displays the asymptotic distribution of eigenvalues of non-symmetric Sturm–Liouville operators. For flows with piecewise smooth velocity profiles, we propose a new framework for eigenvalue problems with interior singularities. Examples are presented to show the scope of our criteria.