Linear Instability of the Prandtl Equations via Hypergeometric Functions and the Harmonic Oscillator

Abstract

We establish a deep connection between the Prandtl equations linearised around a quadratic shear flow, confluent hypergeometric functions of the first kind, and the Schr\"odinger operator. Our first result concerns an ODE and a spectral condition derived in [10], associated with unstable quasi-eigenmodes of the Prandtl equations. We entirely determine the space of solutions in terms of Kummer's functions. By classifying their asymptotic behaviour, we verify that the spectral condition has a unique, explicitly determined pair of eigenvalue and eigenfunction, the latter being expressible as a combination of elementary functions. Secondly, we prove that any quasi-eigenmode solution of the linearised Prandtl equations around a quadratic shear flow can be explicitly determined from algebraic eigenfunctions of the Schr\"odinger operator with quadratic potential. We show finally that the obtained analytical formulation of the velocity align with previous numerical simulations in the literature.