A computer‐assisted instability proof for the Orr‐Sommerfeld equation with Blasius profile

Abstract

AbstractThe Orr‐Sommerfeld equation is one of the governing equations of hydrodynamic stability. Mathematically, it constitutes a non‐selfadjoint eigenvalue problem. Depending on its spectrum being contained in the right complex half‐plane or not, the underlying flow is stable or unstable under some given perturbation. Here, we focus on the Blasius profile modelling a flow along a wall. We present a computer‐assisted method for computing eigenvalue enclosures for such non‐selfadjoint problems. As a specific result, for a particular parameter constellation in the Orr‐Sommerfeld equation (often used as test example in the engineering literature), we enclose an eigenvalue in a circle which is completely contained in the left half‐plane. This constitutes the first rigorous proof of instability for the Orr‐Sommerfeld equation with Blasius profile.