Abstract

Spectral Enclosures for the Orr-Sommerfeld Equation M. Plum and J. Lahmann One of the governing equations of hydrodynamic stability is the {\it Orr-Sommerfeld} equation $$\left . \begin{array}{rcl} (-D^2+a^2)^2u+iaR[V\cdot(-D^2+a^2)u+V''\cdot u)&=&\lambda(-D^2+a^2)u \,\, \text{on}\, I\\ u=u'&=&0 \,\, \text{on}\, \partial I \end{array} \right \}(1) $$ where $I$ is a real interval, $D=d/dx$, and $V \in C^2(I)$ is the profile of an underlying flow with Reynolds number $R$, which is perturbed by a single-mode perturbation with wave number $a > 0$. Depending on the whole spectrum of (some suitable operator realization of) (1) being contained in the right complex half-plane or not, the flow is stable or unstable under the perturbation.\\ We focus on the {\it Blasius} profile $V$ where $I=[0, \infty)$ and $V=f'$, with $f$ denoting the unique solution of Blasius' boundary value problem $$f'''+ff''=0, f(0)=f'(0)=0, f'(\infty)=1 .$$ In the lecture, we will calculate the essential spectrum of (1) in closed form, using Kato's theorem on stability of the essential spectrum under relative compact perturbations, present a computer-assisted method for computing {\it eigenvalue enclosures} for non-selfadjoint problems such as (1). It is based on a fixed-point formulation of (1) and on eigenvalue bounds for a certain auxiliary {\it selfadjoint} problem, for which well known variational methods can be used. As a specific result, for a certain parameter constellation (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.

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