Write the <Rayleigh equation for inviscid shear flow> as
$$
\phi''-\alpha^2\phi-
\frac{U''}{U-c}\phi=0,
\qquad \phi(0)=\phi(1)=0.
$$
Multiply by $\phi^*$, integrate, and use <integration by parts>:
$$
\int_0^1(|\phi'|^2+\alpha^2|\phi|^2)\,dz
+\int_0^1\frac{U''(U-c^*)}{|U-c|^2}|\phi|^2\,dz=0.
$$
Its imaginary part is
$$
c_i\int_0^1\frac{U''}{|U-c|^2}|\phi|^2\,dz=0.
$$
For an unstable mode $c_i>0$, the integral can vanish only if $U''$ changes sign somewhere in the flow. Thus the velocity profile must have an inflection point. This is <Rayleigh's inflection-point theorem>; it is necessary, not sufficient, for inviscid instability.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 331 / 1 / i / a / Solution

 Solution

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