Rayleigh's inflection point criterion

Remarks

Write $c = c_r + i c_i$ with $c_i > 0$, so that $U - c \neq 0$ on $[y_1, y_2]$. Divide the equation by $U - c$, multiply by the complex conjugate $\bar\varphi$ and integrate by parts using the boundary conditions: $$\int_{y_1}^{y_2} \left( \abs{\varphi'}^2 + \alpha^2 \abs{\varphi}^2 \right) dy + \int_{y_1}^{y_2} \frac{U'' \abs{\varphi}^2}{U - c} \, dy = 0.$$ The imaginary part gives $c_i \int U'' \abs{\varphi}^2 / \abs{U - c}^2 \, dy = 0$. If $U''$ kept one sign, this would force $U'' \abs{\varphi}^2 = 0$ everywhere, and then the real part would give $\int ( \abs{\varphi'}^2 + \alpha^2 \abs{\varphi}^2 ) \, dy = 0$, so $\varphi = 0$.

The Rayleigh equation is the limit $\mathrm{Re} \to \infty$ of the Orr–Sommerfeld equation, that is, the linearisation of the Euler equations. Plane Couette flow $U(y) = y$ and plane Poiseuille flow $U(y) = 1 - y^2$ have no inflection point, so they are stable in the inviscid theory. Plane Poiseuille flow is nevertheless unstable for large finite Reynolds number, so the criterion says nothing about viscous instability. The criterion is necessary but not sufficient. Fjørtoft later gave a sharper necessary condition: $U''(U - U(y_s)) < 0$ somewhere in the flow.

Going deeper

For a student

The criterion is the first result of inviscid hydrodynamic stability. Its proof is an energy-type identity: multiply the Rayleigh equation by the conjugate amplitude and integrate. The same technique, applied to the Taylor–Goldstein equation for stratified flows, gives the Miles–Howard criterion and Howard's semicircle theorem. Flows with an inflection point, such as the profile $U(y) = \tanh y$ of a mixing layer, are typically unstable at high Reynolds number through an inviscid mechanism. Flows without one, such as boundary layers with a favourable pressure gradient, become unstable only through viscous effects, as Tollmien–Schlichting waves.