Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 81 / 2 / iv

Complete the square in the dispersion relation:If the imaginary part of the square root has magnitude $b$, thenOne root grows exactly when $b>R_0/2$. For $k\ne 0$, squaring the equivalent positive inequality gives $n^2{U}^2>c_0^2$. Thus positive flow is unstable whenever $\boxed{U>c_0/n}$. This proves the entire stated range, not just a small neighborhood of threshold. Zero wavenumber has only a neutral uniform-area change and a damped velocity mode.

Near threshold write $nU=c_0(1+\delta )$ and follow the neutral branch as $\nu =k{c}_0+\delta {\nu }_1+O({\delta }^2)$. Substitution yieldsTaking its imaginary part gives the growth rateArea increases reduce friction, causing a velocity increase that feeds the area disturbance through mass conservation. The pressure-driven base flow supplies the energy; the positive friction coefficient does not by itself prevent instability of that maintained flow.