= Solution
Complete the square in the <dispersion relation>:
$$
\nu=-\frac{iR_0}{2}\pm\sqrt{k^2c_0^2-\frac{R_0^2}{4}+inkR_0U}.
$$
If the imaginary part of the square root has magnitude $b$, then
$$
b^2=\frac12\left[\sqrt{\left(k^2c_0^2-\frac{R_0^2}{4}\right)^2+n^2k^2R_0^2U^2}-\left(k^2c_0^2-\frac{R_0^2}{4}\right)\right].
$$
One root grows exactly when $b>R_0/2$. For $k\ne0$, squaring the equivalent positive inequality gives $n^2U^2>c_0^2$. Thus \b[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 $\nu=kc_0+\delta\nu_1+O(\delta^2)$. Substitution yields
$$
(2kc_0+iR_0)\nu_1=ikR_0c_0.
$$
Taking its imaginary part gives the growth rate
$$
\boxed{\operatorname{Im}\omega=\frac{\delta R_0k^2c_0^2}{2(k^2c_0^2+R_0^2/4)}+O(\delta^2).}
$$
Area 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.
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