= Friction-induced instability of a collapsible-tube flow
{title2=$\nu^2+iR_0\nu-k^2c_0^2-inkR_0U=0$}
Let $R(\alpha)\propto\alpha^{-n}$, with $n>0$, and use disturbances $e^{i(kx-\omega t)}$ about uniform <velocity> $U$. The intrinsic <frequency> is $\nu=\omega-kU$. Linearized <mass conservation> and the decrease in friction as area grows give the displayed <dispersion relation>. For nonzero $k$ and positive flow, one branch grows exactly when $nU>c_0$. At $nU=c_0$, one branch is neutral, $\nu=kc_0$, while the other remains damped, $\nu=-kc_0-iR_0$. The maintained external-<pressure gradient> supplies the energy; positive friction alone does not imply linear stability of a driven flow.
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