= Spatial attraction of a uniform downhill tube flow
{title2=$(\alpha-q^2/\alpha^3)\alpha_x=(g\sin\theta/c_0^2)[1-qR(\alpha)/\beta]$}
For $p=P_0+\rho c_0^2\alpha^2/2$ and constant flux $c_0A_0q$, a downhill <collapsible tube flow> obeys the displayed steady equation. Uniform area satisfies $qR(\alpha_1)=\beta$. Since $R'<0$, linearizing about this root gives downstream exponential decay precisely when $\alpha_1^2<q$. This is spatial stability of a steady inlet-value problem, not a guarantee of temporal stability of the driven flow. The <wave speed> is $c_0\alpha$, so the stable spatial branch is supercritical.
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