= Collapsible tube flow
A one-dimensional <collapsible tube flow> model couples <cross-sectional area> $\alpha$, mean axial <velocity> $u$ and wall deformation. For constant <fluid> <mass density> $\rho$, use $P$ for physical <pressure> and write the <kinematic pressure> $p=P/\rho$ in the <momentum> equation:
$$
\alpha_t+(\alpha u)_x=0,\qquad
u_t+uu_x=-p_x-R(\alpha)u.
$$
The first equation is <mass conservation>. The second is axial <momentum conservation> with a distributed frictional <acceleration> $-R(\alpha)u$. Its <pressure> convention is essential: using physical $P$ instead would require $-\rho^{-1}P_x$.
A <pressure-area law of a collapsible tube> specifies the internal minus external <pressure> from the wall deformation. Decreasing <area> generally raises the resistance coefficient $R$, while a positive derivative of the restoring wall law gives real small-amplitude <tube wave speeds>. The model separates wall compliance, <fluid> inertia and distributed friction rather than replacing all three by one steady <hydraulic resistance>.
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