 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: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.