 Collapsible tube flow

A one-dimensional collapsible tube model couples area $\alpha$, mean axial velocity $u$ and internal pressure $p$: ${\alpha }_t+(\alpha u{)}_x=0$ and $u_t+u{u}_x=-p_x-R(\alpha )u$. A wall relation specifies internal minus external pressure as a function of area. Decreasing area generally increases the friction coefficient $R$, while positive wall stiffness gives real small-amplitude wave speed.