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.

For steady uniform-density inviscid flow with pressure law $p=p_0+k(r-R)$ and a prescribed thickness $h$, mass conservation and the Bernoulli equation give $h/R=1-s^{-1/2}+\lambda (1-s^2)/4$, where $s=u/V$ and $\lambda =2\rho V^2/(kR)$. The function has a unique maximum at $s={\lambda }^{-2/5}$, producing the displayed choking bound. Below it there are two positive roots; continuity from the upstream state selects the lower-speed root for $\lambda <1$ and higher-speed root for $\lambda >1$. At $\lambda =1$, the maximum is zero and any positive prescribed thickness is incompatible with this model.

A positive collapsible tube flow is subcritical when its characteristic speeds $u+c$ and $u-c$ have opposite signs. A downstream boundary can then send information upstream. For a quadratic pressure-area law and fixed dimensionless flux $q$, this corresponds to $A/A_0>\sqrt{q}$.

A positive collapsible tube flow is supercritical when both characteristic speeds $u\pm c$ are positive relative to the wall. Information then propagates downstream on both branches. For a quadratic pressure-area law and fixed dimensionless flux $q$, the condition is $A/A_0<\sqrt{q}$.

An elastic jump is a short transition from a fast, narrow supercritical collapsible tube flow to a slower, wider subcritical flow. Neglecting body force and distributed friction across the jump conserves flux and the momentum flux shown above. Distinct upstream and downstream areas therefore obey ${\alpha }_2^3+{\alpha }_1{\alpha }_2^2+{\alpha }_1^2{\alpha }_2=3q^2/{\alpha }_1$. The flux function has its unique minimum at $\alpha =\sqrt{q}$, so an upstream state below this value has one conjugate state above it. The jump dissipates mechanical energy; it does not conserve a Bernoulli constant.

For $p=P_0+\rho c_0^2{\alpha }^2/2$ and constant flux $c_0{A}_{0}q$, a downhill collapsible tube flow obeys the displayed steady equation. Uniform area satisfies $qR({\alpha }_1)=\beta$. Since $R^{\prime }<0$, 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.

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, $\nu =k{c}_0$, while the other remains damped, $\nu =-k{c}_0-i{R}_0$. The maintained external-pressure gradient supplies the energy; positive friction alone does not imply linear stability of a driven flow.

The pressure-area law of a collapsible tube describes transverse mechanical equilibrium of a deformable tube wall. On a stable branch ${\tilde{P}}^{\prime }(\alpha )>0$. Linearizing mass and momentum conservation about a stationary uniform tube gives the displayed wave speed. External pressure is prescribed separately and is not part of the elastic wall stiffness.

Linearizing area conservation and the axial momentum equation of a collapsible tube flow gives waves travelling relative to the fluid at speeds $\pm c$, where the displayed relation uses the area derivative at fixed external pressure. For $p=P_0+\rho c_0^2(A/A_0{)}^2/2$, it gives $c=c_{0}A/A_0$.