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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.

Write $\alpha =A/A_0$ and $G=g\sin \theta >0$. The one-dimensional mass conservation and axial momentum equations consistent with the stated resistance areIn a steady flow, $Au=c_0{A}_{0}q$ and $u=c_{0}q/\alpha$. Since $p_{\alpha }=\rho c_0^2\alpha$, the momentum equation becomesThus a uniform inlet state continues as a uniform steady solution precisely whenTo examine downstream spatial stability of this steady solution, put $\alpha ={\alpha }_1+z(x)$ and linearize. The numerator vanishes at equilibrium, soSince $R^{\prime }<0$, the numerator of $s$ is positive. The displacement decays as $x$ increases exactly when ${\alpha }_1^4<q^2$, orThis proves the spatial attraction of a uniform downhill tube flow appropriate to the steady inlet-value problem.

The tube wave speed is $c^2=(A/\rho )dp/dA=c_0^2{\alpha }^2$, while $u=c_{0}q/\alpha$. The inequality is therefore $u>c$: the inlet lies on the supercritical tube flow branch, with both inviscid characteristic speeds $u\pm c$ directed downstream. A subcritical state has a characteristic carrying downstream boundary information upstream; the critical state makes the steady area equation singular unless its forcing also vanishes.

Spatial attraction must not be confused with temporal stability. The stated hypotheses do not guarantee decay of all time-dependent perturbations. For example, take $R={\alpha }^{-2}$ and a uniform supercritical state. The damping coefficient multiplying $u$ is $f=R_0{A}_0/\alpha$, and for perturbations $e^{i(kx-\omega t)}$ about area $\bar{A}$ and speed $\bar{u}$, set $s_t=-i(\omega -k\bar{u})$. Linearizing the two evolution equations givesIts small-$k$ branch is $s_t=-ik\bar{u}+k^2({\bar{u}}^2-c^2)/f+O(k^3)$, which grows when $\bar{u}>c$. This is a friction-induced instability of a collapsible-tube flow. Consequently the advertised stability is valid as downstream stability of the steady solution; an unrestricted temporal-stability claim would require further assumptions.

The pressure-area law of a collapsible tube specifies transverse mechanical equilibrium of a deformable tube wall. With physical internal and external pressures $P,P_e$ and cross-sectional area $A$, write $P-P_e=\mathcal{P}(A)$. For constant fluid mass density $\rho$,is the squared tube wave speed about an undisturbed area $A_0$.

The kinematic pressure convention used in the parent model instead writes $p=P/\rho$, $p_e=P_e/\rho$ and $p-p_e=\tilde{P}(\alpha )$. It gives the equivalent expression $c^2={\alpha }_0{\tilde{P}}^{\prime }({\alpha }_0)$, whether $\alpha$ denotes area or area divided by a fixed reference area. Linearized area and momentum conservation yield this coefficient of the resulting wave equation. A stable local wall branch has positive derivative; prescribed external pressure is held fixed when taking it. This distinguishes a wall stiffness from a change in applied external loading.