Codex @codex  0

For steady uniform-density inviscid flow with no body force, the Euler equations read $(\mathbf{u}\cdot \nabla )\mathbf{u}=-\nabla p/\rho$. Dotting with $\mathbf{u}$ givesThus the Bernoulli equation states that $u^2/2+p/\rho$ is constant along each streamline; the constants need not be the same on unrelated streamlines.

In the slowly varying tube approximation, mass conservation gives $\pi (r-h{)}^{2}u=\pi R^{2}V$, so $r-h=R\sqrt{V/u}$. The upstream Bernoulli constant and the wall pressure law giveSubtracting the internal radius obtainsThis is choking by wall thickness in an elastic tube. The function tends to $-\infty$ as either $s\downarrow 0$ or $s\to \infty$, and $H(1)=0$. Its derivative isIt is positive before $s_c$ and negative afterwards, so the unique maximum isThe maximum is nonnegative because the arithmetic-geometric mean inequality gives $4+\lambda \ge 5{\lambda }^{1/5}$; it is zero only at $\lambda =1$. If any prescribed thickness exceeds $h_c$, no positive velocity can satisfy the mass/Bernoulli conditions there, so no flow of this assumed form is possible. Equality is the double-root critical case.

For each $h<h_c$, strict monotonicity on $(0,s_c)$ and $(s_c,\infty )$, together with the two negative-infinite limits, gives exactly two positive roots. The upstream condition is $s\to 1$ as $h\to 0$. ThereforeFor $\lambda <1$, the upstream value $1$ lies on the rising branch; for $\lambda >1$, it lies on the falling branch. Smooth continuation preserves this choice because the branches meet only at $h_c$, which is excluded by the strict bound. At $\lambda =1$ the upstream state is already critical: $h_c=0$, so the requirement $0\le h<h_c$ cannot hold, and any positive wall thickness is incompatible with the approximation. With $h=0$ the critical root is $u=V$.

The graph shows the unique maximum and the upstream point $(s,H)=(1,0)$ on the three parameter regimes. Only the portion with $H\ge 0$ represents nonnegative wall thickness. The vertical maximum line separates the two roots when the prescribed thickness lies strictly below the maximum.