3.II.18C

Use Euler's equation to derive Bernoulli's theorem for the steady flow of an inviscid fluid of uniform density $\rho$ in the absence of body forces.

Such a fluid flows steadily through a long cylindrical elastic tube having circular cross-section. The variable $z$ measures distance downstream along the axis of the tube. The tube wall has thickness $h(z)$, so that if the external radius of the tube is $r(z)$, its internal radius is $r(z)-h(z)$, where $h(z) \geqslant 0$ is a given slowly-varying function that tends to zero as $z \rightarrow \pm \infty$. The elastic tube wall exerts a pressure $p(z)$ on the fluid given as

$p(z)=p_{0}+k[r(z)-R]$

where $p_{0}, k$ and $R$ are positive constants. Far upstream, $r$ has the constant value $R$, the fluid pressure has the constant value $p_{0}$, and the fluid velocity $u$ has the constant value $V$. Assume that gravity is negligible and that $h(z)$ varies sufficiently slowly that the velocity may be taken as uniform across the tube at each value of $z$. Use mass conservation and Bernoulli's theorem to show that $u(z)$ satisfies

$\frac{h}{R}=1-\left(\frac{V}{u}\right)^{1 / 2}+\frac{1}{4} \lambda\left[1-\left(\frac{u}{V}\right)^{2}\right], \quad \text { where } \quad \lambda=\frac{2 \rho V^{2}}{k R}$

Sketch a graph of $h / R$ against $u / V$. Show that if $h(z)$ exceeds a critical value $h_{c}(\lambda)$, no such flow is possible and find $h_{c}(\lambda)$.

Show that if $h<h_{c}(\lambda)$ everywhere, then for given $h$ the equation has two positive solutions for $u$. Explain how the given value of $\lambda$ determines which solution should be chosen.

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