Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 81 / 2 / ii / Solution

Steady mass conservation makes $\alpha u=Q$. For uniform area, $u=U=Q/{\alpha }_0$ is constant and the pressure-area law of a collapsible tube gives $p_x=p_e^{\prime }(x)$. The acceleration vanishes, so momentum balance requiresThe external pressure must therefore decrease linearly along a positive-flow tube. Its additive constant is free; the constant transmural pressure is $\tilde{P}({\alpha }_0)$.