3.II.18C

State the form of Bernoulli's theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid in the absence of gravity.

Water of density $\rho$ is driven through a tube of length $L$ and internal radius $a$ by the pressure exerted by a spherical, water-filled balloon of radius $R(t)$ attached to one end of the tube. The balloon maintains the pressure of the water entering the tube at $2 \gamma / R$ in excess of atmospheric pressure, where $\gamma$ is a constant. It may be assumed that the water exits the tube at atmospheric pressure. Show that

$R^{3} \ddot{R}+2 R^{2} \dot{R}^{2}=-\frac{\gamma a^{2}}{2 \rho L} .$

Solve equation ( $\dagger$ ), by multiplying through by $2 R \dot{R}$ or otherwise, to obtain

$t=R_{0}^{2}\left(\frac{2 \rho L}{\gamma a^{2}}\right)^{1 / 2}\left[\frac{\pi}{4}-\frac{\theta}{2}+\frac{1}{4} \sin 2 \theta\right]$

where $\theta=\sin ^{-1}\left(R / R_{0}\right)$ and $R_{0}$ is the initial radius of the balloon. Hence find the time when $R=0$.

*Typos? Please submit corrections to this page on GitHub.*