(i) Starting from Euler's equation for an incompressible fluid show that for potential flow with $\mathbf{u}=\nabla \phi$,

$$\frac{\partial \phi}{\partial t}+\frac{1}{2} u^{2}+\chi=f(t)$$

where $u=|\mathbf{u}|, \chi=p / \rho+V$, the body force per unit mass is $-\nabla V$ and $f(t)$ is an arbitrary function of time.

(ii) Hence show that, for the steady flow of a liquid of density $\rho$ through a pipe of varying cross-section that is subject to a pressure difference $\Delta p=p_{1}-p_{2}$ between its two ends, the mass flow through the pipe per unit time is given by

$$m \equiv \frac{d M}{d t}=S_{1} S_{2} \sqrt{\frac{2 \rho \Delta p}{S_{1}^{2}-S_{2}^{2}}}$$

where $S_{1}$ and $S_{2}$ are the cross-sectional areas of the two ends.