From Euler's equations describing steady inviscid fluid flow under the action of a conservative force, derive Bernoulli's equation for the pressure along a streamline of the flow, defining all variables that you introduce.

Water fills an inverted, open, circular cone (radius increasing upwards) of half angle $\pi / 4$ to a height $h_{0}$ above its apex. At time $t=0$, the tip of the cone is removed to leave a small hole of radius $\epsilon \ll h_{0}$. Assuming that the flow is approximately steady while the depth of water $h(t)$ is much larger than $\epsilon$, show that the time taken for the water to drain is approximately

$$\left(\frac{2}{25} \frac{h_{0}^{5}}{\epsilon^{4} g}\right)^{1 / 2}$$