Inviscid fluid issues vertically downwards at speed $u_{0}$ from a circular tube of radius a. The fluid falls onto a horizontal plate a distance $H$ below the end of the tube, where it spreads out axisymmetrically.

Show that while the fluid is falling freely it has speed

$$u=u_{0}\left[1+\frac{2 g}{u_{0}^{2}}(H-z)\right]^{1 / 2}$$

and occupies a circular jet of radius

$$R=a\left[1+\frac{2 g}{u_{0}^{2}}(H-z)\right]^{-1 / 4},$$

where $z$ is the height above the plate and $g$ is the acceleration due to gravity.

Show further that along the plate, at radial distances $r \gg a$ (i.e. far from the falling jet), where the fluid is flowing almost horizontally, it does so as a film of height $h(r)$, where

$$\frac{a^{4}}{4 r^{2} h^{2}}=1+\frac{2 g}{u_{0}^{2}}(H-h)$$