| course |
course_year |
question_number |
tags |
title |
year |
Fluid Dynamics |
IB |
18 |
|
4.I.7C |
2003 |
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
$$\frac{a^{4}}{4 r^{2} h^{2}}=1+\frac{2 g}{u_{0}^{2}}(H-h)$$