Figure Q$3.$b

Figure Q$3.$d$($a$)$ Sketch the diagram of a 'forced vortex' and a 'free vortex' including the relation between tangential velocity $v$ and radial position $r.$ In addition, give two examples of the occurrence of a 'free vortex' and two examples of a 'forced vortex'.

$($b$)$ Noting the diagram and the symbols used in Figure Q$3.$b$,$ derive an expression for the mass, $m,$ in the small layer at the outer radius.

Figure Q$3.$b

$($c$)$ By applying the force balance on the small layer at the outer radius, show that the pressure gradient can be expressed as

$\frac{P}{r}=\frac{v^2}{r}$

$($d$)$ Consider a cylindrical tank, partially filled with a liquid and rotating at angular speed $$ around its axis. At steady state, the shape of the free surface of the liquid is given by the following expression

$z-z_0=\frac{{}^2{r}^2}{2g}$

where $z$ is the free surface height, $g$ the gravitational constant and $r$ the radial coordinate $($see Figure Q$3.$d$)$

Figure Q$3.$d

By using the following relation

$dP=\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{r}}dr+\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{z}}dz$

$($i$)$ Obtain an expression for the pressure difference $P_2-P_1$ between points $1$ and $2.$

$($ii$)$ Obtain an expression for the pressure difference $P_3-P_2$ between points $3$ and $2.$

$($iii$)$ Use the expressions of $P_2-P_1$ and $P_3-P_2$ to derive the relation of the free surface shape.

$($e$)$ Consider the rotating tank described in part d$.$ If the tank diameter is $25cm$ and the angular velocity is $500rpm,$ calculate the distance $h($see Figure Q$3.$d$)$ between the highest and lowest point reached by the free surface of the liquid.$($Answer $,$h$=2\mathrm{.}18)$m