$($a$)$ A cone and plate viscometer was used to determine the viscosity of three different liquids, here denoted as A$,$ B and C$.$ Figure Q$3$a shows the measured torques, $T,$ as a function of the cone's angular velocity, $,$ for the three liquids. Note that the torque required to maintain a constant angular velocity is given by

$T=\frac{2}{3}\frac{r_{\text{m}\text{a}\text{x}}^3}{}$

where $r_{\text{m}\text{a}\text{x}}$ is the radius of the plate, $$ the liquid viscosity and $$ the angle between the cone surface and the plate wall. Based on the torque measurements shown in Figure Q$3$a$,$ determine which liquid is Newtonian, which is shear$-$thinning $($i$.$e$.$ pseudo$-$plastic$)$ and which is shear$-$thickening $($i$.$e$.$ dilatant$).$ You must justify your answer.

$$

$($b$)$ A fluid clutch comprises two flat parallel plates, face$-$to$-$face with a small variable axial separation, as shown in the schematic in Figure Q$3$b$.$ The space between the plates is kept entirely filled with a Newtonian clutch fluid of viscosity $=1\mathrm{.}2$Pas. The clutch plates are both $0\mathrm{.}25m$ in diameter. The driving plate is rotated with a constant angular velocity of $4500rad{s}^{-1}.$ The driven plate will reach $2750rad{s}^{-1}$ if the transmitted torque is $1000Nm.$ Calculate the separation distance between the plates that is necessary to achieve these conditions. Note that the equation for the torque, $T,$ required to maintain a constant relative angular velocity, ${}_{\text{r}\text{e}\text{l}},$ between the two parallel plates of the clutch is:

$T=2\frac{}{h}\frac{r_{\text{m}\text{a}\text{x}}^4}{4}{}_{\text{r}\text{e}\text{l}}$

where $r_{\text{m}\text{a}\text{x}}$ is the radius of the parallel plates and $h$ the separation distance between the plates.$($Answer$,$h$=0\mathrm{.}8$mm$)$

$($c$)$ Oil is siphoned from the bottom of an atmospheric $($open$)$ feeding tank to a lower level tank through a $2cm$ diameter pipe $40m$ in length. The bottom of the feeding tank is $1\mathrm{.}5m$ above the exit point of the pipe, which is in turn $12m$ below the highest point reached by the pipe. The pipe exit is open to atmosphere. The length of pipe between the exit and the highest point is $20m$ and siphoning ceases when the absolute pressure at the highest point falls below $20kN{m}^{-2}.$ Assume a Fanning friction factor $f=0\mathrm{.}002$ and an oil density $=900kg{m}^{-3}.$

$($i$)$ Draw a fully labelled diagram of the system.

$($ii$)$ Calculate the limiting discharge velocity $-$ namely, the average velocity of the fluid when the fluid absolute pressure at the highest points is $20kN{m}^{-2}.($Answer$,$v$=2\mathrm{.}68)$

$($iii$)$ Calculate the depth of fluid in the feeding $($top$)$ tank when the siphon stops. Clarify all the assumptions made to derive your result. If needed, assume a value for the answer to Part ii$).$

Figure Q$3$a$.$ Torque, $T,$ vs angular velocity, $,$ for the three liquids A$,$ B and C$.$Answer$=4\mathrm{.}7$m from bottom of feeding tank