$($a$)$ A $7\mathrm{.}5cm$ diameter fire hose is maintained at a gauge pressure of $P_0$ by a fire engine pumping water. There is a nozzle at the end of the hose, reducing the diameter down to $3\mathrm{.}5cm.$ The hose is held horizontally and discharges into atmosphere.

$($i$)$ Draw a fully labelled diagram of the hose, nozzle section and horizontal water jet. Include a control volume enabling a momentum balance of the calculation of the force required to hold the hose.

$($ii$)$ Derive an expression that relates the average velocity of the water in the hose $(u_0)$ to the average velocity of the water at the nozzle outlet $(u_1),$ the diameter of the hose $(d_0)$ and the diameter of the nozzle outlet $(d_1).$

$($iii$)$ Calculate the average velocity of the water at the nozzle outlet $(u_1)$ and the water volume flow rate in the hose, assuming a gauge pressure, $P_0,$ in the hose of $100$kPa.

$($b$)$ Water is flowing in a bent pipe, as shown in Figure Q$1.$ The gauge pressure of water in the pipe is $8$ bar. The angle between the inlet and outlet sections is $=45.$ The pipe diameter is $0\mathrm{.}32m$ at the bend inlet and $0\mathrm{.}25m$ at the bend outlet. The volumetric flow rate is $100L{s}^{-1}.$

$($i$)$ Calculate the $x-$ and $y-$components of the average fluid velocity at the bend inlet, $u_1,$ and at the bend outlet, $u_2.$

$($ii$)$ Calculate the $x-$ and $y-$components of the pressure force at the bend inlet, $A_1{P}_1,$ and at the bend outlet, $A_2{P}_2.$

$($iii$)$ Apply the momentum balance within the bend and calculate the $x-$ and $y-$components of the resultant force exerted by the pipe walls on the water. Assume the gravity force to be negligible.

Figure Q$1.$ Diagram of the bend.

Figure Q$1.$ Diagram of the bend