Question $3$

$($a$)$ A horizontal pipeline bends through an angle of $45$ as shown in the diagram

below. Oil flows through the pipe from left to right. The diameter of the pipe

changes from $0\mathrm{.}6m$ before the bend to $0\mathrm{.}35m$ after it$.$ Water enters the bend

at a rate of $0\mathrm{.}6\frac{m^3}{s}.$ The fluid pressure at the entrance to the bend is equivalent

to $16m$ of water. Assuming that there is no loss of energy and that the oil has

relative density of $0\mathrm{.}82,$ calculate the force exerted on the bend by the oil. $[16$

marks$]$

$($b$)$ A sharp crested rectangular weir is located in a channel with its crest $0\mathrm{.}3m$

above the channel bed $($Figure Q$3$b$).$ The weir's $C_d=0\mathrm{.}65.$ It is necessary for

the weir to be able to measure flows of up to $0\mathrm{.}12\frac{m^3}{s}$ without causing the

upstream water level to rise more than $0\mathrm{.}5m$ above the channel bed. What

minimum value of $B$ is required to ensure that this condition is met? $[4$ marks$]$