Question Two $(11$ Marks$)$

$2\mathrm{.}1$ An unknown fluid flows through a smooth pipe of radius $\backslash ($ R $\backslash )$ such that the Reynolds number is $10000,$ based on the centre$-$line velocity. Assuming a power law velocity profile, calculate:

a$.$ the average velocity of the flow, $\backslash ($ V $\backslash ),$ as a percentage of the centre$-$line velocity, $\backslash ($ U $\backslash )$;

b$.$ at which position you would place a Pitot tube in order to measure the average velocity in the pipe? Give your answer as a percentage of the pipe radius, $\backslash ($ R $\backslash ).$