Problem $3(5$ marks$).$ A liquid is flowing at an average velocity $\backslash ($ v $\backslash )$ through a cylindrical pipe with diameter $\backslash ($ D $\backslash )$ with a contracted area with a gap $\backslash ($ d $\backslash )($see Figure $3).$ The pressure drop $\backslash (\backslash$Delta P $\backslash )$ is being measured across the contraction and is suspected of being a function of $\backslash ($ v$,$ d$,$ D $\backslash )$ and the fluid density $\backslash (\backslash$rho $\backslash ).$ The density of the fluid is given as $\backslash (1200\backslash$mathrm$\{$~kg$\}\backslash$mathrm$\{$~m$\}^\{-3\}\backslash ).$ The pressure drop was measured as a function of the gap $\backslash ($ d $\backslash )$ below. The pipe is $2$ in diameter and v was given as $\backslash (0\mathrm{.}5\backslash$mathrm$\{$~m$\}\backslash$mathrm$\{$~s$\}^\{-1\}\backslash ).$

Figure $3$: schematic of pipe flow with a contraction.

a$)$ Determine how the pressure drop varies with the geometry $\backslash (($d$,$ D$)\backslash ),$ velocity $\backslash ($ v $\backslash )$ and fluid density $\backslash (\backslash$rho $\backslash )$ in terms of dimensionless parameters.

b$)$ Using the dimensionless parameters, derive a method to plot the data and give $\backslash (\{\}^\{2\}\backslash )$ relationship to predict the pressure drop quantitatively.