$1.$ A cylindrical plug is placed at the exit of a water pipe to control the flow. The pipe has a diameter of $0\mathrm{.}25$ m $,$ and the plug has a diameter of $0\mathrm{.}2$ m $.$ The upstream pressure in the pipe is $3\mathrm{.}5$ MPa $,$ and the flow rate through the pipe is $\backslash (1\mathrm{.}5\backslash$mathrm$\{$~m$\}^\{3\}/\backslash$mathrm$\{$s$\}\backslash ).$ The water density is $\backslash (1000\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$

a$.$ Assuming the flow is steady and incompressible, calculate the mean velocity of the water flowing through the pipe at the upstream section. $(7$ pts $)$

b$.$ Calculate the mean velocity of the water flowing through the annular gap between the pipe and the plug. $(7$ pts$)$

c$.$ Determine the force required to hold the plug in place. $(11$ pts$)$

d$.$ If the plug is suddenly removed and the flow rate remains the same, what would happen to the flow velocity at the outlet and pressure at the inlet of the pipe $($the outlet remains open to the atmosphere$)?$ Would they increase, decrease, or remain the same? Explain. $(5$ pts$)$