Water at $20C($with density $=998k\frac{g}{m^{\text{'}}}$ and dynamic viscosity $=1\mathrm{.}002{10}^{-1}k\frac{g}{m},$ is flowing through a $10$ m long pipe with an internal diameter of $0\mathrm{.}1$ m $.$ The flow velocity is $2\frac{m}{s}.$ The pipe material is smooth, and the flow is assumed to be turbulent.

$($a$)$ Calculate the Reynolds number for the flow.

$($b$)$ Use an empirical formula to estimate the Darcy$-$Weishach friction factor for turbulent flow in a smooth pipe.

$($c$)$ Determine the pressure drop over the length of the pipe using the Darcy$-$Weistach equation.