$($a$)$ Determine the velocity profile in a section of the laminar flow of an incompressible fluid, in steady state and hydrodynamically developed in a pipe of circular cross$-$section. $($b$)$ Determine the expression for the pressure drop between two sections $(1)$ and $(2)$ of the flow that are at different vertical levels. Note: Use the Navier$-$Stokes equation to derive the governing equation.