A straight horizontal duct extends in the $z$ direction for a length $L$ and has a square cross

section, bordered by the lines $x=+-B$ and $y=+-B.$ An incompressible Newtonian fluid

achieved a fully developed velocity profile in the duct. One of the students claimed that the

velocity distribution is given by

$v_z=\frac{(P_0-P_L)B^2}{4L}[1-(\frac{x}{B}{)}^2][1-(\frac{y}{B}{)}^2]$

Write down the final expression of the appropriate component $(\frac{x}{y}\frac{?}{z}$ of the Navier$-$Stokes

equation in the differential form.

Does this equation satisfy the relevant boundary conditions $($Yes$/$No$)$ and the relevant

differential equation $($Yes$/$No$)?$ Provide proper justification for each choice.