Two infinite, horizontal, parallel plates are at a distance $)=(2\mathrm{.}5cm$ apart and contain a viscous fluid $($viscosity $=1\mathrm{.}4Pa.s$ and $g=12\mathrm{.}6k\frac{N}{m^3}).$ The upper plate moves with a velocity U $)=(0\mathrm{.}006\frac{m}{s}$ while the bottom plate is fixed. The fluid contained between the plates moves under a pressure gradient. Assuming laminar flow:

$($a$)$ Name the type of flow regime described here.

$[3$ Marks$]$

$($b$)$ Sketch and label the shear stress and velocity profile.

Note: Thereare a number of possible correct options depending on the balance between the pressure gradient and the upper plate velocity. You need only sketch the solution to one option but clearly state the case it applies to$.$

$[8$ Marks$]$

$($c$)$ Derive an expression for the shear stress and velocity profiles. Clearly state why you have eliminated terms from the Navier$-$Stokes equations to allow for this derivation. See appendix for Navier$-$Stokes equations.

$[12$ Marks$]$

$($d$)$ A U$-$tube manometer $($fluid in manometer has a $g=15\mathrm{.}7k\frac{N}{m^3})$ connected between two points along the bottom plate indicates a differential reading of $2\mathrm{.}5$ mm $.$ At what distance from the bottom plate does the maximm velocity in the gap between the two plates occur? See figure $3$ below.

$[12$ Marks$]$