Consider a polymer $($with density $$ and viscosity $)$ flowing in between two parallel plates in a vertical position. Both plates are stationary at $x=0$ and $x=h.$ A downward pressure gradient is applied $(-$dp$/$dz$)$ which is constant across the z$-$direction, which is also aided by gravity acting on the negative $z-$direction.

a$.$ Starting with the Navier$-$Stokes equations, find the simplified equation that defines the fluid velocity $v_z.$ State your assumptions to achieve this simplified equation. $(5$ pts$)$

b$.$ If $h=4\mathrm{.}0cm$ and the liquid has a $=1640k\frac{g}{m^3}$ and a viscosity $=1\mathrm{.}5Pa*s,$ what is the maximum downward velocity, $v_{z,\text{m}\text{a}\text{x}},$ when $-\frac{dp}{dz}=22,000P\frac{a}{m}.(2$ pts$)$

$---$Be careful here $-$ make sure you consider which direction your pressure gradient is in$!$