Problems $(70\backslash \%)$

$3.$ Problem $1(20$ points$)$: Two horizontal, infinite, parallel plates are spaced a distance d apart. A viscous liquid is contained between the plates. The bottom plate is fixed, and the upper plates moves parallel to the bottom plate with a velocity V$.$ Because of the no $-$ slip boundary condition, the liquid motion is caused by the liquid being dragged along by the moving boundary. There is no pressure gradient in the direction of flow. Remember, this is the so called simple Couette flow. Assuming steady $-$ state conditions,

a$.$ Determine the velocity distribution between the plates

b$.$ Determine an expression for the flowrate passing between the plates for a unit width. Express your answer in terms of $\backslash ($ d $\backslash )$ and $\backslash ($ V $\backslash ).$

c$.$ Determine the pressure distribution between the plates.

d$.$ If $\backslash ($ d$=4\backslash$mathrm$\{$~mm$\},$ V$=0\mathrm{.}0008\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\},\backslash$rho$=2000\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}3\backslash )$ and pressure at the moving plate is atmospheric $101$ kPa $,$ find the velocity and pressure distributions as well as the pressure at the fixed plate.