Problem $2.$ An incompressible isothermal fluid is placed between two infinite parallel plates separated by a distance $\backslash ($ L $\backslash )$ as shown in the figure below. The bottom plate is fixed and the top plate moves with a constant velocity $\backslash ($ V$\_\{0\}\backslash ).$ We assume that all the initial transient behavior has died out, and that the flow field is $($i$)$ steady, $($ii$)$ planar$/$columnar$,($iii$)$ fully$-$developed, and laminar. The flow field is due to a combination of the motion of the top plate and the pressure gradient in the $\backslash ($ x $\backslash )-$direction. The boundary conditions for this situation are the no$-$slip and no$-$penetration conditions at the plate. Assume gravity acts in the negative $\backslash ($ z $\backslash )-$direction $($into the page$).$

$[2$ pts$]($a$)$ Write down the explicit mathematical expressions for the no$-$slip and nopenetration boundary conditions for this situation to complete the governing equations.

$[1$ pts$]($b$)$ Determine the components of gravity: $\backslash ($ g$\_\{$x$\},$ g$\_\{$y$\},$ g$\_\{$z$\}\backslash )$ with proper sign$($s$)$ for this situation.

$[1$ pts$]($c$)$ Use assumption $($i$)[$Steady$]$ to reduce the equations given above.

$[1$ pts$]($d$)$ Next, use assumption $($ii$)[$Planar$/$Columnar$]$ to further reduce the equations given above.

$[1$ pts$]($e$)$ Finally, use assumption $($iii$)[$fully$-$developed flow$]$ to further reduce the equations given above.

Do parts parts $($c$)-($e$)$ in the order they are given. Be sure to label which assumption you use to eliminate which term by drawing an arrow through it with the assumption number next to it like we do in class. Do not solve the resulting simplified equations yet!

$[2$ pts$]($f$)$ Apply the appropriate boundary condition$($s$)$ to solve the reduced continuity equation and use your result to further simplify the momentum equations down to a single equation.

$[2$ pts$]($g$)$ Derive an expression for the velocity profile.