$4\mathrm{.}4-2.$ Shell Momentum Balance for Flow Between Parallel Plates. A fluid of constant density is flowing in laminar flow at steady state in the horizontal $x$ direction between two flat and parallel plates. The distance between the two plates in the vertical $y$ direction is $2y_0.$ Using a shell momentum balance, derive the equation for the velocity profile within this fluid and the maximum velocity for a distance $Lm$ in the $x$ direction. $[$Hint: See the method used in Section $4\mathrm{.}4$B to derive Eq$.(4\mathrm{.}4-9).$ One boundary condition used is $d\frac{v_{}}{d}y=0$ at $y=0.]$

PLEASE solve$/$workout for $1-3$ on a sheet of PAPER!

$1.$ derive the equation for the velocity profile $($separate$,$integrate, and simply$)$

$2.$ derive the equation for the average velocity

$3.$ derive the equation for flow rate, Q

Ans. $v_x=\frac{p_0-p_L}{2L}{y}_0^2|1-(\frac{y}{Y_0}{)}^2|$