$4\mathrm{.}4-2.$Solve $1-3$ after solving question.

$1.$ Derive the equation for the velocity profile $($separate$,$ intergrate, and simplify$).$

$2.$Derive the equation for average velocity.

$3.$Derive the rquation for flow rate Q$.$

Question: 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 $d\frac{v_x}{d}y=0$ at $y=0).$

$180$

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