$4\mathrm{.}4-2.$ 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 $d\frac{v_x}{d}y=0$ at $y=0.]$

$180$

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

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

$2.$Dervive the equation for average velocity

$3.$Dervive the equation for flow rate Q