Problem $2$

Consider a viscous incompressible flow with constant fluid properties between two parallel plates with an imposed pressure gradient. The pressure variation over the length $L$ is given by $p=p_0-p_1.$ The orientation of the plates is illustrated below.

$z$ into the page and gravity acts in $z$ direction.

$($a$)$ Determine the average velocity, $\frac{?}{\text{b}\text{a}\text{r}}(u),$ in terms of the maximum velocity $u_{\text{m}\text{a}\text{x}}.$

$($b$)$ Determine the pressure field $p.$

$($c$)$ Determine the wall shear stress.

$($d$)$ If the head loss over the length $L$ is given by

$h_f=\frac{P}{g}=f(\frac{L}{h})\frac{\frac{?}{\text{b}\text{a}\text{r}}(u{)}^2}{2g}$

determine the friction factor, $f,$ as a function of the Reynolds number, $R{e}_h.$