$($a$)$ Laminar flow is controlled by fluid viscosity and pipe roughness is immaterial. In turbulent flow it controlled by friction and pipe roughness is of key importance. Why?

$($b$)$ Starting with Prandt's $\frac{1}{7^{th}}$ power law:

$\frac{w}{u_{\text{m}\text{a}\text{x}}}=(1-\frac{r}{R}{)}^{\frac{1}{7}}$

Show that for fully$-$developed turbulent flow, the average velocity,$,$ can be expressed as:

$\frac{?}{\text{b}\text{a}\text{r}}(u)=\frac{49}{60}{u}_{\text{m}\text{a}\text{x}}$

$($c$)$ Laminar flow has a distinctive parabolic velocity profile whereas turbulent flow has a truncated velocity profile. Why is this observed?

$($d$)$ Fluid flows in a $50$ mm diameter pipeline with a flowrate of $0\mathrm{.}01\frac{m^3}{s}.$

Calculate Reynolds number, Re$.$

What is $u_{\text{m}\text{a}\text{x}}$ in the pipe?

Data

Fluid density $=1100k\frac{g}{m^3}$

Fluid viscosity $={10}^{-3}N\frac{s}{m^2}$