Q $5-7$: Determining fluid flow through pipes and tubes has great relevance in many areas of engineering and science. In engineering, typical applications include the flow of liquids and gases through cooling systems.

The resistance to flow in such conduits is parameterized by a dimensionless number called the friction factor. For turbulent flow, the Colebrook equation provides a means to calculate the friction factor,

$0=\frac{1}{\sqrt[\phantom{2}]{f}}+2\mathrm{.}0log(\frac{}{3\mathrm{.}7D}+\frac{2\mathrm{.}51}{R_e\sqrt[\phantom{2}]{f}}{)}^{\text{'}}$

where $=$ the roughness $(m),D=$ diameter $(m),$ and $R_e=$ the Reynolds number,

$R_e=\frac{VD}{}$

where $=$ the fluid's density $(k\frac{g}{m^3}),V=$ its velocity $(\frac{m}{s}),$ and $=$ dynamic viscosity $(N.\frac{s}{m^2}).$

For this case, the parameters are $=1\mathrm{.}23k\frac{g}{m^3},=1\mathrm{.}79{10}^{-5}N.\frac{s}{m^2},D=0\mathrm{.}005m,V=40\frac{m}{s},$ and $=0\mathrm{.}0015mm.$ Note that friction factors $f$ range from about $0\mathrm{.}008$ to $0\mathrm{.}08.$

$($a$)$ Articulate the engineering vs mathematical problem. $(3)$

$($b$)$ Prepare this function to apply Fixed Point Method $(3)$

$($c$)$ Write code to find a solution that $(14)$

$($i$)$ uses vectors and for loop

$($ii$)$ plots the convergence over $20$ iterations with initial guess $f=1$

$($iii$)$ prints the final answer