Problem $2$

A Bingham fluid flows of viscosity, $,$ and yield stress, ${}_o,$ through a horizontal pipe of radius, $R,$ and length, $L.$ The flow is driven by a constant pressure drop, $\frac{P}{L}$ along the length of the pipe.

a$)$ Derive an expression for the velocity profile in the pipe.

There is a critical radius, $r_c,$ within which the fluid moves as a slug, so the velocity profile will be composed of two parts. Within $r_c,$ the velocity will be a constant and outside of $r_c,$ the profile will depend upon $r.$

b$)$ Determine the volumetric flow rate for flow through the pipe using the velocity profile from part $($a$).$

c$)$ If$,{}_o=1000\frac{N}{m^2},R=0\mathrm{.}002m,L=1m,$ and $=0\mathrm{.}1N\frac{s}{m^2},$ use the expression from $($b$)$ to determine the pressure drop required to deliver a volumetric flow rate of $2{10}^{-7}\frac{m^3}{s}?$

Please include pictures with your solutions and any helpful advice to solve the problem.