Question$1$:

a$.$ Prove that the velocity profile for a laminar flow in an infinite circular pipe, if the

fluid behaves as a power$-$law fluid, is given by:

$v_z(r)=\frac{n}{n+1}(\frac{d\frac{p}{d}z}{2k}{)}^{\frac{1}{n}}[((R_p{)}^{\frac{n+1}{n}})-((r{)}^{\frac{n+1}{n}})]$

The viscosity law for a power$-$law fluid is given by:

${}_r=k(-\frac{dv}{dr}{)}^n$

Where $k$ and $n$ are constants.

b$.$ From the velocity profile above, estimate the average $($bulk$)$ velocity of fluid.

c$.$ Compare your solution with the worked example of "Laminar flow in an infinite

circular pipe" in your student guide and discuss the differences in terms of velocity

profile.

Hint: Start your solution using the z$-$component of the equation of motion in cylindrical

coordinates in terms of shear stress $().$

Answer $($a$)($b$)$ and $($c$)$