Question $4/$Vraag $4$

$[15]$

A non$-$Newtonian fluid flows in a horizontal, circular pipe. Derive an expression for the velocity profile $u(r)$ as a function of the pressure gradient.

The shear stress for the fluid is described by the Ostwald$-$de Waele relationship, with constant flow consistency index K$,$ where the shear stress is given by:

$=K(\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{r}}{)}^n$

On a single graph, sketch the velocity profiles for the following fluids:

shear thinning $)<mi>(</mi><mn>1</mn>$

Newtonian $(n=1)$

shear thickening $)>(1$