Rheological Properties $(10$ points$)$

One of the major assumptions we made about this model is the Newtonian behavior of whole blood. In truth, this is not accurate, particularly for smaller blood vessels. Whole blood contains red blood cells, white blood cells, and several dissolved solids in the plasma. It is a shear$-$thinning fluid with a yield stress $({}_y)$ that can be modeled using the Casson equation:

$T^{0\mathrm{.}5}=T_y^{0\mathrm{.}5+}{k}^{0\mathrm{.}5}(\frac{du}{dr})0\mathrm{.}5$

Where $k$ is the viscosity coefficient. One feature of this is that once the shear stress is less than or equal to the yield stress, then the profile looks more like plug flow in the center of the vessel, but Hagen$-$Poiseuille near the vessel wall:

The effect is that the velocity is constant from the center outward until $r=r_{\text{c}\text{r}\text{i}\text{t}\text{i}\text{c}\text{a}\text{l}},$ where shear stress is equal to the yield stress.

Derive an expression for velocity $(u)$ as a function of $r,P,R,k,L$ and ${}_y.$ There should be two expressions at the end of this: $r_{\text{c}\text{r}\text{i}\text{t}\text{i}\text{c}\text{a}\text{l}}rRr$ and $r_{\text{c}\text{r}\text{i}\text{t}\text{i}\text{c}\text{a}\text{l}}rR.$ The second expression should $be$ solved for first, $asat$ the smaller radius, velocity $is$ constant. $(Hint$: set the shear $in$ the Casson model equal $to$ the shear $in$ the laminar flow model$)$