Tip #$1$: Review all $5$ pages carefully before you start working on the problem.

Tip #$2$: Check your units in your solution.

Consider a fully developed, isothermal, steady$-$state, unidirectional $($in $$ direction$),$ laminar flow of an incompressible non$-$Newtonian fluid between two concentric cylinders $($see illustration below$).$ The outer cylinder rotates steadily about its axis at an angular velocity of $[ra\frac{d}{s}]$ and the inner cylinder is stationary $($i$.$e$.,$ the inner cylinder does not rotate$).$ The cylinders are long compared to the radii $($i$.$e$.,L>3R).$ The fluid flow is only driven by the rotation of the outer cylinder.

The fluid is a Power Law liquid, with a shear stress $-$ shear rate relationship represented by the following equation $($for the unidirectional laminar flow$)$:

shear stress $=-K($ shear rate ${)}^n$

Where $K$ is the fluid consistency index and $n$ is the flow behavior index.

The Navier Stokes equations $($i$.$e$.,$ momentum equations$)$ and shear stress components for the cylindrical coordinate system are listed on the next page.

Momentum equation in $r$ direction:

$(\frac{del{u}_r}{\text{d}\text{e}\text{l}\text{t}}+u_r\frac{del{u}_r}{\text{d}\text{e}\text{l}\text{r}}+\frac{u_{}}{r}\frac{del{u}_r}{\text{d}\text{e}\text{l}}+u_z\frac{del{u}_r}{\text{d}\text{e}\text{l}\text{z}}-\frac{u_{}^2}{r})=-\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{r}}-[\frac{1}{r}\frac{\text{d}\text{e}\text{l}(r{}_{rr})}{\text{d}\text{e}\text{l}\text{r}}+\frac{1}{r}\frac{\text{d}\text{e}\text{l}({}_r)}{\text{d}\text{e}\text{l}}+\frac{\text{d}\text{e}\text{l}({}_{zr})}{\text{d}\text{e}\text{l}\text{z}}-\frac{{}_{}}{r}]+g_r$

Momentum equation in $$ direction:

$(\frac{del{u}_{}}{\text{d}\text{e}\text{l}\text{t}}+u_r\frac{del{u}_{}}{\text{d}\text{e}\text{l}\text{r}}+\frac{u_{}}{r}\frac{del{u}_{}}{\text{d}\text{e}\text{l}}+u_z\frac{del{u}_{}}{\text{d}\text{e}\text{l}\text{z}}-\frac{u_r{u}_{}}{r})=-\frac{1}{r}\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}}-[\frac{1}{r^2}\frac{\text{d}\text{e}\text{l}(r^2{}_r)}{\text{d}\text{e}\text{l}\text{r}}+\frac{1}{r}\frac{\text{d}\text{e}\text{l}({}_{})}{\text{d}\text{e}\text{l}}+\frac{\text{d}\text{e}\text{l}({}_z)}{\text{d}\text{e}\text{l}\text{z}}-\frac{{}_r-{}_r}{r}]+g_{}$

Momentum equation in z direction:

$(\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{t}}+u_r\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{r}}+\frac{u_{}}{r}\frac{del{u}_z}{\text{d}\text{e}\text{l}}+u_z\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{z}})=-\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{z}}-[\frac{1}{r}\frac{\text{d}\text{e}\text{l}(r{}_{rz})}{\text{d}\text{e}\text{l}\text{r}}+\frac{1}{r}\frac{\text{d}\text{e}\text{l}({}_z)}{\text{d}\text{e}\text{l}}+\frac{\text{d}\text{e}\text{l}({}_{zz})}{\text{d}\text{e}\text{l}\text{z}}]+g_z$

Also, the law of viscosity for the Power Law fluid with constant density may be expressed as:

${}_{rr}=-K[2(\frac{del{u}_r}{\text{d}\text{e}\text{l}\text{r}}{)}^n]$;$,{}_{}=-K[2(\frac{1}{r}\frac{del{u}_{}}{\text{d}\text{e}\text{l}}{)}^n+(\frac{u_r}{r}{)}^n]$;$,{}_{zz}=-K[2(\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{z}}{)}^n]$

${}_r={}_r=-K[(r\frac{\text{d}\text{e}\text{l}(\frac{u_{}}{r})}{\text{d}\text{e}\text{l}\text{r}}{)}^n+(\frac{1}{r}\frac{del{u}_r}{\text{d}\text{e}\text{l}}{)}^n]$;$,{}_z={}_z=-K[(\frac{1}{r}\frac{del{u}_z}{\text{d}\text{e}\text{l}}{)}^n+(\frac{del{u}_{}}{\text{d}\text{e}\text{l}\text{z}}{)}^n]$

${}_{zr}={}_{rz}=-K[(\frac{del{u}_r}{\text{d}\text{e}\text{l}\text{z}}{)}^n+(\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{r}}{)}^n]$

a$)$ State all pertinent assumptions; also, use the continuity equation. $[30$ points$]$

b$)$ Develop a mathematical model $[$differential equation$($s$)+$ boundary conditions$]$ that will represent the flow of this fluid between the two concentric cylinders. Start with the momentum equations shown above. $[40$ points$]$

c$)$ Solve the mathematical model developed in $($b$)$ and obtain an algebraic expression that will represent the velocity profile $u_{}(r)$ of the Power Law fluid. Show your work. $[40$ points$]$

d$)$ BONUS: If the exponent $\text{'}n\text{'}$ in the obtained solution in part $($c$)$ is set to $n=1,$ does your solution reduce itself to a velocity profile that could be obtained for a Newtonian fluid? Start with Navier Stokes equations for Newtonian fluids, and show your work. $[25$ points$]$

e$)$ BONUS: Develop an expression for the rotational volumetric flow rate of fluid, $Q[\frac{m^3}{s}],$ using the velocity profile for the Power Law fluid obtained in part $($c$).[20$ points$]$

f$)$ BONUS: Again, check if your answer in part $($e$)$ is correct by reducing the obtained expression to the well$-$known results for a Newtonian fluid $($i$.$e$.,$ set $n=1).$ Use the velocity profile obtained in part d$,$ and show your work. $[15$ points$]$

See the next page for part $"g"$ of the problem.

g$)$ Create a graph $u_{}(r)$ versus $r,($i$.$e$.,$ plot velocity profile$)$ for the following power law fluid $[10$ points$]$:

${}_r=-0\mathrm{.}6(\frac{del{u}_{}}{\text{d}\text{e}\text{l}\text{r}}{)}^{0\mathrm{.}7}$;$=1000[\frac{kg}{m^3}]$;$R=0\mathrm{.}05[m]$;$=1\mathrm{.}2[ra\frac{d}{s}]$

Use Excel to create this graph; label the axes. Check the Reynolds number for this case.