# Non-Newtonian flow in annulus

## Input(s)

r: Inner Radius (ft)

R: Outer Radius ( $$f t$$ )

n: Power Law Constant 2 (dimensionless)

$$\kappa$$: Dilatational Viscosity $$(\mathrm{cP})$$

vo: Input Velocity (ft/s)

$$\rho$$: Fluid Density $$\left(\mathrm{lb} / \mathrm{ft}^{3}\right)$$

## Output(s)

$$v_{z}$$: Output Velocity (ft/s)

w: Mass Flow Rate (lb/s)

## Formula(s)

$\begin{gathered} \mathrm{v}_{\mathrm{z}}=\mathrm{v}_{\mathrm{o}} *\left(\frac{\left(\left(\frac{\mathrm{r}}{\mathrm{R}}\right)^{1-\left(\frac{1}{\mathrm{n}}\right)}\right)-1}{\left(\kappa^{1-\left(\frac{1}{\mathrm{n}}\right)}\right)-1}\right) \\ \mathrm{W}=\left(\frac{2 * \pi * \mathrm{R}^{2} * \rho * \mathrm{v}_{\mathrm{o}}}{\left(\kappa^{1-\left(\frac{1}{\mathrm{n}}\right)}\right)-1}\right) *\left(\frac{1-\left(\kappa^{3-\left(\frac{1}{\mathrm{n}}\right)}\right)}{3-\left(\frac{1}{\mathrm{n}}\right)}-\left(\frac{1-\kappa^{2}}{2}\right)\right) \end{gathered}$

## Reference(s)

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley & Sons, Chapter: 8, Page: 258.

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