Momentum flux distribution of flow through an annulus

Input(s)

$\mathbf{p}_{\mathbf{o}}$: Pressure at Initial Point $(\mathrm{Pa})$

$\mathbf{p}_{\mathbf{L}}$: Pressure at Point $\mathrm{L}(\mathrm{Pa})$

R : Radius $(\mathrm{m})$

L: Length $(\mathrm{m})$

r: Cylindrical Shell of Thickness (m)

K: Ratio of Inner Pipe's Radius to Outer Pipe's Radius (fraction) $\left(\mathrm{m}^{2} \mathrm{~kg} \mathrm{~s}^{-2} \mathrm{~K}^{-1}\right.$ )

Output(s)

$\tau_{\text {rz }}$: Momentum Flux Distribution $(\mathrm{Pa})$

Formula(s)

$$ \tau_{\mathrm{rz}}=\left(\mathrm{p}_{\mathrm{o}}-\mathrm{p}_{\mathrm{L}}\right) * \frac{\mathrm{R}}{2 * \mathrm{~L}} *\left(\frac{\mathrm{r}}{\mathrm{R}}-\frac{1-\mathrm{K}^{2}}{2 * \ln \left(\frac{1}{\mathrm{~K}}\right)} *\left(\frac{\mathrm{R}}{\mathrm{r}}\right)\right) $$

Reference(s)

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley \& Sons, Chapter: 2, Page: 55.