Pressure distribution in a creeping flow around a sphere

Input(s)

$\boldsymbol{p}_{\boldsymbol{o}}$: Pressure in the Plane $\mathrm{z}=0$ far away from the Sphere $(\mathrm{Pa})$

$\rho$: Density $\left(\mathrm{kg} / \mathrm{m}^{3}\right)$

$\mathrm{g}$: Gravitational Acceleration $\left(\mathrm{m} / \mathrm{s}^{2}\right)$

z: Direction $(\mathrm{m})$

$\mu$: Viscosity $(\mathrm{kg} /(\mathrm{ms}))$

$v_{\infty}$: Velocity as $r$ Goes to Infinity $(\mathrm{m} / \mathrm{s})$

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

r: Cylindrical Shell of Thickness (m)

Output(s)

p: Pressure Distribution $(\mathrm{Pa})$

Formula(s)

$$ \mathrm{p}=\mathrm{p}_{\mathrm{o}}-(\rho * \mathrm{~g} * \mathrm{z})-\left(\left(\frac{3}{2}\right) *\left(\mu * \frac{\mathrm{v}_{\infty}}{\mathrm{R}}\right) *\left(\frac{\mathrm{R}}{\mathrm{r}}\right)^{2}\right) * \cos (\theta) $$

Reference(s)

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley \& Sons, Chapter: 2, Page: 59.