Average velocity of fluids in flow of two adjacent immiscible fluids

Input(s)

$\boldsymbol{P}_{\boldsymbol{o}}$: Pressure at Initial Point $(\mathrm{Pa})$

$\boldsymbol{P}_{L}$: Pressure at Point L $(\mathrm{Pa})$

$\boldsymbol{b}$: Distance $(\mathrm{m})$

$\boldsymbol{\mu}_{\boldsymbol{I}}$: Viscosity of Phase I, Denser, More Viscous Fluid $(\mathrm{kg} /(\mathrm{ms}))$

$\boldsymbol{\mu}_{\boldsymbol{I I}}$: Viscosity of phase II, Less Dense, Less Viscous Fluid $(\mathrm{kg} /(\mathrm{ms}))$

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

Output(s)

$v_{z I}$: Average Velocity for Phase I $(\mathrm{m} / \mathrm{s})$

$\boldsymbol{v}_{z I I}$: Average Velocity for Phase II $(\mathrm{m} / \mathrm{s})$

Formula(s)

$$ \begin{aligned} & v_{z I}=\frac{\left(P_{o}-P_{L}\right) * b^{2}}{12 * \mu_{I} * L} *\left(\frac{7 * \mu_{I}+\mu_{I I}}{\mu_{I}+\mu_{I I}}\right) \\ & v_{z I I}=\frac{\left(P_{o}-P_{L}\right) * b^{2}}{12 * \mu_{I I} * L} *\left(\frac{\mu_{I}+7 * \mu_{I I}}{\mu_{I}+\mu_{I I}}\right) \end{aligned} $$

Reference(s)

Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (2002). Transport Phenomena (Second ed.). John Wiley \& Sons, Chapter: 2, Page: 58.