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.

Related

Average velocity of flow through an annulus

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