Velocity profile of fluids in flow of two adjacent immiscible fluids

Input(s)

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

$\boldsymbol{P}_{\boldsymbol{L}}$: Pressure at Point L $(\mathrm{Pa})$ b: Half Plane Thickness (m)

$\mathrm{x}$: Vertical Distance $(\mathrm{m})$

L: Length (m)

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

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

Output(s)

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

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

Formula(s)

$$ \begin{aligned} & \mathrm{v}_{\mathrm{zI}}=\frac{\left(\mathrm{P}_{\mathrm{o}}-\mathrm{P}_{\mathrm{L}}\right) * \mathrm{~b}^{2}}{2 * \mu_{\mathrm{I}} * \mathrm{~L}} *\left(\frac{2 * \mu_{\mathrm{I}}}{\mu_{\mathrm{I}}+\mu_{\mathrm{II}}}+\frac{\mu_{\mathrm{I}}-\mu_{\mathrm{II}}}{\mathrm{mu}_{\mathrm{I}}+\mu_{\mathrm{II}}} * \frac{\mathrm{x}}{\mathrm{b}}-\left(\frac{\mathrm{x}}{\mathrm{b}}\right)^{2}\right) \\ & \mathrm{v}_{\mathrm{ZII}}=\frac{\left(\mathrm{P}_{\mathrm{o}}-\mathrm{P}_{\mathrm{L}}\right) * \mathrm{~b}^{2}}{2 * \mu_{\mathrm{II}} * \mathrm{~L}} *\left(\frac{2 * \mu_{\mathrm{II}}}{\mu_{\mathrm{I}}+\mu_{\mathrm{II}}}+\frac{\mu_{\mathrm{I}}-\mu_{\mathrm{II}}}{\mu_{\mathrm{I}}+\mu_{\mathrm{II}}} * \frac{\mathrm{x}}{\mathrm{b}}-\left(\frac{\mathrm{x}}{\mathrm{b}}\right)^{2}\right) \end{aligned} $$

Reference(s)

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley \& Sons, Chapter: 2, Page: 57.