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.