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Time-dependent flow in simple branch hydraulic systems

The principal branch of the Lambert W function is employed in the field of mechanical engineering, in the study of time dependent transfer of Newtonian fluids between two reservoirs with varying free surface levels, using centrifugal pumps. The Lambert W function provided an exact solution to the flow rate of fluid in both the laminar and turbulent regimes: Q turb = Q i ζ i W 0 Q lam = Q i ξ i W 0 {\displaystyle {\begin{aligned}Q_{\text{turb}}&={\frac {Q_{i}}{\zeta _{i}}}W_{0}\left\\Q_{\text{lam}}&={\frac {Q_{i}}{\xi _{i}}}W_{0}\left\end{aligned}}} where Q i {\displaystyle Q_{i}} is the initial flow rate and t {\displaystyle t} is time.