\[Re = \frac{\rho U D}{\mu} = \frac{UD}{\nu}\]

where \(Re\) is the Reynolds number
\(U\) is the freestream velocity
\(\rho\) is the density of the fluid
\(\mu\) is the dynamic viscosity of the fluid
\(\nu\) is the kinematic viscosity of the fluid
\(D\) is the diameter of the pipe.

\[U= \frac{Q}{A}\]

where \(Q\) is the volumetric flow rate, and \(A\) is the cross-sectional area of the pipe.

\[\frac{l_{e,laminar}}{D} = 0.06Re\]

where \(l_{e,laminar}\) is the entrance length for laminar flow.

\[\frac{l_e,turbulent}{D} = 4.4Re^{\frac{1}{6}}\]

where \(l_{e,turbulent}\) is the entrance length for turbulent flow.