Reynolds Number

\[\text{Re} = \frac{Dv\rho}{\mu} = \frac{\text{inertial force}}{\text{viscous force}}\] where \[\begin{aligned} \text{inertial force} &= ma = \rho V\frac{dv}{dt} \\ \text{viscous force} &=\tau A = \mu A \frac{dv}{dy}\end{aligned}\] Hence, \[\text{Re} = \frac{\rho V\dfrac{dv}{dt}}{\mu A \dfrac{dv}{dy}} = \frac{\rho Vdy}{\mu Adt} = \frac{Dv\rho}{\mu}\]