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Reynolds Number (Re)

A dimensionless number characterizing the flow regime (laminar or turbulent) of a fluid. Re = (density * velocity * characteristic length) / viscosity.
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The statement of the theorem

Let u(x,t)\textbf{u}(\textbf{x}, t) be the velocity field of an incompressible Newtonian fluid (∇∙u=0\nabla \bullet \textbf{u} = 0) with density ρ\rho and kinematic viscosity ν\nu. The flow dynamics are governed by the momentum equation: \n\nρ(∂u∂t+(u⋅∇)u)=−∇p+μ∇2u+f\rho \left( \frac{\partial \textbf{u}}{\partial t} + (\textbf{u} \cdot \nabla) \textbf{u} \right) = -\nabla p + \mu \nabla^2 \textbf{u} + \textbf{f} \n\nwhere μ=ρν\mu = \rho \nu is the dynamic viscosity. \n\nWe define the characteristic scales: a length scale LL, a characteristic velocity scale U=∣∣u∣∣charU = ||\textbf{u}||_{char}, and the fluid properties ρ\rho and ν\nu. The Reynolds number, ReRe, is rigorously defined as the dimensionless ratio of the characteristic inertial forces to the characteristic viscous forces:\n\nRe=ρULμ=ULνRe = \frac{\rho U L}{\mu} = \frac{U L}{\nu} \n\nThis ratio quantifies the relative magnitude of the non-linear advective terms (inertia) versus the linear diffusive terms (viscosity) in the governing partial differential equation.
Source: Wikipedia