Reynolds number

The Reynolds number is the dimensionless ratio Re = ρvL/η, where ρ is the fluid density, v a characteristic velocity, L a characteristic length, and η the dynamic viscosity. It measures the relative importance of inertial forces to viscous forces in a flow.

For Re ≪ 1, viscosity dominates: flow is laminar, reversible, and governed by the Stokes equations. Bacteria, dust in still air, and blood in fine capillaries live here. For Re ≫ 1, inertia dominates: flow becomes turbulent, irreversible, and chaotic. Swimming fish, rivers, and airliners live here. The transition is gradual but historically pinned near Re ≈ 2300 for pipe flow, the value Osborne Reynolds established in his 1883 dye experiment.

Introduced by Osborne Reynolds in 1883; generalised and popularised across all flow geometries over the following decades.