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The Reynolds number Re = vD/ν is a dimensionless number that compares inertial forces to viscous forces in a flowing fluid. It predicts whether flow will be laminar (smooth) or turbulent (chaotic).

Named after Osborne Reynolds who demonstrated the transition in his famous 1883 experiment using dye in a glass tube.

For pipe flow, the critical thresholds are:

• Re < 2,300: Laminar — fluid moves in smooth parallel layers
• 2,300 – 4,000: Transition — flow is unstable, may switch between laminar and turbulent
• Re > 4,000: Turbulent — chaotic mixing, eddies, higher energy loss

These thresholds are for circular pipes. For open channels and other geometries, different critical values apply.

Kinematic viscosity ν = μ/ρ, where μ is dynamic viscosity and ρ is density.

Units: m²/s or cSt (centistokes). 1 cSt = 10⁻⁶ m²/s.

Common values: water at 20°C ≈ 1 cSt; hydraulic oil VG 32 at 40°C ≈ 32 cSt; air at 20°C ≈ 15.6 cSt (much higher than water due to low density despite low dynamic viscosity).

When fluid enters a pipe, the velocity profile is initially flat (uniform). As viscous effects propagate from the wall, the profile gradually develops into the fully-developed shape (parabolic for laminar).

The entry length is the distance needed:

• Laminar: Le ≈ 0.06 × Re × D (can be very long at high Re)
• Turbulent: Le ≈ 4.4 × Re^1/6 × D (relatively short, 10–60 diameters)

Temperature strongly affects viscosity, especially for oils. Hydraulic oil viscosity drops approximately 50% for every 20°C increase.

Example: ISO VG 46 oil is ~46 cSt at 40°C but ~200 cSt at 0°C and ~15 cSt at 80°C. This dramatically changes the Reynolds number — cold-start conditions may have very different flow regimes than operating temperature.

Water viscosity changes less dramatically: ~1.8 cSt at 0°C, ~1 cSt at 20°C, ~0.3 cSt at 80°C.