Viscosity and flow regimes

Viscosity: definition

Dynamic viscosity μ (Pa·s) characterises a fluid’s internal resistance to shear. For simple laminar flow between two plates, the tangential stress τ follows Newton’s law:

τ = μ * (dv/dy)

A fluid that obeys this linear law is said to be Newtonian (water, air); otherwise, it is non-Newtonian (toothpaste, blood, sludge).

The Reynolds number

The Reynolds number Re is a dimensionless number that compares inertial forces with viscous forces:

Re = ρ * v * D / μ

where D is a characteristic length (diameter for a pipe). It is used to predict the flow regime:

Regime	Re value (pipe)
Laminar	Re < 2000
Transient	2000 <= Re <= 4000
Turbulent	Re > 4000

Laminar flow and Poiseuille’s law

In laminar flow within a cylindrical pipe of radius R and length L, the volumetric flow rate is given by Poiseuille’s law:

Qv = (π * R⁴ * δP) / (8 * μ * L)

This law shows a very strong dependence on the radius (to the power of 4): halving the radius of a pipe reduces the flow rate by a factor of 16 for the same pressure drop.

Common pitfall

The critical Reynolds number (approximately 2000) is indicative, not a strict threshold: the transition depends on the roughness of the walls and disturbances at the inlet, and may vary significantly depending on the experimental conditions.