Prandtl number#

Named after: Ludwig Prandtl (1875-1953).

$$\text{Pr} \stackrel{\text{def}}{=} \frac{\nu}{\alpha} \sim \frac{\text{momentum diffusion}}{\text{heat diffusion}}$$

Measures momentum diffusivity relative to thermal diffusivity. It indicates the relative thickness of velocity and thermal boundary layers.

Name	Symbol	SI units	Dimension
kinematic viscosity	$\nu$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$
thermal diffusivity	$\alpha$	$\mathrm{m}^{2}\,\mathrm{s}^{-1}$	$\text L^{2}\,\text T^{-1}$

Heat transfer

Range	Regime	Description
0 – 0.1	liquid metal	Thermal diffusivity dominates over momentum diffusion. Thermal boundary layers are much thicker than velocity boundary layers, typical of liquid metals.
0.1 – 10	gas	Momentum and thermal diffusivities are comparable. Velocity and thermal boundary layers have similar thicknesses, typical of many gases.
10 – ∞	oil or viscous liquid	Momentum diffusion dominates over thermal diffusion. Thermal boundary layers are thinner than velocity boundary layers, typical of oils and viscous liquids.

Name	Symbol	SI units	Dimension
kinematic viscosity	\(\nu\)	\(\mathrm{m}^{2}\,\mathrm{s}^{-1}\)	\(\text L^{2}\,\text T^{-1}\)
thermal diffusivity	\(\alpha\)	\(\mathrm{m}^{2}\,\mathrm{s}^{-1}\)	\(\text L^{2}\,\text T^{-1}\)