Prandtl number

Builds on Dimensionless numbers in fluid mechanics Viscosity

Definition

Consider a fluid with density $\rho$ , dynamic viscosity $\mu$ , specific heat capacity at constant pressure $c_p$ and thermal conductivity $k$ . Let $\nu = \mu / \rho$ be its kinematic viscosity and $\alpha = k / (\rho c_p)$ its thermal diffusivity. The Prandtl number of the fluid is $\mathrm{Pr} = \frac{\nu}{\alpha} = \frac{c_p \mu}{k}.$

Remarks

It is a dimensionless number: $\nu$ and $\alpha$ both have dimension $\mathsf{L}^2 \mathsf{T}^{-1}$ . Example: air at room temperature has $\mathrm{Pr} \approx 0.7$ , and water has $\mathrm{Pr} \approx 7$ . Non-example: the ratio $\mu / k$ is not the Prandtl number; it is not dimensionless, since the factor $c_p$ is missing. For $\mathrm{Pr} \gg 1$ the thermal boundary layer is thinner than the velocity boundary layer, and for $\mathrm{Pr} \ll 1$ (as in liquid metals) it is thicker.

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Dimensionless numbers in fluid mechanics concept
Viscosity concept
Prandtl number definition

Tag	`EPND9BNW82`
URI	`m2://EPND9BNW82`
Kind	definition
Writing	W0 stub
MSC	76M55, 76R05, 80A19
Defines	Prandtl number
Uses	Dimensionless numbers in fluid mechanics, Viscosity
Changed	3 November 2026

Prandtl number

Definition

Remarks

Reading path

Further reading

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