Prandtl number

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One of the characteristics of similarity of heat processes in fluids and gasses. The Prandtl number depends only on the thermodynamical state of the medium, and is defined by

$$ \mathop{\rm Pr} = \frac \nu {a} = \ \frac{\mu c _ {p} } \lambda , $$

where $ \nu = \mu / \rho $ is the kinematic coefficient of viscosity, $ \mu $ is the dynamic coefficient of viscosity, $ \rho $ is the density, $ \lambda $ is the coefficient of heat conductivity, $ a = \lambda / \rho c _ {p} $ is the coefficient of thermal diffusion, and $ c _ {p} $ is the specific heat capacity of the medium at constant pressure.

The Prandtl number is connected with other characteristics of similarity, the Péclet number and the Reynolds number, by the relation $ \mathop{\rm Pr} = \mathop{\rm Pe} / \mathop{\rm Re} $.

The Prandtl number is named after L. Prandtl.


The Prandtl number is sometimes also called the Darcy–Prandtl number.


[a1] N. Curle, H.J. Davies, "Modern fluid dynamics" , II , v. Nostrand-Reinhold (1971)
[a2] S. Chandrasekhar, "Hydrodynamics and hydrodynamic stability" , Dover, reprint (1981) pp. Chapt. VII
[a3] C.-S. Yih, "Stratified flows" , Acad. Press (1980)
[a4] L.D. Landau, E.M. Lifshitz, "Fluid mechanics" , Pergamon (1959) pp. 203, 208 (Translated from Russian)
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Prandtl number. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Material from the article "Prandtl number" in BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article