# Prandtl number

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.

#### Comments

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

#### References

[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) |

**How to Cite This Entry:**

Prandtl number.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Prandtl_number&oldid=48274