# Reynolds number

One of the non-dimensional numbers which provide criteria of similarity for flows of viscous liquids and gases. It characterizes the relation between the inertial forces and the forces of viscosity: $$ \mathrm{Re} = \frac{\rho \, \nu \, l}{\mu} $$ where $\rho$ is the density, $\mu$ is the dynamical coefficient of viscosity of the liquid or gas, $\nu$ is the typical rate of flow, and $l$ is the typical linear dimension.

The Reynolds number also determines the mode of flow of a liquid in terms of a critical Reynolds number, $\mathrm{Re}_{\mathrm{cr}}$. When $\mathrm{Re} < \mathrm{Re}_{\mathrm{cr}}$, only laminar liquid flow is possible, whereas when $\mathrm{Re} > \mathrm{Re}_{\mathrm{cr}}$ the flow may become turbulent (cf. Turbulence, mathematical problems in).

The Reynolds number is named after O. Reynolds.

#### References

[a1] | G.K. Batchelor, "An introduction to fluid dynamics" , Cambridge Univ. Press (1974) pp. Sect. 4.7 |

[a2] | M.I. Vishik, A.V. Fursikov, "Mathematical problems of statistical hydromechanics" , Kluwer (1988) pp. Chapts. 3; 4; 6 (Translated from Russian) |

[a3] | L.D. Landau, E.M. Lifshitz, "Fluid mechanics" , Pergamon (1959) (Translated from Russian) |

**How to Cite This Entry:**

Reynolds number.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Reynolds_number&oldid=53754