# Rayleigh number (for the mushy zone of a solidifying alloy)

## Description

In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free convection or natural convection. When the Rayleigh number is below a critical value for that fluid, heat transfer is primarily in the form of conduction; when it exceeds the critical value, heat transfer is primarily in the form of convection.

The Rayleigh number is named after Lord Rayleigh and is defined as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity. Hence the Rayleigh number itself may also be viewed as the ratio of buoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivities.

For most engineering purposes, the Rayleigh number is large, somewhere around 10^6 to 10^8.

The Rayleigh number can be also used as a criterion to predict convectional instabilities, such as A-segregates, in the mushy zone of a solidifying alloy. The mushy zone Rayleigh number is defined as shown here.

A-segregates are predicted to form when the Rayleigh number exceeds a certain critical value. This critical value is independent of the composition of the alloy, and this is the main advantage of the Rayleigh number criterion over other criteria for prediction of convectional instabilities, such as Suzuki criterion.

Torabi Rad et. Al. showed that for steel alloys the critical Rayleigh number is 17. Pickering et. al explored Torabi Rad’s criterion, and further verified its effectiveness. Critical Rayleigh numbers for lead–tin and nickel-based super-alloys were also developed.

Related formulas## Variables

Ra | Rayleigh number (dimensionless) |

Δ_ρ | density delta (kg/m^{3}) |

ρ_{0} | density (kg/m^{3}) |

g | Standard gravity |

K | mean permeability (of the initial portion of the mush) (m^{2}) |

L | characteristic length scale (m) |

α | thermal diffusivity (m^{2}/s) |

ν | kinematic viscosity (m^{2}/s) |