Editing Rayleigh number (section)

==Derivation==
The Rayleigh number describes the behaviour of fluids (such as water or air) when the mass density of the fluid is non-uniform. The mass density differences are usually caused by temperature differences. Typically a fluid expands and becomes less dense as it is heated. Gravity causes denser parts of the fluid to sink, which is called [[convection]]. Lord Rayleigh studied<ref name=":0"/> the case of [[Rayleigh–Bénard convection|Rayleigh-Bénard convection]].<ref>{{Cite journal |last1=Ahlers |first1=Guenter |last2=Grossmann| first2=Siegfried |last3=Lohse |first3=Detlef |date=2009-04-22 |title=Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection |journal=Reviews of Modern Physics |volume=81 |issue=2 |pages=503–537 |arxiv=0811.0471 |doi=10.1103/RevModPhys.81.503|bibcode=2009RvMP...81..503A |s2cid=7566961 }}</ref> When the Rayleigh number, Ra, is below a critical value for a fluid, there is no flow and heat transfer is purely by [[thermal conduction|conduction]]; when it exceeds that value, heat is transferred by natural convection.<ref name=":2"/>

When the mass density difference is caused by temperature difference, Ra is, by definition, the ratio of the time scale for diffusive thermal transport to the time scale for convective thermal transport at speed <math>u</math>:<ref name=":1">{{Cite journal |last1=Squires |first1=Todd M. |last2=Quake |first2=Stephen R. |date=2005-10-06 |title=Microfluidics: Fluid physics at the nanoliter scale |journal=Reviews of Modern Physics |volume=77 |issue=3 |pages=977–1026 |doi=10.1103/RevModPhys.77.977 |url=https://authors.library.caltech.edu/1310/1/SQUrmp05.pdf |bibcode=2005RvMP...77..977S}}</ref>

<math display="block">\mathrm{Ra} = \frac{\text{time scale for thermal transport via diffusion}}{\text{time scale for thermal transport via convection at speed}~ u}.</math>

This means the Rayleigh number is a type<ref name=":1"/> of [[Péclet number]]. For a volume of fluid of size <math>l</math> in all three dimensions{{clarify|reason=e.g. Does this mean a cube of side-length l?|date=December 2020}} and mass density difference <math>\Delta\rho</math>, the force due to gravity is of the order <math>\Delta\rho l^3g</math>, where <math>g</math> is acceleration due to gravity. From the [[Stokes's law|Stokes equation]], when the volume of fluid is sinking, viscous drag is of the order <math>\eta l u</math>, where <math>\eta</math> is the [[dynamic viscosity]] of the fluid. When these two forces are equated, the speed <math>u \sim \Delta\rho l^2 g/\eta</math>. Thus the time scale for transport via flow is <math>l/u \sim \eta/\Delta\rho lg</math>. The time scale for thermal diffusion across a distance <math>l</math> is <math>l^2/\alpha</math>, where <math>\alpha</math> is the [[thermal diffusivity]]. Thus the Rayleigh number Ra is

<math display="block">\mathrm{Ra} = \frac{l^2/\alpha}{\eta/\Delta\rho lg} = \frac{\Delta\rho l^3g}{\eta\alpha} = \frac{\rho\beta\Delta T l^3g}{\eta\alpha}</math>

where we approximated the density difference <math>\Delta\rho=\rho\beta\Delta T</math> for a fluid of average mass density <math>\rho</math>, [[Coefficient of thermal expansion|thermal expansion coefficient]] <math>\beta</math> and a temperature difference <math>\Delta T</math> across distance <math>l</math>.

The Rayleigh number can be written as the product of the [[Grashof number]] and the [[Prandtl number]]:<ref name=":1"/><ref name=":2"/>
<math display="block">\mathrm{Ra} = \mathrm{Gr}\mathrm{Pr}.</math>