# Marangoni number

The Marangoni number (Ma) is the dimensionless number that balances thermal transport via flow (convection) due to a gradient in surface tension, with thermal diffusion.[1] The number is named after Italian scientist Carlo Marangoni, although its use dates from the 1950s[1][2] and it was neither discovered nor used by Carlo Marangoni. The most common application is to a layer of liquid, such as water, when there is a temperature difference ${\displaystyle \Delta T}$ across this layer. This could be due to the liquid evaporating or being heated from below. There is a surface tension at the surface of a liquid that depends on temperature, typically as the temperature increases the surface tension decreases. Thus if due to a small fluctuation temperature, one part of the surface is hotter than another, there will be flow from the hotter part to the colder part, driven by this difference in surface tension, this flow is called the Marangoni effect. This flow will transport thermal energy, and the Marangoni number compares the rate at which thermal energy is transported by this flow to the rate at which thermal energy diffuses.

For a liquid layer of thickness ${\displaystyle L}$, viscosity ${\displaystyle \eta }$ and thermal diffusivity ${\displaystyle \alpha }$, with a surface tension ${\displaystyle \gamma }$ which changes with temperature at a rate ${\displaystyle \partial \gamma /\partial T}$, the Marangoni number can be calculated using the following formula[3]:

${\displaystyle \mathrm {Ma} =-(\partial \gamma /\partial T).{\frac {L.\Delta T}{\eta .\alpha }}}$

When Ma is small thermal diffusion dominates and the there is no flow, but for large Ma, flow (convection) occurs, driven by the gradients in the surface tension. This is called Bénard-Marangoni convection.

## References

1. Pearson, J. R. A. (1958). "On convection cells induced by surface tension". Journal of Fluid Mechanics. 4 (5): 489–500. doi:10.1017/S0022112058000616. ISSN 0022-1120.
2. Block, Myron J. (1956). "Surface Tension as the Cause of Bénard Cells and Surface Deformation in a Liquid Film". Nature. 178 (4534): 650–651. doi:10.1038/178650a0. ISSN 0028-0836.
3. Pr. Steven Abbott. "Marangoni Number Calculator". stevenabbott.co.uk. Retrieved 2 March 2019.