The Marangoni number (Ma) is, as usually defined, the dimensionless number that compares the rate of transport due to Marangoni flows, with the rate of transport of diffusion.
Diffusion is of whatever is creating the gradient in the surface tension.
advective transport rate, due to surface tension gradient
diffusive transport rate, of source of gradient
{\displaystyle \mathrm {Ma} ={\dfrac {\mbox{advective transport rate, due to surface tension gradient}}{\mbox{diffusive transport rate, of source of gradient}}}}
[1] Then the relevant diffusion process is that of thermal energy (heat).
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 Marangoni number for a simple liquid of viscosity
is the only length scale in the problem, which in practice implies that the liquid be at least
The transport rate is usually estimated using the equations of Stokes flow, where the fluid velocity is obtained by equating the stress gradient to the viscous dissipation.
A surface tension is a force per unit length, so the resulting stress must scale as
As Ma is a type of Péclet number, it is a velocity times a length, divided by a diffusion constant,
, Here this is the diffusion constant of whatever is causing the surface tension difference.
A common application is to a layer of liquid, such as water, when there is a temperature difference
This could be due to the liquid evaporating or being heated from below.
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.
, the Marangoni number can be calculated using the following formula:[3]
When Ma is small thermal diffusion dominates and there is no flow, but for large Ma, flow (convection) occurs, driven by the gradients in the surface tension.