Marangoni Convection Analysis
Theory: surface-tension differences drive flow
Overview
Gently warming a thin oil layer from below produced hexagonal flow cells. The temperature difference seems too small for buoyant convection.
Very likely Marangoni convection. Oil's surface tension falls as temperature rises, so any temperature unevenness on the surface pulls the surface from hot to cold regions, creating flow. Buoyant convection favours thick layers, scaling with thickness cubed, but surface-tension convection scales only linearly with thickness. So in thin layers surface-tension convection starts first, at far smaller temperature differences. Bénard's famous hexagons are in fact often due to Marangoni convection.
Marangoni and Rayleigh numbers
$\partial\sigma/\partial T$ is the surface-tension temperature coefficient and $d$ the layer thickness. Convection onset is roughly Marangoni number about 80 or Rayleigh number about 669 (solid bottom, free top). Their ratio scales with thickness squared.
Free-surface boundary condition
In CFD, the Marangoni effect enters through a free-surface condition balancing shear stress against the surface-tension gradient; $s$ is along the surface.
Tears of wine
The 'tears of wine' — liquid creeping up the inside of a glass and running back down like tears — were explained in 1855 by James Thomson (Lord Kelvin's brother) as a surface-tension effect: alcohol evaporates from the thin wall film, raising its surface tension and pulling liquid up. In 1865 the Italian Marangoni studied the phenomenon in detail, and his name stuck. In 1958 Pearson showed many of Bénard's hexagonal cells are driven by surface tension rather than buoyancy, and the subject spread to space experiments, crystal growth and welding.
Worked examples
Example 1: onset temperature difference
For a layer heated from below, the temperature differences for Marangoni and buoyant onset compare as follows:
| Liquid, thickness | Marangoni | Buoyancy | Starts first |
|---|---|---|---|
| Silicone oil 1 mm | 0.94 K | 55 K | Marangoni |
| Silicone oil 2 mm | 0.47 K | 6.9 K | Marangoni |
| Silicone oil 5 mm | 0.19 K | 0.44 K | Marangoni |
| Silicone oil 10 mm | 0.094 K | 0.055 K | Buoyancy |
| Water 2 mm | 0.038 K | 5.8 K | Marangoni |
| Water 10 mm | 0.008 K | 0.046 K | Marangoni |
The crossover thickness is 7.6 mm for silicone oil and 24.7 mm for water. Water surfaces, however, are easily contaminated by surfactants that cancel tension differences, so in practice Marangoni flow is often weaker.
Example 2: a weld pool
For a steel weld pool with surface-tension coefficient 4×10⁻⁴ N/mK, surface temperature difference 1,000 K and pool size 2 mm, surface shear is 200 Pa and velocity scale about 1.2 m/s. The Péclet number is about 400, so heat is carried by flow rather than conduction.
The weld pool flows that fast inside?
Yes — so penetration shape is set by the flow. Near-pure iron has lower surface tension when hotter, so the surface flows outward from the centre, carrying heat outward for shallow, wide penetration. But a little sulfur or oxygen reverses the sign of the temperature coefficient: the surface flows inward, carrying heat downward for deep, narrow penetration. That's one reason penetration varies between material lots under identical welding conditions. In CFD of welding or metal additive manufacturing, how this boundary condition is applied greatly affects results.
Analysis workflow
- Establish the surface-tension temperature coefficient (including sign) and effects of contamination or impurities.
- Compare Marangoni and Rayleigh numbers to see which dominates.
- Apply the shear-stress condition on the free surface and refine the near-surface grid.
- Where surface deformation is large, combine with free-surface tracking such as VOF.
- Check surface velocity and temperature with tracer particles or IR cameras.
“Same settings, different penetration”
In automated TIG welding with unchanged settings, penetration suddenly became shallower. Investigation found a new stainless-steel lot with lower sulfur content. With less sulfur the surface-tension temperature coefficient turns negative, weld-pool surface flow goes outward, and penetration becomes shallow and wide. CFD with the coefficient's sign flipped reproduced the difference well. A small number in a material certificate changes weld quality through the direction of flow.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Buoyancy only in thin layers | Flow missed | Compute Marangoni number |
| Fixed coefficient sign | Penetration shape wrong | Include composition effects |
| Ignoring contamination | Flow overestimated | Verify experimentally |
| Coarse surface grid | Surface velocity underestimated | Refine near the surface |
| Assuming a flat surface | Errors with large deformation | Track with VOF etc. |
I'd like to learn related topics.
Try buoyant convection with the convection cells tool. Related analyses include natural convection and conduction, pool boiling CFD, heat pipes, welding thermal analysis and additive manufacturing thermal analysis.
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