Do you mean numerical simulation only, or analytical approaches also? Why do you think that a thermocapillary effect is relevant hear? Is material you are considering is porous?
For a comparison, I would propose to use Rosenthal solution permitting calculate the temperature at the certain depth. So if you know the arc power and linear velocity, you can easily calculate a penetration depth. If you know something about the porosity, it not should be a big problem to calculate a capillary's contribution. Probably, you will have a difficulty to reach required thermal conductivity and /or surface tension data...
Alexandr, many thanks for the answer. It is clear for me that you are not planning to study welding in the deep penetration mode (welding by laser or electron beam) and are focused exclusively on arc welding, where the classical deep penetration mode is not implemented.
Welding in deep penetration mode is the result of intense hydrodynamics of the melt. Rosenthal's solution does not answer the questions of hydrodynamics of the welding process.
Thermocapillary recirculation is realized in conduction welding and on the back side of the weld pool in deep penetration welding (far away from the front wall of the cavity).
I am sure that the thermocapillary effect (with a shear flow structure !) is related to deep penetration welding, because I have researched this problem in detail for many years and performed a detailed verification study based on a wide range of experiments. Publications with fragments of this study are presented on my RG page.