No, with finite element methods you can approach that. It's just a test for your computational capability, not for the law.

Sudden change of wetting angle destroys Hydrostatic equilibrium of capillary liquid column, and the second law of thermodynamics fails

Inserting a coloured picture is not mathematical or physical evidence.

There is a mismatch between what I want to express and what you said.

the question is not 'failure of second law of thermodynamics' but what triggers convection - even very small temperature gradients are sufficient.

The problem lies in whether there is a solution for the height of the liquid level and the solution for the steam pressure.

2. Without a solution, the system is not in equilibrium, and the second law of thermodynamics is invalid.

3. My judgment has not been resolved. Because the infiltration angle at point C is discontinuous, there are other judgments.

4. If you think there is a solution, you can research it out, which is your paper.

No, we don't need to prove anything right now because you haven't made a claim of substance. You want to disprove the second law, so you need really strong evidence and not a diffuse claim. We have no such agenda.

The invasion angle of point c in the figure is abrupt, which is a mathematical singularity. The system has no stable solution. You won't use the Second law of thermodynamics to fight against mathematics, will you?!

https://www.researchgate.net/publication/371968304_Mathematical_proof_that_the_second_law_of_thermodynamics_is_not_applicable_to_the_double_gas_cycle

This communication claims the existence of a perpetuum mobile because surfaces with different curvatures have different vapour-pressures. According to the author, the liquid from the higher-vapour-pressure-surface-region consciously evaporates and condenses on the region of the lower-vapour-pressure-region.

This claim ignores the effects of gravity. In a gravitational field, the pressure of the vapour in the gas phase depends on the height. At higher elevations (at point A in the Figure), the pressure of the vapour is lower; simultaneously, the vapor-pressure of the concave liquid surface is also lower. While, at lower elevations (at point B in the Figure), the pressure of the vapour is higher; simultaneously, the vapor-pressure of the convex liquid surface is also higger. The liquid surface at any point is in equilibrium with the above vapour phase, and the vapour phase itself is in gravitational equilibrium. Hence, the dotted line transport process in the Figure does not exist.

The laws of thermodynamics are still valid.