Weak contribution to levelset interface

Dear all,

I would like to add a weak contribution (for surface tension) to a moving levelset interface for the following application:
I want to model a fluid with a free surface, on which there should be a mass source. The fluid surface is deformed by fluid flow and surface tension.

I already tried to model this with a moving mesh approach but these simulations did not run stable enough, if ever. Furthermore I thought about two-phase-laminar-flow. Here a surface tension at the interface is possible, but no mass source.

That's why I decided to track the interface between air and fluid with levelset. Here I can define: interface velocity = fluid velocity + velocity representing the mass source

Does anyone know how to add a weak contribution to a moving levelset interface? Or is there another way to implement the surface tension (for example how is it realized in the two-phase-laminar-flow)?

I would be very happy if anyone could help me!