Well I think the resultant stresses at the interface of fluid and air/gas should also be in equilibrium.

continuitiy and NS euqation. then a free surface model should be used to identify the free surface. generally, we have VOF and levelset

Continuity and Momentum equation plus all the boundary conditions which include a kinematic and a dynamic balance equation at the interface.

You should also consider if you can neglect thermal effect otherwise you are going to complicate the equations adding the energy equation.

There are several references online, you have just to define your problem depending on geometry and kind of excitation.