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Boundary condition on the free surface

The physical requirement that there can be no mass flux through the free surface, gives a kinematic boundary condition that has to be fulfilled at the surface. The condition states that the boundary has to move as a material surface, and is written as  


where $eta(x,y,t)$ is the location of the free surface.

A dynamic condition has also to be fulfilled at the surface. This condition states that the momentum across the free surface has to be conserved,


where $tau_{ij}$ is the stress tensor begin{equation}tau_{ij}=p delta_{ij} -  mu(frac{partial u_i}{partial x_j} + frac{partial u_j}{partial x_i}) end{equation}


is the surface tension, the surface curvature, patm the atmospheric pressure, ni is the unit normal vector directed outwards of the surface. If the surface tension is neglected and the flow is considered non viscous, the dynamic condition can be simplified to


The corresponding boundary condition for , Eq.(3) becomes