boundary condition on free surface for the conservation mass equation

Hi! 

I am trying to set up boundary conditions for the free surface for the
mass eqaution, which is takes from the fluid dynamics in 2D. The
boundary is material-air. 

The equation is dr/dt + div(ru)=0, where r is density and u is velosity. 

Material is compressible. 

All my attempts lead to the solution when the porosity on that
boundary and inside of the material (in some region close to boundary)
becomes negative. Deeper inside of the material solution becomes
realistic. I guess it because of the wrong bc's.

The material is compressed by the internal hydrostatic stresses. 

If someone could suggest anything it would be great! 

Thank you,
Boris 
 







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