[ilia.nikiforov]

Originally Posted by FMDenaro

That means


Div v =0


dv/dt + div (vv) + Grad phi = 0

That is a first order hyperbolic system of equation in the velocity field, the "pressure" phi being only a lagrangian multiplier (just apply the curl to the momentum). Again, only one condition is required on the boundary. Since the fluid is ideal, you cannot set the physical non-slip condition and only the normal velocity is prescribed to fulfill the continuity equation.
To have two BCs you need a second order equation, that is the momentum equation with the diffusive term in the RHS.

Thank you very much for your help, looks like I need to review systems of PDEs in general, thank you for pointing me in the right direction.