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Assume we have a cylindrical container that has a lower part filled with some liquid, say, water, and an upper part filled with another liquid, say, air. Assume that the interface between them is a section of a plane defined by $z = 0$. Assume that both liquids are described by Euler equations.

In that case, we have four partial differential equations for their evolution. I will denote all variables for air with subindex $a$ and for water with subindex $w$.

$$ \frac{\partial \rho_a}{\partial t} + \nabla\cdot(\rho_a \mathbf{v}_a) = 0 $$ $$ \rho_a \left( \frac{\partial \mathbf{v}_a}{\partial t} + (\mathbf{v}_a \nabla) \mathbf{v}_a \right) = -\nabla p_a $$

$$ \frac{\partial \rho_w}{\partial t} + \nabla\cdot(\rho_w \mathbf{v}_w) = 0 $$ $$ \rho_w \left( \frac{\partial \mathbf{v}_w}{\partial t} + (\mathbf{v}_w \nabla) \mathbf{v}_w \right) = -\nabla p_w $$

Is it possible from these equations to reason that the pressure on the interface is the same? In other words,

$$ p_a (z = 0 ) = p_w (z = 0). $$

In order to do that, do we have to assume that one liquid cannot penetrate another one? If not, is there some kind of boundary condition that is implied if we assume the hypothesis about the impossibility of such penetration?

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