Boundary conditions errantly applied to pressure across flui

[TheCanadian]

I am trying to decipher if an error occurred in a calculation given in this paper.

It is understandable that if two compressible fluids of different uniform densities have a common interface (e.g. Figure 1), then to be in equilibrium and supported against gravity, there must be a pressure gradient across the boundary of the two fluids. Thus on either side of the boundary is a different pressure (i.e. pressure discontinuity). Based on reading the authors' assumptions, each fluid has a constant density and pressure (and thus sound speed) at equilibrium, but these values differ between the two fluids. Using simply (16) from the paper, I have calculated that the pressure difference across the boundary in equilibrium is given by: $p_2 - p_1 = \frac {c^2}{4}(\rho_{01} - \rho_{02}) > 0$ (where $g$ is a constant acceleration given by a static field) and thus non-zero.

In the paper, the authors solve the problem and essentially derive equations (21) and (25) which are evaluated on either side of the boundary. When plugging (21) into (25), what I fail to understand is why the pressure can be taken outside of the summation/difference across the boundary as indicated in (26)? They now treat the pressure, $p$, as a given value yet it only has meaning on either side of the boundary as $p_1$ or $p_2$ to my knowledge. Thus is there an error in the paper? What is the definition of this $p$? In the ensuing equations, the authors distinguish between the densities on either side of the boundary (i.e. $\rho_{01}$ and $\rho_{02}$), but make no such distinctions about the pressure, which is now simply $p$ instead of $p_1$ and/or $p_2$ and I am failing to recognize the obvious reason(s) why.

 

[Chestermiller]

Who says that the pressure has to be discontinuous at the interface? Where did you get this idea? Only if the boundary is curved (so that surface tension comes into play) or when viscous stresses are significant can the pressure be discontinuous at the interface.

 

[TheCanadian]

Who says that the pressure has to be discontinuous at the interface? Where did you get this idea? Only if the boundary is curved (so that surface tension comes into play) or when viscous stresses are significant can the pressure be discontinuous at the interface.

I agree, the pressure should be constant on the interface and thus continuous. My only reason to question this was based on the paper stating the speed of sound is constant on either side of the interface. But if density is constant on either side of the interface, this would imply pressure is similarly constant on either side, although a pressure gradient is needed for force balance. Hence why the assumption of discontinuous pressure along the boundary which was alarming and suggesting a possible error. I suspect that the authors were not exactly suggesting the speed of sound is uniform throughout either fluid and this may have simply resulted in my misinterpretation of their wording. In the correct case, believe this would indicate a non-uniform temperature and thus a non-uniform pressure in either fluid to support this constant density in either fluid, where the density (and thus temperature) would be discontinuous along the interface, but not pressure.