A high curvature relationship occurs at the beginning and the end of our two-fluid experiment.

A tiny bubble has such a high curvature/small radius that it is difficult for random vibrations to deform it.
As the water-based fluid flows into the oil-based fluid, into the crests (fingers) of the unstable waves at the boundary, their tip radii get very small and their curvature very large, thus damping out vibrations there.

In both cases (beginning and end), the curvature at the boundary of the two-fluid system is so large that the boundary cannot be made to deform. In this case, there exists no relationship across the boundary. Therefore, no recognizable existence there.