Limitations on the existence of shock solutions in a two-fluid system

It is explained why shock solutions exist for only a limited range of values of the parameters of the problem of an immiscible, inviscid and incompressible two-layer supercritical flow of stable stratification confined on top and bottom by horizontal walls. For a given α, the non-dimensional depth of the lower layer, and r, the ratio of the density of the upper fluid to that of the lower fluid, the regions where shock solutions exist can be demarcated in q₁ – q₂ space, where q₁ and q₂ are the non-dimensional discharges in the lower and upper layers respectively.