Stability of a two-layer system of inhomogeneous heavy barotropic fluids

Basing on the static energy criterion for a bounded domain of an arbitrary shape and with regard for the boundary conditions at all parts of the boundary, the stability of a two-layer system of inhomogeneous barotropic fluids in the uniform gravity field is studied for arbitrary distributions of their densities and elastic properties over depth. Almost coinciding with each other (up to the strictness of one of the two inequalities), equally valid for an arbitrary number of layers, the necessary and sufficient conditions for stability are obtained, that represents a new exhaustive result for the problem considered. Additionally (with compressibility admitted) possible influence of viscosity (which may be anisotropic), and also the case when the layers consist of solid elastic materials, are considered. In the case of instability, the lower estimates for the greatest rate of disturbances growth are obtained.