Self-gravitating instability of triple superposed fluids of different densities with plane interfaces

Abstract

The problem of two superposed fluids is formulated and the self-gravitating hydrodynamic basic equations are solved. Upon appropriate boundary conditions, a general eigenvalue relation is derived and discussed. The stability states are identified as ρ' < ρ, ρ' = ρ and ρ' > ρ where ρ' and ρ are the densities of the self-gravitating superposed fluids. The analytical results are confirmed numerically and it is found that ρ'/ρ is stabilizing according to restrictions. A physical interpretation has been declared to some new parameters.In part B the governed equations of the perturbed and unperturbed states are solved. Their solutions satisfy certain conditions across the fluids plane interfaces. A cumbersome stability criterion is derived based on the linear perturbation technique of normal mode analysis. Several limiting cases could be recovered with some simplifications. The (in-)stability restrictions are identified for several different values of the densities ratios of the three superposed fluid layers.