Please cite this article in press as: Mukhamediev ShA, et al. Stability of a two-layer system of inhomogeneous heavy barotropic fluids. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.07.005 ARTICLE IN PRESS G Model JAMM-2355; No. of Pages 7 Journal of Applied Mathematics and Mechanics xxx (2016) xxx–xxx Contents lists available at ScienceDirect Journal of Applied Mathematics and Mechanics jou rn al h om epa ge : www.elsevier.com/locat e/jappmathmech Stability of a two-layer system of inhomogeneous heavy barotropic fluids  Sh. A. Mukhamediev, E.I. Ryzhak ∗ , S.V. Sinyukhina O. Schmidt Institute of Physics of the Earth of the Russian Academy of Science, Moscow, Russia a r t i c l e i n f o Article history: Received 2 December 2015 Available online xxx a b s t r a c t 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. © 2016 Elsevier Ltd. All rights reserved. 1. Introduction The issues of stability of a two-layer system of fluids in the uniform gravity field are studied. The reservoir occupied by the fluids is supposed to be of arbitrary prescribed shape. The fluids considered are compressible and inviscid (barotropic), whereas corresponding distributions of both their densities and elastic properties over depth being arbitrary. Models of that type having numerous applications in physics of atmosphere and ocean, astrophysics, geophysics, and technical physics attracted and still attract attention of researchers for more than a hundred years. In Rayleigh’s pioneering work 1 two different problems on stability of incompressible inviscid fluids in a gravity field were considered. In the first problem, it was supposed that two homogeneous fluids fill, correspondingly, upper and lower half-spaces separated by a horizontal plane. In the second problem a fluid is supposed to be inhomogeneous, filling the whole space, the density distribution over depth being exponential. Under these assumptions, Rayleigh succeeded in substantiating the following result: the system of incompressible inviscid fluids with density inversion (the density below is less than that above) is unstable. Subsequently and up to the present time Rayleigh’s model was reproduced with some minor variations in a great number of studies aimed at analysis of various physical phenomena. It is obvious that, for many applications of the model, the assumption of fluids incompressibility is inappropriate. The problem of stability/instability of a system of compressible fluids was posed by Chandrasekhar (e.g. Ref. 2) and considered later (Refs 2–5 and some others) in a rather particular setting: the infinite horizontal dimensions of the system along with the absence of boundary conditions at infinity, the homogeneity of elastic properties within each layer, a special character of density distribution, and low compressibility. Nevertheless even in such a setting the problem still remained unsolved (see a discussion in Ref. 5 containing a review of a number of previous studies). The method of analysis proposed by Rayleigh 1 and used by his followers is restricted in principle, that consequently leads to certain limitations of the results obtained. Let us discuss the restrictions mentioned, bearing in mind that we loosened them (using both physically more adequate setting of the problem and in principle different method of its analysis) and finally succeeded in solving the problem completely.  Prikl. Mat. Mekh. Vol. 80, No. 3, pp.375–385, 2016. ∗ Corresponding author. E-mail address: e i ryzhak@mail.ru (E.I. Ryzhak). http://dx.doi.org/10.1016/j.jappmathmech.2016.07.005 0021-8928/© 2016 Elsevier Ltd. All rights reserved.