ar X iv : p at t - so l / 9 90 80 05 v 1 1 1 A ug 1 99 9 Longwave Interface Instability In Two - Fluid Vibrational Flow

We consider longwave mode of the interface instability in the system comprising of two immiscible fluid layers. The fluids fill out plane horizontal cavity which is subjected to horizontal harmonic vibration. The analysis is performed within the framework of ”high frequency of the vibration” approximation and the averaging procedure. The nonlinear equation (having the form of Newton’s second law) for the amplitude of interface deformation is obtained by means of multiple scales method. It is shown that (in addition to previously detected quasistationary periodic solutions) the equation has a class of quasistationary solitary solutions. In experimental works by Bezdeneznykh et al. [1] and by Wolf [5] for a long horizontal reservoir filled with two immiscible viscous fluids, an interesting phenomenon was found at the interface: the horizontal vibrations lead to the formation of a steady relief. This formation mechanism has a threshold nature; it is noteworthy that such a wavy relief appears on the interface only if the densities of the two fluids are close enough, i.e. it does not appear for the liquid/gas interface (free surface). The interface is absolutely unstable if the heaviest fluid occupies the upper layer; i.e., the horizontal vibration does not prevent the evolution of Rayleigh-Taylor instability, in contrast to the vertical one which under certain conditions suppresses its evolution. A theoretical description of this phenomenon was provided by Lyubimov & Cherepanov [3] within the framework of a high frequency (of the vibration) approximation and an averaging procedure; they found that a horizontal vibration leads to a quasistationary state i.e., a state where the mean motion is absent but the interface oscillates with a small amplitude (of the order of magnitude of the cavity displacement) with respect to the steady relief. They also obtained the general equations and boundary conditions for English translation from ”Hydrodynamika”, Perm, 1998, pp. 191 196 e-mail: [email protected]