Abstract
The problem of propagation of internal waves for an ideal incompressible fluid was considered. The hydrodynamic system consisted of three layers of a finite thickness that did not mix and were bounded with a solid cover from above and a solid bottom from below. The surface tension force acted on the interfaces of liquid media.The problem was formulated in a dimensionless form. The nonlinearity coefficient equal to the ratio of the characteristic amplitude to the characteristic wavelength was used as a small parameter.Solutions of the linear problem were sought in the form of progressive waves. On the basis of these solutions, the dispersion relation was obtained as a condition of solvability of the system of linear differential equations. Existence of two characteristic modes (the real roots of the dispersion relation) was revealed. The graphs of the roots of the dispersion relation were analyzed depending on various physical and geometric parameters of the system. It has been established that thickness of the layers did not affect dispersion of the waves while the change of the surface tension and the ratio of densities had a significant effect on the wave propagation conditions. Wave packets were considered in a linear formulation which was a superposition of harmonic waves of close lengths. It was found that amplitude of the envelope of the wave packet on the lower contact surface remained sinusoidal while it varied on the upper contact surface according to a more complicated law.The problem of propagation of internal waves along the surface of three liquid layers can simulate a strongly stratified thermocline in the ocean. The study of influence of surface tension can also be used to develop new technologies associated with the use of three liquid layers that do not mix.
Highlights
Fluids with discrete or continuous density stratification are encountered in numerous applied problems
The conditions of internal wave propagation are quite sensitive in relation to physical and geometric parameters of the system
This paper has described internal waves in the presence of a flow and obstacles but did not take into account formation of internal waves under the influence of gravitation and capillary forces
Summary
Fluids with discrete or continuous density stratification are encountered in numerous applied problems. Perturbation of such an inhomogeneous fluid causes propagation of internal waves with interesting characteristic properties. Mathematical formulas of such problems consist of a system of nonlinear differential partial derivative equations. In a three-layer fluid, specific classes of nonlinear waves, so called envelope solitons, can spread. They are still poorly studied both analytically and numerically. All this shows urgency of the problem of studying the internal waves in a three-layer fluid. The knowledge of the effect of surface tension can be applied in the development of new technologies associated with the use of three non-mixing liquid layers
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