Lunova M. Modeling of internal wave processes in layered fluids

The thesis is devoted to the investigation of the conditions of propagation and interaction of internal weakly nonlinear wave packets in a three-layer hydrodynamic system "half-space – layer – layer with a rigid lid" using the asymptotic method of multiscale expansion. The thesis deals with the formulation of a weakly nonlinear problem of propagation and interaction of wave packets in the system "half-space – layer – layer with a rigid lid". Using the method of multiscale expansion to the third order, the first three linear approximations of the problem of propagation and interaction of waves along the contact surfaces in the system are obtained. The problem of the first approximation is solved and a dispersion relation is obtained, which has two pairs of linearly independent solutions. The solutions of the second approximation problem are found. The conditions for the solvability of the second and third approximations are derived. Two evolution equations of the envelopes of internal upper and lower wave-packets are obtained in the form of a nonlinear Schrödinger equation. Taking into account the effect of surface tension, the analysis of the dependence of the frequencies of the center of wave-packet up to the physical and geometric parameters of the system is completed. The transition of the system from first degenerate case to the second one leads to a qualitative symmetry of graphs of the frequencies of center of wave-packet via to the value of density of middle layer. The effect of the surface tension on one of the contact surfaces or on two surfaces simultaneously leads to an increase of the absolute value of the frequency of the center of the wave-packet. The response waves’ amplitudes dependence on the system parameters is investigated. It is found that with increasing thickness of the upper layer, the amplitudes of the response waves limit to the fixed value. In the case of the equal jumps in density on the contact surfaces, the absolute values of the amplitudes of the response waves coincide. The study of the form of the elevation of the contact surfaces in the framework of the solution of the first approximation problem had showed the dependence of the amplitude of the elevation of both, the upper and lower, contact surfaces on the density of the middle layer. Taking into account the second-order approximation, the analysis of the form of the elevation of the contact surfaces have revealed the parameters of system where waves with sharpened combs and blunted soles are formed or waves with blunted combs and sharpened soles are formed. Diagrams of modulation stability for different values of the thickness of upper layer on the plane "layer density – wave number" were constructed, and curves that separate the region of modulation stability from instability for capillary and gravitational waves were found. It is revealed that the areas of modulation instability substantially expand with increasing of the upper layer thickness. The expressions for quantitative and qualitative estimation of energy of three layers of fluid and total energy of the hydrodynamic system "half-space – layer – layer with a rigid lid" in the form of integrals on time and vertical space variables from the product of velocity potentials derivations on time and horizontal space variables are derived. The analysis of the influence of the geometrical parameters of the system on the energy characteristics revealed the following effects and regularities: the presence of the progressive gravitational wave on the upper contact surface leads to an increase of the total energy of the system to the limit value, while the total energy of the system for capillary waves has a decreasing nature; in the case of gravitational-capillary waves, if the progressive wave on the upper contact surface is not taken into account and the progressive wave on the lower contact surface is taken into account, then the energy of the system has a decreasing nature with increasing thickness of the upper layer and, also, gravitational waves energy is larger then capillary waves energy. In general, the results of the investigation have a theoretical nature. To a large extent, it take into account the modern problems of wave processes modeling that arise in practical activity. The presence of a large number of interconnected physical and geometric parameters of the system leads to the need for detailed analysis and interpretation of the results. It create a holistic picture of the wave process and, to the fullest extent, can be applied to study wave processes in an ice-covered ocean and with a layered fluid structure that occurs near the mouth of rivers and in the open ocean during ice melt. The study of the influence of surface tension has practical application in the development of new technologies using layered media that do not mixed, which has the potential to be used in creating innovative solutions for various fields of applied science.