Waves on the Free Surface of a Layer of a Two-Phase Mixture of Unlimited Depth

About the author:

Nina N. Butakova, Cand. Sci. (Phys.-Math.), Associate Professor, Department of Fundamental Mathematics and Mechanics, University of Tyumen; n.n.butakova@utmn.ru

Abstract:

This article considers the problem of the propagation of waves on the free surface of a two-phase mixture of infinite depth. The author finds the asymptotic solution of the problem in the linear approximation. Multiphase media characterized by the presence of macroscopic inclusions (in comparison with molecular scales) are present in nature, widely distributed in technological processes. Disperse mixtures consisting of two phases are the simplest of them. Mathematical modeling of such media is complicated by the need to take into account the effects of interfacial interaction; that significantly increases the number of parameters in the equations.

Therefore, the main difficulty in constructing a mathematical model is the creation of a closed system of equations. This paper uses the scheme of joint phase deformation proposed by H. A. Rakhmatullin (as a closing condition, the assumption of equal pressure in phases is used). The author considers a nonlinear boundary value problem of the propagation of surface waves along a layer of a two-phase mixture of infinite depth; as the carrier phase, an incompressible ideal liquid is chosen, the dispersed phase is solid non-deformable particles. The author assumes that the length of the surface wave is much greater than its height. The asymptotic solution of the problem in the linear approximation with respect to a small amplitude parameter in the form of damped traveling waves is found. The pressure, the velocity of the wave motion of the dispersed and carrier phase, and the shape of the free surface are determined. The paper shows that the perturbation of the concentration of the second phase is of a higher order of smallness in comparison with the wave perturbations of the phase and pressure velocities. Dispersion relations are obtained to determine the phase velocity and the damping decrement of the wave. Calculations are performed to illustrate the results obtained.

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