Vol 88, No 3 (2024)

Regular quaternion equations of the spatial Hill problem (a variant of the limited three-body problem (Sun, Earth, Moon (or another low-mass moving cosmic body under study)) are obtained, when the distance between two bodies with finite masses is considered very large, in four-dimensional Kustaanheimo-Stiefel variables (KS-variables) within the framework of the elliptical and circular spatial bounded three-body problem, as well as the regular quaternion equations of the planar Hill problem in two-dimensional Levi-Civita variables. In these equations, the variables are KS-variables or Levi-Civita variables and the energy of relative motion of the body under study, or a variable that converts for the circular Hill problem into a constant of motion of this body (the Jacobi integration constant), as well as the planetocentric distance of the Sun and real time associated with a new independent variable by the Sundman differential transformation of time or other more complex differential ratio. These equations are supplemented by the equation of the Earth’s orbit in polar coordinates and the equation for the true anomaly characterizing the Earth’s position in the orbit. The first integral of the obtained equations in KS-variables in the case of a circular problem is established. Another first partial integral in the general case is a bilinear relation connecting KS-variables and their first derivatives. Three new forms of regular equations of the spatial Hill problem in quaternion osculating elements (slowly changing quaternion variables) are proposed. The proposed regular quaternion equations have an oscillatory form or the form of equations with slowly changing variables, which makes it possible to effectively use analytical and numerical methods of oscillation theory and methods of nonlinear mechanics in the study of the Hill problem.

The floating ice sheet determines the dynamic interaction between the ocean and the atmosphere, affects the dynamics of the sea surface and subsurface waters, since the ice sheet and the entire mass of liquid under it participate in the general vertical movement. The paper investigates the phase structure of wave fields arising at the interface between ice and a flow of homogeneous liquid of finite thickness when flowing around a localized pulsating source of disturbances. The ice sheet is modeled by a thin elastic plate, the deformations of which are small, and the plate is physically linear. An integral representation of the solution is obtained, and the results of calculations of dispersion dependencies and phase patterns for various parameters of wave generation are presented. It is shown that the main parameters determining the characteristics of the amplitude-phase structure of wave disturbances of the ice sheet surface are ice thickness, flow velocity, and pulsation frequency. Numerical calculations demonstrate that when the flow velocities, ice thickness, and frequency change, there is a noticeable qualitative restructuring of the phase patterns of the excited long-range wave fields at the ice-liquid interface.

The work numerically simulates the flow of a polydisperse gas suspension in a channel. The carrier medium was described as a viscous, compressible, heat-conducting gas. The mathematical model implemented a continuum technique for the dynamics of multiphase media, taking into account the interaction of the carrier medium and the dispersed phase. For each component of the mixture, a complete hydrodynamic system of equations of motion for the carrier phase and dispersed phase fractions was solved. The dispersed phase consisted of particles with different sizes of dispersed inclusions. For the carrier medium, homogeneous Dirichlet boundary conditions were specified on the side surfaces of the channel. For fractions of the dispersed phase, boundary conditions for slippage. The influence of the boundary conditions of the flow of the carrier medium on the dynamics of gas suspension fractions has been revealed.

A theoretical study of “geometric” SP-evanescent (head) waves propagating in an isotropic homogeneous half-space or half-plane with a free boundary shows that these waves can satisfy the condition of absence of effort on the boundary plane if and only if the Lamé parameter λ is vanishingly small, which makes the existence of head waves of this type practically impossible. The analysis is based on the Helmholtz representation for the displacement field in combination with the decomposition of the stress and strain tensor into spherical and deviatoric parts.

The obtained result about the non-existence of this type of evanescent waves can find application in theoretical geophysics in the study of seismic wave fields in the vicinity of earthquake epicenters, as well as in non-destructive acoustic diagnostic methods.

Propagation of harmonic Lamb waves in plates made of functionally graded materials (FGM) with transverse inhomogeneity is studied by the modified Cauchy six-dimensional formalism. For arbitrary transverse inhomogeneity a closed form dispersion equation is derived. Dispersion relations for materials with different kinds of inhomogeneity are obtained and compared.