Vol 87, No 3 (2023)

Boundary value problems are considered for harmonic, biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on the plane. The problems are reduced to solving linear integral equations on boundary contours, which are assumed to be smooth. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which integral equations are expressed. The algorithm uses the periodicity of functions defined on closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed degree. Applications to solving problems of hydrodynamics, filtration and other problems of theoretical physics are considered.

The influence of the surface electric charge on the character and properties of wave motion along the free surface of a viscous homogeneous liquid has been investigated by analytical asymptotic methods. Expressions describing the dispersion dependences of the wave motion components are obtained. The phase and group velocities of the structures forming the wave motion are determined.

For a plane flow of a vibrationally excited dissociating diatomic gas the necessary conditions of the existence of growing (neutral) inviscid perturbations, similar to the Rayleigh criterion of a “generalized” inflection point, are obtained. The corresponding formulas are presented for cases with a certain physical interpretation. In particular, the model of a vibrationally excited one-component gas is considered as the initial stage of thermal dissociation, as well as a wide spread model with one dissociation-recombination reaction. The case of a binary molecular-atomic mixture with a vibrationally excited molecular component and a “frozen” gas-phase dissociation-recombination reaction is considered as an intermediate one. Comparative numerical calculations were carried out, which showed, in particular, that under conditions of developed dissociation, the use of the criterion of the “generalized” inflection point does not take into account the specifics of the process. The wave numbers and phase velocities of the I and II inviscid modes calculated on its basis may differ significantly from the results obtained using the new necessary condition.

The problem of constructing asymptotics of the internal gravity waves far fields arising from an impulsive localized source of perturbations in a stratified fluid of finite depth rotating as a whole is solved. In the approximation of constant buoyancy frequency, uniform and nonuniform asymptotics of solutions are constructed to describe far wave fields, which are expressed in terms of the Airy function and its derivative. The exact and asymptotic results are compared, and it is shown that at times longer than several buoyancy periods and at distances of the order of the liquid layer thickness, the obtained asymptotics allow one to describe the amplitude-phase structure of far wave fields.

For power elliptical body, the drag force in the high-speed rarefied gas flow is calculated based on several local models. The solution of the variational problem determines the degree of minimum resistance in the generatrix of the body, depending on coefficient of the ellipticity and different elongation in a wide range of Reynolds numbers.

The patterns of the wave of thermal interaction of water drops in a high-temperature molten lead, are studied. Due to the boiling of water on the surface of molten lead, both liquids (phases) are separated by a vapor film. A one-dimensional model of interacting and interpenetrating continuums is used, which describes the dynamics of each fluid by introducing a special field characterized by its own velocity, temperature, and volume fraction. Wave velocity is determined by the equality of phase velocities and temperatures in the Chapman-Jouguet plane. The parameters at the pressure peak are calculated from the conditions at the discontinuity which are the boundary conditions for integrating the conservation equations in the zone of interaction of water droplets with the melt. The resulting structure of the thermal detonation wave is characterized by the fact that the maximum pressure is at some distance from the shock wave.