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Vol 87, No 3 (2023)

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High-Accuracy Numerical Schemes for Solving Plane Boundary Problem for a Polyharmonic Equation and Their Application to Problems of Hydrodynamics

Petrov A.


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.

Prikladnaâ matematika i mehanika. 2023;87(3):343-368
pages 343-368 views

Towards Heat Transfer Critical Conditions for Flow of Fluids with a Nonmonotonic Dependence of Viscosity on the Temperature in Annular Channel

Kireev V.N., Mukhutdinova A.A., Urmancheev S.F.


The present work is devoted to mathematical modeling of the features of the flow of fluids with a nonmonotonic dependence of viscosity on temperature, which is inherent in some solutions and melts of polymers, as well as in a number of liquid metal alloys. For a given pressure drop, the critical conditions of heat transfer on the channel walls are found, which determine the fluid flow rate in the process of establishing a flow associated with the formation of a localized high-viscosity region.

Prikladnaâ matematika i mehanika. 2023;87(3):369-378
pages 369-378 views

Wave Motion in a Surface Electric Charged Viscous Homogeneous Liquid

Ochirov A., Chashechkin Y.D.


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.

Prikladnaâ matematika i mehanika. 2023;87(3):379-391
pages 379-391 views

On Some Regularities of Realization of Electrostatic Instability of Charged Surface of Liquid in Basin of Finite Dimensions

Grigor’ev A.I., Shiryaeva S.O., Koromyslov V.A.


Physical regularities of realization of electrostatic instability of flat charged surface of non-compressible viscous conducting liquid are considered, in pool of finite dimensions, where spectrum of emerging capillary waves is discrete. It has been shown that the critical conditions for the onset of electrostatic instability of an uncompressible viscous conductive liquid, in a basin of finite dimensions, coincide with those for a limitless surface of an infinitely deep ideal uncompressible liquid (coincide with the conditions for realizing Tonks–Frenkel instability). This allows the experimental verification of the criterion for realizing Tonks–Frenkel instability to use basins of finite dimensions, avoiding fundamental errors.

Prikladnaâ matematika i mehanika. 2023;87(3):392-408
pages 392-408 views

Necessary Conditions for Development of Inviscid Instabilities in a Vibrationally Excited Dissociating Gas

Grigoryev Y.N., Ershov I.V.


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.

Prikladnaâ matematika i mehanika. 2023;87(3):409-422
pages 409-422 views

Simulation of Rising Bubble Dynamics

Zotova A.N., Kandaurov A.A., Troitskaya Y.I., Sergeev D.A.


A direct numerical simulation of the rising of an initially quiescent air bubble in water without flow has been carried out. For comparison with the experiment, a complicated initial shape of the bubble, corresponding to the experimental one, was taken. The changes in the shape of the bubble during rising, obtained as the result of numerical simulation, are close to the experimental deformations of the bubble. For comparison with the results of numerical simulations available in the literature, we simulated rising bubble, which initially had a spherical shape. It was found that during rising, the shape of the bubble is first close to elliptical and oscillates, but then it becomes more complicated – a “tail” appears in the lower part of the bubble. This regime of the rising bubble dynamics is confirmed by the results of numerical simulation published in the literature.

Prikladnaâ matematika i mehanika. 2023;87(3):423-431
pages 423-431 views

Far Fields Asymptotics of Internal Gravity Waves from a Pulse Localized Source in a Rotating Stratified Medium

Bulatov V.V., Vladimirov I.Y.


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.

Prikladnaâ matematika i mehanika. 2023;87(3):432-441
pages 432-441 views

Direct and Inverse Problems of Dynamics of Surface Waves Caused by Flow around Underwater Obstacle

Knyazkov D.Y., Baydulov V.G., Savin A.S., Shamaev A.S.


The paper presents algorithms and results of calculations of the dynamics of the surface layer of a liquid under the action of currents that have emerged from the depth. Several approaches to modeling the velocity field in a horizontal flow round a fixed underwater obstacle are investigated. Formulas for calculating the velocity field on the free surface of an ideal homogeneous liquid are proposed. A computer program has been developed that makes it possible to simulate the interaction of a stratified fluid flow with an underwater obstacle. The possibility of using asymptotic formulas for the far-field approximation to calculate the velocity field in a uniformly stratified fluid is studied.

Prikladnaâ matematika i mehanika. 2023;87(3):442-453
pages 442-453 views

Power Elliptic Bodies of Minimum Resistance in the Gas Flow

Nguyen V.L.


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.

Prikladnaâ matematika i mehanika. 2023;87(3):454-460
pages 454-460 views

Numerical Investigation of the Mass Transfer of Dispersed Particles during the Passage of a Shockwave in a Mono and Polydisperse Gas Suspension

Gubaidullin D.A., Tukmakov D.A.


The paper numerically simulates the propagation of a shock wave through a gas suspension. The carrier medium was described as a viscous, compressible, heat-conducting gas. The mathematical model implemented a continuum method for the dynamics of multiphase media, taking into account the interaction of the carrier medium and the dispersed phase. The mass transfer of disperse inclusions suspended in the gas, caused by the interaction of the shock wave with monodisperse gas suspensions and with gas suspensions having a multi-fractional composition, was modeled. Differences in the mass transfer of particles depending on the particle size are revealed. It was also found that the process of mass transfer of dispersed inclusions in a monodisperse gas suspension differs from a similar process for a fraction of a polydisperse gas suspension having the same particle size and the same volume content.

Prikladnaâ matematika i mehanika. 2023;87(3):461-474
pages 461-474 views

Investigation of a Thermal Detonation Wave in a Mixture of  Water Drops with Molted Lead

Melikhov V.I., Melikhov O.I., Bashar S.


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.

Prikladnaâ matematika i mehanika. 2023;87(3):475-488
pages 475-488 views

Problem of Longitudinal Vibrations of a Viscoelastic Rod of Maxwell Type

Korzyuk V.I., Rudzko J.V., Kolyachko V.V.


In this paper, we study well-posedness in the sense of Hadamard of the Cauchy problem for a one-dimensional hyperbolic system of partial differential equations describing the longitudinal vibrations of a viscoelastic rod of Maxwell type with constant cross-section. We discuss some properties of the system and its solutions: the conservation of modified “energy”, the finite propagation speed, dispersion, and dissipation of solutions.

Prikladnaâ matematika i mehanika. 2023;87(3):489-498
pages 489-498 views

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