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2021 year, number 4
T. A. Bodnar
Biysk Technological Institute, Biysk, Russia
Keywords: steady waves on the surface of a liquid, integral equation, Laurent series, kernel of the Nekrasov equation, conformal mapping
Abstract >>
The Nekrasov integral equation is obtained which describes the steady flow of an ideal incompressible fluid with a free surface above an uneven bottom with a wavy profile. A numerical method has been developed for solving this equation at the coordinates of the bottom profile defined in the plane of the complex variable z = x + iy .

A. A. Cherevko, T. S. Gologush, V. V. Ostapenko
Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
Keywords: twophase filtration, CABARET scheme, optimal control, particle swarm method, arteriovenous malformation, embolization
Abstract >>
The joint flow of blood and embolization composition inside an arteriovenous malformation is modeled with the use of a onedimensional model of twophase filtration based on real clinical data. Numerical simulations are performed by a monotonic modification of the CABARET scheme. Optimal regimes of embolization for real patients are found with the use of a modified particle swarm method, which is a numerical method of global optimization.

S. A. Kinelovskii
Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
Keywords: polymorphism, shock wave, boron nitride, phase transition
Abstract >>
The model relating the polymorphic transformation of a crystalline substance under shock wave loading to the change in its elastic energy is further considered, now by an example of boron nitride. The results obtained show that the model provides a reliable description of the martensite phase transition in boron nitride subjected to a shock wave. It is found that the shock adiabat of boron nitride regardless of its structure has an inflection at the shock wave velocity D » 6.2 km/s. The nature of this inflection is not clear yet.

V. Yu. Liapidevskii, A. A. Chesnokov, V. E. Ermishina
Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
Keywords: equations of multilayer shallow water, internal solitary waves, Boussinesq approximation
Abstract >>
A nonhomogeneous system of onedimensional conservation laws is proposed, which describes propagation of largeamplitude bottom (subsurface) internal waves in multilayer stratified shallow water in the Boussinesq approximation. The model can be applied to layered flows of a stably stratified fluid and is hyperbolic for moderate velocity shear in the layers. Steady solutions of equations of motion are studied, and conditions for the formation of solitary waves of the first mode are formulated. The model is verified by means of comparing the results predicted by this model with the results of actual observations and calculations of twodimensional equations of motion. Propagation of unsteady nonlinear wave packets in a multilayer fluid is numerically simulated.

V. Sh. Shagapov^{1}, E. V. Galiakbarova^{2}, Z. R. Khakimova^{2}
^{1}Mavlyutov Institute of Mechanics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa, Russia ^{2}Ufa State Petroleum Technical University, Ufa, Russia
Keywords: well, probe, fracture, hydraulic fracturing, harmonic waves pressure, pulse signal
Abstract >>
The paper considers the evolution of a pulse signal in the annular gap between a diagnostic probe and an open well surrounded by a lowpermeability fractured formation. Fractures are located along the well, and the well and the fracturedporous medium are filled with the same acoustically compressible fluid. The problem is solved numerically by the fast Fourier transform. Dispersion equations are obtained that describe the propagation of damped traveling waves in the gap, taking into account fluid filtration through longitudinal fractures. Analysis was performed of the influence of the filtration characteristics of the reservoir, hydraulic fractures, and the width of the gap between the probe body and the borehole wall on the phase velocity, the attenuation coefficient, and the evolution of pulse signals.

N. G. Musakaev^{1}, S. L. Borodin^{1}, M. K. Khasano^{2}
^{1}Tyumen Department of the Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Scences, Tyumen, Russia ^{2}Sterlitamak Branch, Bashkir State University, Sterlitamak, Russia
Keywords: porous medium, filtering, gas hydrate, phase transition, mathematical model
Abstract >>
This paper describes a mathematical model and features of gas hydrate formation during the injection of natural gas with a given composition into a porous reservoir containing the same gas and water in the initial state. Numerical solutions of the axisymmetric problem are constructed, which describe the distributions of the main parameters in the reservoir both with the frontal surface and with the volumetric region of phase transitions. The influence of the parameters of the injected gas and porous medium on the mode and rate of gas hydrate formation is investigated.

O. A. Simonov^{1}, L. N. Filimonova^{2,3}
^{1}Tyumen Research Center, Siberian Branch, Russian Academy of Sciences, Tyumen, Russia ^{2}Tyumen Department of the Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Tyumen, Russia ^{3}Tyumen State University, Tyumen, Russia
Keywords: porous medium, free convection, water density maximum
Abstract >>
A numerical study of the cooling of a watersaturated porous medium in a cylindrical heatinsulated vessel with a vertical cooling element is performed. The effect of convective heat transfer on the cooling a porous medium saturated with water is carried out, taking into account the density inversion at various values of the porous medium permeability. It is shown that the effect of a maximum water density during the convective motion of water in porous media in the vicinity of zero temperature should be taken into account. The presence of a maximum water density in highly permeable soils restructures the flow and reduces the convective transfer rate, which slows down the cooling of the system.

M. V. Isupov, A. Yu. Litvintsev
Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
Keywords: inductively coupled plasma, induction discharge, electric arc, plasmatron
Abstract >>
This paper describes the generation of an induction discharge of atmospheric pressure with ferromagnetic enhancement of magnetic coupling between an inductor and a discharge. It is shown that intensifying a magnetic flux that connects the inductor and the plasma makes it possible to efficiently generate an electrodeless discharge in a lowfrequency (10100 kHz) range and greatly simplifies the use of the inductively coupled plasma. This study also presents new experimental data on the dependence of the electric field strength and the thermal efficiency of a lowfrequency induction discharge of atmospheric pressure on the discharge current strength and the flow rate of a plasmaforming gas (argon or air). It is shown how the criterion for generating a lowfrequency induction discharge is related to its thermal efficiency is shown, and methods for reducing the heat losses of a lowfrequency induction discharge of atmospheric pressure are analyzed.

S. V. Khabirov
Mavlyutov Institute of Mechanics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa, Russia
Keywords: vortex gas flows, group analysis, optimal system of subalgebras, invariant solutions, simple waves
Abstract >>
The ideal gas submodel invariant with respect to time translation and space translation along one direction has four integrals in the case of vortex motion. For the stream function and specific volume, a system of nonlinear differential equations of the third order with one arbitrary element containing the equation of state and arbitrary functions of integrals. Equivalence transformations for an arbitrary element were found. The problem of group classification was solved. The optimal system of dissimilar subalgebras for the Lei algebras from group classification was obtained. Examples of invariant solutions describing vortex gas flow with variable entropy, including a point source or stock are considered. Analogs of simple waves were obtained for some twodimensional subalgebras.

J.B. Carrat^{1,2}, N. D. Shmakova^{1}, A. V. Cherdantsev^{1,3}, N. V. Gavrilov^{1}, E. V. Ermanyuk^{1}
^{1}Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia ^{2}Novosibirsk State University, Novosibirsk, Russia ^{3}Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
Keywords: water entry, inclined impact, water trapping, capillary waves, synthetic schlieren method
Abstract >>
An inclined impact of a spherical segment with a curvature radius of 106.5 mm onto a water layer 20 mm thick is studied up to the instant of the bodyliquid contact. The range of the impact angle with respect to the horizontal plane is 9015oC, and the vertical component of velocity for all angles is 10 or 20 mm/s. The measurements are performed by a synthetic schlieren method. The marker displacements are measured by a PTV algorithm. Distributions of the liquid layer thickness in space and time along two horizontal axes are obtained. It is shown that a change in the impact angle does not alter the dynamics of crater deepening and expansion at the initial stage; the crater remains axisymmetric. It is found that the rear slope of the crater becomes steeper in the case with the maximum deviation of the angle from the vertical line, presumably, because of the local decrease in pressure. The characteristics of capillary waves generated by the impact are independent of the impact angle.

O. I. Gusev^{1,2}, G. S. Khakimzyanov^{1,2}, L. B. Chubarov^{1,2}, D. Dutykh^{3,4}
^{1}Federal Research Center for Information and Computing Technologies, Novosibirsk, Russia ^{2}Novosibirsk National Research State University, Novosibirsk, Russia ^{3}University of GrenobleAlpes, Grenoble, France ^{4}University of Savoy Mont Blanc, Chambery, France
Keywords: surface waves, waterland boundary, runup on shore, nonlinear dispersive shallow water equations, boundary conditions on the waterland boundary, numerical modeling
Abstract >>
This article focuses on the effect of frequency dispersion on the wave runup height and the characteristics of the surface waves reflected from the coastal slope. The calculations are performed within the framework of nonlinear dispersive and dispersionless shallow water models using the original boundary conditions on the movable waterland boundary. We consider the problem of solitary wave runup on flat coastal slopes with parameters close to the characteristics of one of the Kamchatka bays, for which the authors obtained probabilistic assessments of the tsunami hazard. The obtained results show that the maximum runup and amplitudes of the reflected waves are overestimated by the dispersionless model by 100%.

A. G. Knyazeva, E. S. Parfenova
Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, Russia
Keywords: coupled model, particle beam, wave propagation, nonlinear effects, elastic stresses, diffusion, heat conduction, relaxation of heat flux, relaxation of mass flux, chemical reaction
Abstract >>
A model of the initial stage of surface treatment of a material by a particle beam is presented. The model takes into account the mutual influence of elastic, thermal, and diffusion waves, as well as the formation of a new phase in the surface layer of the substrate. Examples of problem solution for different combinations of model parameters are shown; the dynamics of process development under the action of two consecutive pulses is illustrated. Possible partial variants of the model are given.

G. V. Alekseev^{1}, V. A. Levin^{2}, D. A. Tereshko^{1}
^{1}Institute of Applied Mathematics, FarEast Branch, Russian Academy of Sciences, Vladivostok, Russia ^{2}Institute of Automation and Control Processes, FarEast Branch, Russian Academy of Sciences, Vladivostok, Russia
Keywords: heat flux concentrators, design problems, numerical optimization
Abstract >>
Inverse problems of design of special shells that serve for concentrating heat fluxes are considered. With the use of an optimization method and geometric discretization, these problems are reduced to finitedimensional extreme problems, where the thermal conductivity coefficients of the shell sectors play the role of control parameters. The chosen set of control parameters ensures obtaining thermal concentrators, which can be simply implemented and provide high efficiency in the class of devices under consideration.

O. V. Kaptsov, D. O. Kaptsov
Institute of Computational Modeling, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, Russia
Keywords: KadomtsevPetviashvili equation, double waves, solitons
Abstract >>
A group of point transformations permitted by the threedimensional KadomtsevPetviashvili equation is calculated. An example of an invariant solution is given. Exact solutions for the equation under study in the form of double waves are found. The resulting solutions are expressed in terms of elementary functions and describe the interaction of a pair of solitons. Smooth bounded rational solutions are also constructed.

G. S. Yan'kova^{1,2}, A. A. Cherevko^{1,2}, A. K. Khe^{1,2}, O. B. Bogomyakova^{3}, A. A. Tulupov^{2,3}
^{1}Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia ^{2}Novosibirsk State University, Novosibirsk, Russia ^{3}International Tomography Center, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
Keywords: poroelasticity, finite element method, normalpressure hydrocephalus, cerebrospinal fluid, magnetic resonance imaging
Abstract >>
This paper describes the use of a multiphase model of poroelasticity meant for the brain substance and based on medical data to study the displacement of a ventricle wall in the brain and the value of pressure on it. The dependence of these values on the parameters of the model in normalpressure hydrocephalus is studied. It is shown for the brain geometry with different levels of detail that the simplified geometry of the brain allows one to accurately estimate critical pressures and displacements in the case of more complex geometry.

T. V. Markelova^{1,2,3}, M. S. Arendarenko^{1,2}, E. A. Isaenko^{1,2}, O. P. Stoyanovskaya^{1,2}
^{1}Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia ^{2}Novosibirsk National Research State University, Novosibirsk, Russia ^{3}Boreskov Institute of Catalysis, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
Keywords: twophase polydisperse medium, hyperbolic sound waves, dispersion ratio, CFD test
Abstract >>
The problem of the propagation of plane sound waves of small amplitudes in a mixture of an isothermal carrier gas and solid particles of various sizes is formulated on the basis of a multiliquid macroscopic model of the medium. In the model, the dispersed phase is considered as N fractions of monodisperse particles, and the dynamics of each fraction is described using the equations of a continuous medium in which does internal pressure is absent. The fractions exchange momenta with the carrier gas, but not with each other. The whole mixture is acted upon by the total pressure determined by the motion of gas molecules, and the dust particles are considered buoyant. An analytical solution of the problem is obtained using the Fourier method and analysis of variance. In the general case for an arbitrary value of the relaxation time, the solution is found numerically using the developed and published code. In special cases (infinitely small time of velocity relaxation or relaxation equilibrium and infinitely long time of velocity relaxation or frozen equilibrium), the effective velocity of sound in the gasdust medium is determined and used to obtain simple analytical representations of the solution of the problem.

A. L. Kazakov^{1}, L. F. Spevak^{2}
^{1}Matrosov Institute for System Dynamics and Control Theory Siberian Branch, Russian Academy of Sciences, Irkutsk, Russia ^{2}Institute of Mechanical Engineering, Ural Branch, Russian Academy of Sciences, Ekaterinburg, Russia
Keywords: reactiondiffusion system, diffusion wave, exact solution, radial basis functions
Abstract >>
The problem of constructing solutions to a system of two coupled nonlinear parabolic equations of the reactiondiffusion type is considered. Solutions in the form of diffusion waves propagating over zero background with a finite speed are investigated. The theorem on the construction of exact solutions by reduction to the Cauchy problem for a system of ordinary differential equations is proved. A timestep numerical technique for solving the reactiondiffusion system using radial basis function expansion is proposed. The same approach is used to solve systems of ordinary differential equations that determine the exact solutions of the reactiondiffusion system. Numerical analysis and estimation of the accuracy of solutions to a system of ordinary differential equations are carried out. These solutions are applied to verify the obtained timestep solutions of the original system.

Yu. L. Trakhinin
Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
Keywords: magnetohydrodynamics, free boundary problem, surface tension, local theorem of existence and uniqueness
Abstract >>
Data are presented on timelocal solvability of problems with free boundaries for a system of equations of magnetohydrodynamics of an ideal compressible fluid. A problem with a free plasmavacuum boundary and a problem with boundary conditions on a contact discontinuity are considered. A scheme is given for proving the local existence and uniqueness of smooth solutions of these problems with and without account for surface tension.

Y. Amirat^{1}, V. V. Shelukhin^{2,3}
^{1}University of Auvergne, ClermontFerrand, France ^{2}Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia ^{3}Altai State University, Barnaul, Russia
Keywords: NavierStokes equations, CahnHilliard equations, filtration of miscible fluids, twoscale homogenization
Abstract >>
Equations of filtration of a twocomponent miscible fluid in a porous medium are derived with the use of the method of twoscale homogenization of the system of the NavierStokes and CahnHilliard equations. The case of strong miscibility is considered.

D. S. Denisenko
Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
Keywords: stratified flows, trapped waves
Abstract >>
A steady problem of trapped solitary waves in supercritical flows of a stratified fluid above an uneven bottom is considered. For gently sloping lowamplitude obstacles, a family of approximate twoparameter solutions is constructed, which correspond to internal solitary waves in the limit of a zero height of the obstacle. It is numerically demonstrated that the number of approximate solutions significantly depends on the bottom shape.

