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Journal of Applied Mechanics and Technical Physics

2018 year, number 2

Exact Solutions of the Boltzmann Equations with a Source

Yu. N. Grigor'ev1,2, S. V. Meleshko3, A. Suriyawichitseranee3
1Institute of Computational Technologies, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
2Novosibirsk State University, Novosibirsk, 630090, Russia
3School of Mathematics, Nakhon Ratchasima, 30000, Thailand
Keywords: уравнение Больцмана, изотропная функция распределения, функция источника, инвариантные решения, Boltzmann equation, isotropic distribution function, source function, invariant solutions

Abstract >>
Exact solutions of a nonlinear Boltzmann kinetic equation with a source are constructed in the case of an isotropic distribution function and Maxwell model of isotropic scattering. These solutions are constructed with the use of an equivalence group such that one of its transformations uniquely identifies the class of the source functions that are linear in terms of the distribution function; moreover, the transformed equation has a zero right side. As a result, one can explicitly find invariant solutions of the type of the Bobylev-Krook-Wu solutions, in particular, those that admit physical interpretation.

The Model of Electric Connection of a Low-Conducting Liquid in High-Frequency Electric Field

V. A. Il'in, L. A. Ponomarev
Keywords: электроконвекция, слабопроводящая жидкость, переход в хаотическое состояние, electroconvection, low-conductivity fluid, transition to chaotic state

Abstract >>
This paper describes the study of electroconvection of low-conductivity fluid in a capacitor with boundary conditions of adhesion on a solid surface in the case of rapid charge relaxation. The linear stability of fluid equilibrium in a constant electric field is investigated. The model describing the averaged fluid flow in a high-frequency electric field. The nonlinear modes of electroconvection are described. A mode map is constructed. It is established that, depending on the frequency of the external field, the transition to chaotic state occurs through quasiperiodic or intermittent structures.

On the Nonlinear Dynamics of Turbulent Thermals in Shear Flow

L. H. Ingel'1,2
1Taifun Scientific and Production Association, Obninsk, 249038, Russia
2Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, 119017, Russia
Keywords: конвекция, термики, турбулентность, интегральные модели, сдвиговые течения, нелинейность, аналитические решения, convection, thermals, turbulence, integral models, shear flows, nonlinearity, analytical solutions

Abstract >>
The nonlinear integral model of a turbulent thermal is extended to the case of the presence of a horizontal component of its motion with respect to the medium (for example, the rise of a thermal in shear flow). In contrast to traditional models, the possibility of heat release in the thermal is taken into account. For a piecewise constant vertical horizontal velocity profile medium and a constant vertical velocity shear, analytical solutions are obtained which describe different dynamic modes of thermals. The nonlinear interaction between the horizontal and vertical components of thermal motion because each of them influences the rate of entrainment of the ambient medium, i.e., the rate of increase in the size of the thermal and, hence, its mobility. It is shown that the enhancement of the entrainment of the medium due to the interaction of the thermal with the cross flow can lead to a significant decrease in the mobility of the thermal.

Evolution of Two Vortices Near a Solid Surface

D. A. Gadzhiev, A. M. Gaifullin
Central Aerohydrodynamic Institute named after N. E. Zhukovskii, Zhukovskii, 140180, Russia
Keywords: вихрь, отрыв пограничного слоя, диффузия вихря, "подскок" вихря, vortex, boundary layer separation, vortex diffusion, vortex «op up»

Abstract >>
A numerical solution is found for the problem of evolution of two oppositely rotating vortices in a viscous incompressible fluid near a solid surface. The mechanism of vortex dissipation is determined. The trajectory of vortex motion is constructed.

Lift Force Acting on a Heavy Solid in a Liquid-Filled Rotating Cavity with a Time-Varying Rotation Rate

O. A. Vlasova1, V. G. Kozlov1, N. V. Kozlov2
1Perm State Humanitarian Pedagogical University, Perm, 614990, Russia
2Institute of Continuous Media Mechanics, Ural Branch of Russian Academy of Science, Perm, 614013, Russia
Keywords: вращение, модуляция скорости, тяжелое тело, вязкая жидкость, подъемная сила, rotation, velocity modulation, heavy body, viscous fluid, lift force

Abstract >>
The dynamics of a heavy cylindrical body in a liquid-filled horizontal cylindrical cavity with a time-varying rotation rate is experimentally investigated. The body is near the cavity boundary under a centrifugal force and undergoes solid-body rotation together with the liquid and the cavity at a fixed rotation rate. The dependence of the body dynamics on the amplitude and frequency of the rotation rate modulation is investigated. It is found that at a critical amplitude of modulation (at definite frequency), the heavy body repulses from the cavity boundary and comes into a steady state at some distance from the wall. It is found that the average lift force (repulsive one) is generated by the azimuthal oscillation of the body in the rotating frame of reference and manifests itself at a distance comparable to the thickness of the viscous boundary layer. In the experiments, we observed an azimuthal drift of the body due to asymmetric azimuthal oscillations of the body. In the limit of high frequency of the rotation rate modulation, the dependence of the lift force coefficient on the gap between the body and the wall is determined.

Natural Aeroacoustic Oscillations of the Gas Near Two Plates in a Channel

A. P. Konstantinov
Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
Keywords: собственные колебания в неограниченных областях, акустический резонанс, тонкостенные препятствия в каналах и трубах, natural oscillations in unbounded regions, acoustic resonance, thin-walled obstacles in channels and tubes

Abstract >>
Natural aeroacoustic oscillations of the gas near two thin plates arranged in a tandem manner in a rectangular channel are studied in a two-dimensional formulation. A bifurcation of natural frequencies depending on the distance between the plates is detected, and the frequency of natural oscillations is found as a function of the plate length and the distance from the channel walls. The fields of pressure and gas velocities in the examined range of oscillations are constructed.

Vortex Scattering of Monatomic Gas Along Plane Curves

R. F. Shayakhmetova
Institute of Mechanics of Ufa Federal Research Center, Russian Academy of Sciences, Ufa, 450054, Russia
Keywords: газовая динамика, инвариантное решение, проективный оператор, уравнение Абеля, gas dynamics, invariant solution, projective operator, Abel equation

Abstract >>
An invariant submodel of gas dynamics equations constructed on a three-dimensional subalgebra with a projective operator for the case of monatomic gas is under consideration. The submodel is reduced to an Abel equation, with integral curves constructed for it. For a separatrix of a saddle, an approximate solution is studied. Such solutions describe the vortex scattering of gas along plane curves placed on the surface of revolution.

Calculation and Analysis of the Power Supply of a Railgun from the Magnetic Dynamic Generator and the Transformer

G. V. Nosov1, M. G. Nosova2, Yu. Z. Vasil'eva1, E. O. Kuleshova1
1National Research Tomsk Polytechnic University, Tomsk, 634050, Russia
2Tomsk University of Control Systems and Radioelectronics, Tomsk, 634050, Russia
Keywords: магнитогидродинамический генератор, индуктивный накопитель, трансформатор, сверхпроводящая обмотка, рельсотрон, magnetohydrodynamic generator, inductive storage, transformer, superconducting winding, railgun

Abstract >>
The power supply of an electromagnetic accelerator of solids (railgun) from a source of electromagnetic energy with a magnetohydrodynamic generator and a transformer with superconducting windings is calculated and analyzed. The laws and equations for electrical circuits are used, which are solved analytically and numerically. It is shown that a transformer with superconducting windings can be used to accumulate electromagnetic energy from a magnetohydrodynamic generator and to power a railgun in burst mode operation.

Behavior of a Semi-Infinite Ice Cover under a Uniformly Moving Load

L. A. Tkacheva
Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
Keywords: поверхностные волны, изгибно-гравитационные волны, плавающая упругая пластина, дисперсионные соотношения, движущаяся нагрузка, преобразование Фурье, метод Винера-Хопфа, surface waves, flexural-gravity waves, floating elastic plate, dispersion relations, moving load, Fourier transform, Wiener-Hopf technique

Abstract >>
This paper considered the behavior of a semi-infinite ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a load moving at a constant velocity along the edge of the cover at some distance from it. The ice cover is simulated by a thin elastic plate of constant thickness. In a moving coordinate system, the deflection of the plate is assumed to be steady. An analytic solution of the problem is constructed using the Wiener-Hopf technique. The wave forces, the deflection of the plate, and the elevation of the free surface of the fluid at different speeds of the load are considered.

Steady Fluid Flow to a Radial System of Horizontal Wells

P. E. Morozov
Institute of Mechanics and Engineering of the Kazan Scientific Center, Russian Academy of Sciences, Kazan, 420111, Russia
Keywords: полуаналитическое решение, лучевая система, горизонтальные скважины, анизотропный пласт, профиль притока, продуктивность, semi-analytical solution, radial system, horizontal wells, anisotropic reservoir, inflow profile, productivity

Abstract >>
A semi-analytical method for determining the productivity of a radial system of horizontal wells in an anisotropic layer is proposed. Results of calculations of the productivity and distribution of fluid flow along the length of the wellbores of the radial system of horizontal wells using the proposed method are compared with the data of experimental studies based on electrolytic modeling and engineering calculations. The effects of the number of wellbores, their location in the reservoir, and the hydraulic pressure loss on the distribution of the fluid flow along the length of horizontal wellbores are investigated.

Investigations of Heat and Mass Transfer for Thermal Protection Materials in a Long Flight

V. I. Zinchenko, V. D. Goldin, V. G. Zverev
Institute of Applied Mathematics and Mechanics, Tomsk, 634050, Russia
Keywords: гиперзвуковая скорость, аэродинамический нагрев, тепловая защита, сопряженный тепломассообмен, теплозащитные материалы, hypersonic velocity, aerodynamic heating, thermal protection, coupled heat and mass transfer, thermal protection materials

Abstract >>
The problem of unsteady coupled heat and mass transfer in the course of motion of a spherically blunted conical body fabricated with the use of thermal protection materials is considered. Numerical integration is applied to study the characteristics of heat and mass transfer at constant stagnation parameters (Mach number 6, altitude 15 km, and flight time 600 s), which impose severe constraints on the choice of materials for thermal protection. It is demonstrated that the use of advanced ceramic materials ensures an admissible temperature regime and maintaining the initial geometry of the body, including its motion at an angle of attack.

Natural Convection Heat Transfer in 2D and 3D Trapezoidal Enclosures Filled with Nanofluid

P. Akbarzadeh, A. H. Fardi
Faculty of Mechanical and Mechatronics Engineering, Shahrood, Iran
Keywords: естественная конвекция, наножидкости, наклонные трехмерные замкнутые сосуды, поперечные сечения трапециевидной формы, численное решение, natural convection, nanofluids, inclined 3D trapezoidal enclosure, numerical solution

Abstract >>
The purpose of the present study is to investigate the heat transfer performance due to free convection of nanofluids with variable properties inside 2D and 3D channels with trapezoidal cross sections. The governing equations are solved numerically using the finite volume method and the SIMPLER algorithm. In this study, the effect of the nanoparticle volume fraction, Rayleigh number, side wall angles of the trapezoidal section, and axial slope of the 3D channel are examined. The presented results include the average Nusselt number, flow circulation streamlines, and isothermal contours. The heat transfer rate (i.e., Nusselt number) is seen to increase in both 2D and 3D channels with an increase in the Rayleigh number. In 2D trapezoidal enclosures, the Nusselt number decreases with an increase in the nanoparticle volume fraction from zero to 2% and increases if the nanoparticle volume fraction is greater than 2%. In 3D channels, an increase in the axial slope of the channel leads to an increase in the Nusselt number.

Effect of Pulse Shape on Spall Strength

V. I. Smirnov1, Yu. V. Petrov2
1Emperor Alexander I St. Petersburg State Transport University, St. Petersburg, 198103, Russia
2Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, 199178, Russia
Keywords: откол, инкубационное время, структурно-временной критерий, пороговый силовой импульс, импульсная емкость разрушения, spall, incubation time, structure-time criterion, threshold power pulse, pulse fracture capacity

Abstract >>
This paper analyzes the effect of the time-dependent shape of a load pulse on the spall strength of materials. Within the framework of a classical one-dimensional scheme, triangular pulses with sections of signal increment and attenuation and without sections of signal increment are considered. Calculation results for the threshold characteristics of fracture for rail steel are given. The possibility of optimization of fracture by selecting a loading time with the use of an introduced characteristic of dynamic strength (pulse fracture capacity) is demonstrated. The study is carried out using a structure-time fracture criterion

Energy Condition of Plasticity

A. S. Gulgazli
Azerbaijan State University of Oil and Industry, Baku, AZ1010, Azerbaijan
Keywords: пространство напряжений, деформационная анизотрoпия, повторное нагружение, потенциальная энергия, девиатор напряжений, инварианты тензора, stress space, strain anisotropy, repeated loading, potential energy, stress deviator, tensor invariants

Abstract >>
A strength condition is obtained that takes into account the deformation anisotropy and the effect of uniform pressure on the mechanical properties of materials.

One-Dimensional Interaction of a Cylindrical Unloading wave with a Moving Elastic Plastic Boundary

E. A. Gerasimenko1, L. V. Kovtanyuk1, A. A. Burenin2,3
1Institute of Automation and Control Processes, Far East Branch of the Russian Academy of Sciences, Vladivostok, 69004, Russia
2Institute of Engineering Science and Metallurgy, Far East Branch of the Russian Academy of Sciences, Komsomolsk-on-Amur, 681005, Russia
3Komsomolsk-on-Amur State Technical University, Komsomolsk-on-Amur, 681013, Russia
Keywords: большие деформации, упругость, вязкость, пластичность, ударная волна, волна разгрузки, large deformations, elasticity, viscosity, plasticity, shock wave, unloading wave

Abstract >>
The one-dimensional dynamic problem of the interaction of the unloading wave with the elastic-plastic boundary is considered within the theory of large elastic-plastic deformations. It is shown that before the formation of the unloading wave, the increasing pressure gradient leads to quasistatic deformation of the elastic-viscoplastic material filling the round tube, which is retained in the tube due to friction about its wall, resulting in the formation of near-wall viscoplastic flow and an elastic core. The unloading wave is initiated at the moment when the material begins to slip on the inner wall of the tube. Calculations were conducted using the ray method of constructing approximate solutions behind strong discontinuity surfaces, and ray expansions of the solutions behind the cylindrical surfaces of discontinuities were obtained.

Increasing the Fatigue Life of Dissimilar Friction Stir Spot Welded AL/CU Joints by Optimization of Technological Parameters

S. Siddharth, T. Senthilkumar
University College of Engineering, Tiruchirappalli, 620024, India
Keywords: точечная сварка трением с перемешиванием, алюминий, медь, усталость, оптимизация, fiction stir spot welding, aluminium, copper, fatigue, optimization

Abstract >>
Friction stir spot welding is a new technique used in industries for spot joining dissimilar combinations. In this investigation, dissimilar combinations of Al5052 aluminium and C10100 copper are joined by using this technique with variations of important process parameters, such as the tool rotational speed, dwell duration, and plunging depth. A central composite design model is developed for establishing empirical relationships between the process parameters and the fatigue life of the joints (number of cycles to fracture). The analysis of variance is used for determining the significance of the developed model. The response surface methodology is used for maximizing the fatigue strength. By confirmation experiments, the model is validated, and the error is found to be within four percent.

Elastoplastic Model of Rocks with a Linear Structural Parameter

A. F. Revuzhenko, O. A. Mikenina
Mining Institute, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
Keywords: горная порода, упругость, пластичность, самоуравновешенные напряжения, структурный параметр, rocks, elasticity, plasticity, self-balanced stresses, structural parameter

Abstract >>
A closed mathematical model is formulated, which takes into account elastoplastic deformations and the medium capability of accumulating the energy of internal self-balanced stresses. Satisfaction of the diffeomorphism postulate (assumption of displacement field smoothness) is not required; as a result, the strains depend on the stresses and second derivatives of the stresses with respect to the coordinates. The model involves a linear structural parameter. Relations that take into account local bending of the elementary volumes of the medium are derived.

Solution of a Plane Hydrofracture Problem with Stress Contrast

I. O. Gladkov1, A. M. Linkov1,2
1Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, 199178, Russia
2Rzeszow University of Technology, Rzeszow, 35959, Poland
Keywords: гидроразрыв, плоская задача, контраст напряжений, hydraulic fracture, plane problem, stress contrast

Abstract >>
A plane hydrofracture problem for the Khristianovich-Geertsma-de Klerk model is extended and solved in the case where a confining stress closing a fracture is not constant in the direction of its propagation. A method is developed for solving the problem with an arbitrary stress contrast. It is stated that the transition through a contact with positive (negative) contrast occurs with fracture arresting (acceleration), whose intensity is controlled by a dimensionless parameter derived from theoretical considerations and numerical results

Nonlinear Deformation and Stability of a Noncircular Cylindrical Shell under Combined Loading with Bending and Twisting Moments

V. K. Belov, L. P. Zheleznov, T. S. Ognyanova
Siberian Aeronautic Research Institute (SibNIA) named after S. A. Chaplygin, Novosibirsk, 630000, Russia
Keywords: некруговые цилиндрические композитные оболочки, дискретное подкрепление, нелинейное деформирование, устойчивость, метод конечных элементов, noncircular cylindrical composite shells, discrete reinforcement, nonlinear deformation, stability, finite element method

Abstract >>
A previously developed technique is used to solve problems of strength and stability of discretely reinforced noncircular cylindrical shells made of a composite material with allowance for the moments and nonlinearity of their subcritical stress-strain state. Stability of a reinforced bay of the aircraft fuselage made of a composite material under combined loading with bending and twisting moments is studied. The effects of straining nonlinearity, stiffness of longitudinal ribs, and shell thickness on the critical loads that induce shell buckling are analyzed.

Modeling of the Establishment and Stability of Gel Barriers in Main Cracks

A. M. Il'yasov, G. T. Bulgakova
Ufa State Aviation Technical University, Ufa, 450008, Russia
Keywords: магистральная трещина, "сшитый" гель, вязкопластичная жидкость, устойчивость гелевого барьера, main crack, cross-linked gel, viscoplastic liquid, gel barrier stability

Abstract >>
A quasi-one-dimensional nonstationary model for the waterproofing of main cracks using Herschel-Bulkley viscoplastic fluid is presented. The mathematical model developed in a one-dimensional isothermal approximation with the hydrodynamic parameters (pressure and velocity) averaged over the cross section of the crack can be used to determine the optimal technological parameters of the waterproofing process and the size of gel barriers to ensure their stability on exposure to an intense filtration flow after crack waterproofing. The ranges of flow rates which provide waterproofing of a crack of fixed width for the selected gel are determined. The stability of the established gel barriers is estimated.

Dynamic Problem for an Elastic Space with a Moving Semi-Infinite Crack

A. I. Zhornik, V. A. Kirichek
Chekhov Taganrog Institute, Taganrog, 347926, Russia
Keywords: динамическое разрушение, напряжения, перемещения, смешанные граничные условия, факторизация, коэффициент интенсивности напряжений, dynamic fracture, stress, displacement, mixed boundary conditions, factorization, stress intensity factor stresses

Abstract >>
This paper considers the dynamic problem of a semi-infinite crack in an elastic space that suddenly begins to grow with a constant speed. At the initial time, the crack faces at some distance from its tip are subjected to normal tensile concentrated forces which then move along the crack faces at a speed different from the speed of the crack tip. The stress intensity factor is calculated. Various special cases are examined.

Fracture Parameter Minimization of a Circular Disk with Mixed Conditions on its Boundary

V. M. Mirsalimov1,2
1Azerbaijan Technical University, Baku, AZ1073, Azerbaijan
2Institute of Mathematics and Mechanics, National Academy of Sciences, Azerbaijan, Baku, AZ1141, Azerbaijan
Keywords: круговой диск, смешанные граничные условия, трещины, функция перемещений точек границы диска, минимизация параметров разрушения, circular disk, mixed boundary conditions, cracks, displacement function of disk boundary points, fracture parameter minimization

Abstract >>
Minimax criterion is used to carry out the theoretical analysis of normal displacement of points at the boundary of a circular disk weakened by arbitrarily placed rectilinear cracks. This paper presents a criterion and method for solving the problem of fracture of the circular disk with mixed conditions on its boundary. A closed system of algebraic equations is constructed, which allows for minimization of stress intensity factors. The normal displacement of points at the boundary of the circular disk, for which the bearing capacity of the disk increases.