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Journal of Applied Mechanics and Technical Physics

2016 year, number 5

Nonclassical Models of the Theory of Plates and Shells

B. D. Annin1, Yu. M. Volchkov1,2
1Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
2Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: уравнения теории оболочек, слоистые и композитные оболочки, контактные задачи, полиномы Лежандра, equations of the theory of shells, layered and composite shells, contact problems, Legendre polynomials

Abstract >>
Publications dealing with the study of methods of reducing a three-dimensional problem of the elasticity theory to a two-dimensional problem of the theory of plates and shells are reviewed. Two approaches are considered: the use of kinematic and force hypotheses and expansion of solutions of the three-dimensional elasticity theory in terms of the full system of functions. Papers where a three-dimensional problem is reduced to a two-dimensional problem with the use of several approximations of each sought function (stresses and displacements) by segments of the Legendre polynomials are also reviewed.

DOI: 10.1134/S0021894416050011

Bonds between Metals and Nanocomposites Occurring in Explosion Welding

M. P. Bondar1, E. V. Karpov1,2, Ya. L. Lukyanov1
1Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
2Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: сварка взрывом, нанокомпозит, мезоструктурный материал, деформируемость, скорость деформации, изгиб, прочность сварного соединения, explosion welding, the nanocomposite, mesostructural material, deformability, strain rate, bending, welded strength

Abstract >>
This paper describes the study of the influence of a microstructure characterized by directed or chaotic distribution of nanoinclusions and strain rate on the deformability of a nanocomposite. It is revealed that, under identical load conditions, crack formation occurs in nanocomposites whose structural elements are mostly directed in the same way at lower strain rates than in nanocomposites with chaotic distribution of the hardener. It is shown that, as the strain rate increases, the influence of the structural order on the limit deformation reduces due to transition from shear strain to rotational strain. No cracks are formed in bonding metals and nanocomposites by explosion welding. The experimental results obtained in the study of transverse bending of two-layer welded beams and the structure in the vicinity of the weld reveal that the obtained metal - nanocomposite compound has a uniform structure retained in deformation, with fracture occurring in the nanocomposite.

DOI: 10.1134/S0021894416050023

On the Carrying Capacity of Structural Elements

V. I. Van'ko
Bauman Moscow State Technical University, Moscow, 105005 Russia
Keywords: несущая способность, стержни и оболочки, критическая нагрузка, критическое время выпучивания, carrying capacity, bars and shells, critical load, critical buckling time

Abstract >>
Different approaches to the determination of the carrying capacity of bars and shells under elastoplastic deformation and creep deformation conditions.

DOI: 10.1134/S0021894416050035

Experimental Study of the Inelastic Behavior of Materials

R. A. Vasin
Research Institute of Mechanics, Moscow, 119192 Russia
Keywords: пластичность, вязкость, сверхпластичность, определяющие соотношения, теория эксперимента, plasticity, viscosity, superplasticity, constitutive relations, theory of experiment

Abstract >>
This paper presents the results of an experimental study of the elastic and viscoplastic properties of alloys using elastoplastic theory. The need to study complex loading of materials, consider the microstructure evolution during deformation, and develop an appropriate theory of experiment and adequate constitutive relations is highlighted.

DOI: 10.1134/S0021894416050047

Influence of the Cross-Sectional Shape of Tensile Bars on Their Creep Rupture Strength in an Aggressive Environment

A. M. Lokoshchenko, L. V. Fomin
Institute of Mechanics, Moscow, 119192 Russia
Keywords: стержни, длительная прочность, время до разрушения, поврежденность, агрессивная окружающая среда, диффузионный фронт, bars, creep rupture strength, time to rupture, damage, aggressive environment, diffusion front

Abstract >>
The creep rupture strength of tensile bars in an aggressive environment is studied analysis. The time to rupture of the bars is analyzed as function of the shape of their cross section. It is shown that the influence of the aggressive environment on the creep rupture strength of the bars is determined by the diffusion of its elements into the bar material, resulting in a decrease of its creep rupture strength. The diffusion of the aggressive environment into the bar is studied using an approximate method of solving the diffusion equation taking into account the motion of the diffusion front. Bars with different cross-sectional shapes are considered using Rabotnov kinetic theory. It is shown that under the same tensile stress, the bar with a circular cross-section has minimum time to failure.

DOI: 10.1134/S0021894416050059

Version of the Linear Elasticity Theory with a Structural Parameter

A. F. Revuzhenko
Chinakal Institute of Mining, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630091 Russia
Keywords: теория упругости, структура, гладкость поля смещений, дисперсия, elasticity theory, texture, smoothness of the displacement field, dispersion

Abstract >>
It is shown that, in the planar case, the system of constitutive equations of the linear elasticity theory should contain five independent equations. n the classical theory, only three equations are formulated, while the other two are contained in implicit form in the postulate of diffeomorphism, which the assumption of smoothness of the displacement field. A closed elasticity model is constructed without the assumption of diffeomorphism, and it contains a structural parameter having a dimension of length. It is shown that, in a static version, macrodeformations depend on stresses and the second derivatives of stresses by the coordinates, while there is dispersion of longitudinal and transverse waves in the dynamics.

DOI: 10.1134/S0021894416050060

Analyzing the Deformation of a Porous Medium with Account for the Collapse of Pores

V. M. Sadovskii, O. V. Sadovskaya
Institute of Computational Modelling, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, 660036 Russia
Keywords: пористая среда, схлопывание пор, упругий предвестник, пластическая ударная волна, метод сквозного счета, параллельный вычислительный алгоритм, porous medium, pore collapse, elastic precursor, plastic shock wave, shock-capturing method, parallel computational algorithm

Abstract >>
The generalized rheological method is used to construct a mathematical model of small deformations of a porous media with open pores. Changes in the resistance of the material to external mechanical impact at the moment of collapse of the pores is described using the Mises~--- Schleicher strength condition. The irreversible deformation is accounted for with the help of the classic versions of the von Mises - Tresca - Saint-Venant yield condition and the condition that simulates the plastic loss of stability of the porous skeleton. Within the framework of the constructed model, this paper describes the analysis of the propagation of plane longitudinal compression waves in a homogeneous medium accompanied with plastic strain of the skeleton and densification of the material. A parallel computational algorithm is developed for the study of the elastoplastic deformation of the porous medium under external dynamics loads. The algorithm and the program are tested by calculating the propagation of plane longitudinal shock waves of compression and cylindrical cavity of in an infinite porous medium. The calculation results are compared with exact solutions, and it is shown that they are in good agreement.

DOI: 10.1134/S0021894416050072

Optimum Shapes of Axisymmetric Bodies Penetrating into Soil

V. L. Kotov, E. Yu. Linnik, A. A. Tarasova
Research Institute of Mechanics, Nizhny Novgorod, 603950 Russia
Keywords: грунт, тело вращения, оптимизация, минимальное сопротивление внедрению, максимальная глубина проникания, модель локального взаимодействия, метод локальных вариаций, абсолютно оптимальное тело, soil, body or revolution, optimization, minimum penetration resistance, maximum penetration depth, local interaction model, method of local variations, absolutely optimum body

Abstract >>
This paper presents the results of a study of the shapes of axisymmetric bodies with minimum penetration resistance and maximum depth of penetration into the plastic soils. Optimum shapes of bodies of revolution of predetermined length and cross-sectional radius with generatrices represented by line segments are obtained by a modified method of local variations. The problem is solved using a binomial quadratic model of local interaction, including inertial and strength terms containing constant and Coulomb frictions. The resistance forces and the depth of penetration of cones and the obtained bodies of optimal shape are determined at different penetration velocities.

DOI: 10.1134/S0021894416050084

Green's Function for a Prestressed Thermoelasticity Half-Space with an Inhomogeneous Coating

T. I. Belyankova, V. V. Kalinchuk
Southern Scientific Center, Russian Academy of Sciences, Rostov-on-Don, 344006 Russia
Keywords: термоупругость, функционально-градиентный материал, предварительно напряженная термоупругая среда с покрытием, начальные напряжения, предварительный нагрев, трехмерная функция Грина, thermoelasticity, functionally graded material, prestressed thermoelastic medium with a coating, initial stresses, preheating, three-dimensional Green's function

Abstract >>
A mathematical model is developed for an inhomogeneous thermoelastic prestressed half-space consisting of a stack of homogeneous functionally graded layers rigidly attached to a homogeneous base.аEach component of the inhomogeneous medium under an initial mechanical stress and temperature.аConsistent linearization of the constitutive relations of nonlinear mechanics of a thermoelastic medium is performed using the theory of superpositionа of small deformations on finite deformations with the inhomogeneity of the medium taken into account.аIntegral formulas for studying dynamic processes in inhomogeneous prestressed thermoelastic media are derived.

DOI: 10.1134/S0021894416050096

Free Vibrations of Circular Cylindrical Shells with a Small Added Concentrated Mass

G. S. Leizerovich, S. V. Seregin
Komsomolsk-on-Amur State Technical University, Komsomolsk-on-Amur, 681013 Russia
Keywords: круговая цилиндрическая оболочка, присоединенная масса, изгибно-радиальные колебания, расщепление изгибного частотного спектра, circular cylindrical shell, added mass, bending-radial vibrations, splitting of bending frequency spectrum

Abstract >>
The effect of a small added mass on the frequency and shape of free vibrations of a thin shell are studied using shallow shell theory. The proposed mathematical model assumes that mass asymmetry even in a linear formulation leads to coupled radial bending vibrations. The interaction of shape-forming waves is studied using modal equations obtained by the Bubnov-Galerkin method. Splitting of the bending frequency spectrum is found which is caused not only by the added mass, but also shell wave-formation parameters. The ranges of the relative lengths and shell thicknesses are determined in which the interaction of bending and radial vibrations can be ignored.

DOI: 10.1134/S0021894416050102

Viscoelastic Problem for Piecewise Homogeneous Piezoelectric Plates

S. A. Kaloerov, A. A. Samodurov
Donetsk National University, Donetsk, 83001 Ukraine
Keywords: вязкоупругость, упругое включение, пьезопластина, комплексные потенциалы, пьезоэффективность материала, viscoelasticity, elastic inclusion, piezoelectric plate, complex potentials, piezoelectric performance

Abstract >>
An electromagnetic viscoelastic problem is solved for piecewise-homogeneous plates. The problem is reduced to solving a sequence of problems of electromagnetoelasticity with complex potentials. General representations of approximation functions for multiply connected domains and the boundary conditions for their determination are given. An analytical solution of the problem for a plate with one inclusion and an approximate solution to the plate with a finite number of inclusions are obtained. The change in the electromagnetoelastic state is investigated numerically as a function of time, the properties of the plate and inclusion materials, and the distance between the inclusions.

DOI: 10.1134/S0021894416050114

Influence of Hydrogen on the Strength and Ductility of 30KhGSA and EI659 Steels in Quasi-Static and Shock-Wave Tests

V. A. Ogorodnikov, A. A. Yukhimchuk, M. A. Mochalov, A. V. Andramanov, A. Yu. Baurin, A. O. Blikov, I. E. Boitso, S. V. Erunov, I. P. Maksimkin, I. L. Malkov, A. S. Pupkov, E. V. Shevnin
All-Russian Research Institute of Experimental Physics, Sarov, 607188 Russia
Keywords: прочность, пластичность, сталь, водород, квазистатические и ударно-волновые испытания, strength, ductility, steel, hydrogen, quasi-static and shock-wave tests

Abstract >>
The influence of hydrogen on the strength and ductility of 30KhGSA and EI659 steels in quasi-static and shock-wave tests was studied experimentally. The sensitivity of the steels to hydrogenation in air under normal pressure or in hydrogen at a pressure of 25 MPa was determined from the results of tests of samples in the initial state and samples pre-hydrogenated at a temperature of 773 K and a hydrogen pressure of 50 MPa for 24 h. It is found that EI659 steel doped with tungsten and vanadium is less sensitive to to hydrogen than 30KhGSA steel doped with manganese and chromium.

DOI: 10.1134/S0021894416050126

Explosive Loading of Deformable Gas Permeable Axisymmetric Structural Elements

E. G. Glazova, A. Yu. Konstantinov, A. V. Kochetkov, S. V. Krylov
Research Institute of Mechanics, Nizhny Novgorod, 603950 Russia
Keywords: металлические сетки, многослойные камеры, взаимопроникающие континуумы, упругопластическое деформирование, взрыв, схема Годунова, metal meshes, multilayer shells, interpenetrating continua, elastoplastic deformation, explosion, Godunov scheme

Abstract >>
A mathematical model is proposed which describes the interrelated processes of unsteady elastoplastic deformation of stacks of woven metal wire meshes and wave processes in pore gas in a two-dimensional axisymmetric approximation. The nonlinear equations of the dynamics of two interpenetrating continua are solved numerically using a modified Godunov scheme. The problem of explosive loading of a multilayer shell with an internal permeable deformable layer is solved. The results of numerical solutions are compared with experimental data. The influence of the gas-permeable layer on the shell deformation is determined.

DOI: 10.1134/S0021894416050138

Experimental Study of the Effect of a Passive Porous Coating on Disturbances in a Hypersonic Boundary Layer 2. Effect of the Porous Coating Position

S. V. Lukashevich1, S. O. Morozov1,2, A. N. Shiplyuk1,3
1Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
2Novosibirsk State University, Novosibirsk, 630090 Russia
3Novosibirsk State Technical University, Novosibirsk, 630073 Russia
Keywords: эксперимент, гиперзвуковой пограничный слой, вторая мода возмущений, пассивное пористое покрытие, experiment, hypersonic boundary layer, second mode of disturbances, passive porous coating

Abstract >>
The effect of the position of a passive porous coating on natural disturbances in a hypersonic boundary layer is studied experimentally. The experiments are performed in the flow around a sharp cone aligned at a zero angle of attack with the free-stream Mach number M = 5,8, stagnation temperature T0 = 370±5 К, and unit Reynolds numbers Re1∞ = 2,6 x 106, 4,6 x 106, 6,6 x 106, and 107 m-1. The wave characteristics of the boundary layer are calculated with the use of the linear stability theory for flow parameters corresponding to experimental values. A comparison of experimental and predicted results shows that the presence of a porous coating in the region where the second mode is unstable leads to reduction of its amplitude at the measurement point, whereas the presence of a porous coating in the region of second mode stability leads to enhancement of the amplitude.

DOI: 10.1134/S002189441605014X

Flow Structure in a Vortex Chamber

D. G. Akhmetov1, T. D. Akhmetov2,3
1Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
2Budker Institute of Nuclear Physics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
3Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: вихревая камера, измерение поля скорости, линии тока, структура течения, vortex chamber, velocity field measurement, streamlines, flow structure

Abstract >>
The structure of a swirl flow in a vortex chamber is studied. Distributions of the azimuthal and axial components of velocity are obtained almost in the entire volume of the chamber. A pattern of streamlines of this flow is constructed, and the mechanism of the emergence of a reverse flow directed toward the closed end face of the chamber is identified.

DOI: 10.1134/S0021894416050151

Finite Element Analysis of the Temperature Field in Laser Cladding of Ni-Based Powders on Teeth Surfaces of the Helical Gear

L. Chen, S. Liu, R. Tao, D. Liu, D. Lou, P. Bennett
School of Mechanical Engineering, Wuhan, Hubei, 430068 R.P. China
Keywords: лазерное плакирование, поле температур, поверхность зуба, метод конечных элементов, laser cladding, temperature field, teeth surfaces, finite element analysis

Abstract >>
The temperature field formed in the process of laser cladding of worn teeth surfaces of the gear shaft is simulated by the finite element analysis software. Isothermal lines are inside the tooth are obtained. Simulated results are compared with experimental data. Recommendations are given on improvement of the cladding technology to provide a durable continuous coating.

DOI: 10.1134/S0021894416050163

Research on the Skirt Tail Explosively Formed Projectile Stable Shaping Technology

W. Li, X. Wang, W. Li, K. Chen
Nanjing University of Science and Technology, Nanjing, 210094 China
Keywords: формируемый взрывом снаряд, устойчивость полета, хвостовая юбка, лайнер, численное моделирование, explosively formed projectile (EFP), flight stability, skirt tail, liner, numerical simulation

Abstract >>
The effect of the arc-cone liner configuration parameters on explosively formed projectile (EFP) flight stability is studied experimentally. The effect the liner edge structure on the EFP formation is computed numerically. The results show that an EFP formed by a liner whose thickness is 0.046 times the charge caliber has improved flight stability and can perforate a steel armor 0.5 times the charge caliber thick in the case of large stand-off distances. A good tail skirt EFP can be formed by choosing appropriate parameters: the liner-to-charge diameter ratio in the interval 0.96÷0.98, the liner edge thickness equal to 0.0020÷0.0025 times the charge caliber, and the liner edge chamfer of 45÷50°.

DOI: 10.1134/S0021894416050175

Computational Modeling of an MHD Flow of a Non-Newtonian Fluid Over an Unsteady Stretching Sheet with Viscous Dissipation Effects

M. N. Tufail1,2, A. S. Butt1, A. Ali1,3
1Quaid-i-Azam University 45320, Islamabad, 44000, Pakistan
2University of Management and Technology, Sialkot Campus 51310, Pakistan
3Prince Muhammad Bin Fahd University, Al Khobar, 31952, Saudi Arabia
Keywords: неньютоновская жидкость, магнитогидродинамическое течение, вязкостная диссипация, теплоперенос, non-Newtonian fluid, magnetohydrodynamic flow, viscous dissipation, heat transfer

Abstract >>
An investigation has been conducted on the MHD Casson fluid and heat transfer over an unsteady stretching sheet with viscous dissipation effects. With suitable dimensionless variables, partial differential equations are reduced to ordinary differential equations, which are then solved by the homotopy analysis method. Dependences of flow characteristics on various parameters involved into the equations are obtained.

DOI: 10.1134/S0021894416050187

MHD Boundary-Layer Flow of a Non-Newtonian Nanofluid Past a Stretching Sheet with a Heat Source/Sink

M. Madhu, N. Kishan
Osmania University, Hyderabad, Telangana, 500007 India
Keywords: неньютоновская наножидкость, растягиваемая пластина, броуновское движение, термофорез, non-Newtonian nanofluid, stretching sheet, Brownian motion, thermophoresis

Abstract >>
The goal of the present paper is to examine the magnetohydrodynamic effects on the boundary layer flow of the Jeffrey fluid model for a non-Newtonian nanofluid past a stretching sheet with considering the effects of a heat source/sink. The governing partial differential equations are reduced to a set of coupled nonlinear ordinary differential equations by using suitable similarity transformations. These equations are then solved by the variational finite element method. The profiles of the velocity, temperature, and nanoparticle volume fraction are prese graphically, and the Nusselt and Sherwood numbers are tabulated. The present results are compared with previously published works and are found to be in good agreement with them.

DOI: 10.1134/S0021894416050199

On a Magnetohydrodynamic Flow of the Casson Fluid with Partial Slip and Thermal Radiation

S.A. Shehzad1, T. Hayat2,3, A. Alsaedi3, M.A. Meraj1
1COMSATS Institute of Information Technology, Sahiwal, 57000, Pakistan
2Quaid-I-Azam University, Islamabad, 44000, Pakistan
3King Abdulaziz University, Jeddah, 21589, Saudi Arabia
Keywords: магнитогидродинамические течения, условия скольжения, жидкость Кэссона, тепловое излучение, MHD flow, slip condition, Casson fluid, thermal radiation

Abstract >>
A magnetohydrodynamic flow of the Casson fluid over a stretching surface in the presence of the slip condition, heat transfer, and thermal radiation is considered. The effects of the skin friction coefficient and local Nusselt number on flow parameters are analyzed numerically. The present results are compared with the existing limiting solution.

DOI: 10.1134/S0021894416050205

Lie Symmetry Analysis of a Double-Diffusive Free Convective Slip Flow with a Convective Boundary Condition Past a Radiating Vertical Surface Embedded in a Porous Medium

A.A. Afify1,2, Md.J. Uddin3
1Qassim University, Buraidah, 51452, Saudi Arabia
2Helwan University, Ain Helwan, Cairo, 11795, Egypt
3American International University-Bangladesh, Banani, Dhaka, 1213, Bangladesh
Keywords: группы Ли, групповой анализ, тепло- и массообмен, пористая среда, течение с проскальзыванием, конвективные граничные условия, Lie group analysis, heat and mass transfer, porous medium, slip flow, convective boundary condition

Abstract >>
A numerical study of a steady two-dimensional double-diffusive free convection boundary layer flow over a vertical surface embedded in a porous medium with slip flow and convective boundary conditions, heat generation/absorption, and solar radiation effects is performed. A scaling group of transformations is used to obtain the governing boundary layer equations and the boundary conditions. The transformed equations are then solved by the fourth- and fifth-order Runge--Kutta--Fehlberg numerical method with Maple 13.The results for the velocity, temperature, and concentration profiles, as well as the skin friction coefficient, the Nusselt number, and the Sherwood number are presented and discussed.

DOI: 10.1134/S0021894416050217

On Numerical Modeling of Capillary Flow Dynamics in a Microchannel Model by using the Finite Element Method

C.-T. Lee1, C.-C. Lee2, J.-E. Lin3, M.-L. Liu4
1Beijing Institute of Technology-Bryant College, Zhuhai, Guangdong, P.R. China
2Department of Chemistry, Simor Fraser University, British Columbia, BC, Canada
3Department of Mathematical Sciences, George Mason University, Fairfax, Virginia, USA
4Department of Physics, Shantou University, Shantou, Guangdong, P. R. China
Keywords: двухфазное капиллярное течение, микроканал, поверхностное натяжение, метод конечных элементов, two-phase capillary flow, microchannel, surface tension, finite element method

Abstract >>
A simple microchannel model with submillimeter-scale geometries is proposed for studying capillary flows and investigating the dynamics in the channel. The finite element method incorporating surface tension and two-phase flow characteristic is applied. Velocity and pressure fields in the microchannel are presented. It is shown that the capillary-phase front in the microchannel is stirred, suffering small oscillations and retreating from the previous position before traveling again. Such a phenomenon is caused by nonlinear interaction of the capillary flow, surface tension, and boundary conditions.

DOI: 10.1134/S0021894416050229

Analysis of Heat Transfer Over a Plate in a Porous Medium with a Convective Boundary Condition

S. Mukhopadhyay, I. Ch. Mandal
The University of Burdwan, Burdwan-713104,India
Keywords: вынужденное конвективное течение, пористая среда, модель Дарси - Форгеймера - Бринкмана, автомодельные решения, конвективные граничные условия, forced convective flow, porous medium, Darcy--Forchheimer--Brinkman model, similarity solutions, convective boundary condition

Abstract >>
Heat transfer characteristics of an incompressible viscous fluid past a plate embedded into a porous medium with a convective boundary condition are obtained by using the Darcy-Forchheimer-Brinkman model. The governing partial differential equations are reduced to nonlinear ordinary differential equations, and similarity solutions are obtained. The similarity equations are solved numerically. With increasing convective parameter, the surface temperature increases. The rate of heat transfer increases as the Prandtl number increases.

DOI: 10.1134/S0021894416050230

Heat Enhanced by an Exothermic Reaction on a Fully Developed MHD Mixed Convection Flow in a Vertical Channel

C. Jayabalan1, K.K. Sivagnana Prabhu1, R. Kandasamy2
1Department of Mechanical Engineering, R.M.K Engineering College, Anna University, Chennai, India
2Research Centre for Computational Mathematics, FSTPi, Universiti Tun Hussein Onn Malaysia, Parit Raja, 86400, Batu Pahat, Johor, Malaysia
Keywords: вертикальный канал, развитое магнитогидродинамическое течение, режим смешанной конвекции, экзотермическая химическая реакция, vertical channel, fully developed MHD mixed convective flow, exothermic chemical reaction

Abstract >>
The problem of a fully developed MHD mixed convection flow in a vertical channel with the first-order chemical reaction is analyzed. The dimensionless governing ordinary differential equations are solved numerically by using the Maple 18 software. It is observed that dual solutions exist for both velocity and temperature.

DOI: 10.1134/S0021894416050242