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2014 year, number 2

A New Approach to the Group Analysis of One-Dimensional Stochastic Differential Equations

M. A. Abdullin1, S. V. Meleshko2, F. S. Nasyrov1
1Ufa State Aviation Technical University, Ufa, 450000 Russia
2Suranari Technological University, Nakhon Ratchasima, 30000 Thailand
Keywords: stochastic differential equations, symmetry, group analysis

Abstract >>
Stochastic evolution equations are investigated using a new approach to the group analysis of stochastic differential equations. It is shown that the proposed approach reduces the problem of group analysis for this type of equations to the same problem of group analysis for evolution equations of special form without stochastic integrals.

Thermogravitational Mechanism of Alignment of the Front of Chemoconvection Patterns with an Exothermic Chemical Reaction

D. A. Bratsun
Perm' State Humanitarian Pedagogical University, Perm', 614990 Russia
Keywords: chemoconvection, Rayleigh instability, Rayleigh-Taylor instability, neutralization, heat release

Abstract >>
This paper studies the stability of the front of chemoconvective finger patterns spontaneously formed in a two-layer system of reacting liquids placed in a narrow gap between two solid plates. The mathematical model of this process includes a system of equations that take into account chemical reaction, diffusion, and convection, written in the Hele-Shaw approximation. Numerical analysis of the model shows that with increasing intensity of heat release, the envelope of the salt fingers is smoothed and chemoconvection occurs in a narrow region adjacent to the interface.

Hydraulic Jumps in Two-Layer Flow with a Free Surface

S. L. Gavrilyuk1, M. Yu. Kazakova2,3
1Institute of Industrial Thermal Systems University of Aix-Marseille, Marseille, 13453 France
2Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
3Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: two-layer shallow water equations, conservation laws, Rankine-Hugoniot relations, conservative equations

Abstract >>
This paper presents a closure relation which describes hydraulic jumps in two-layer flows with a free surface over a flat bottom. This relation is derived from the momentum equations for each layer, which, subject to the condition of conservation of the total momentum and mass of each layer, become conservative in a sense. It is shown that use of this relation provides a reduction in the total energy at the jump.

Approximate Symmetries and Solutions of the Kompaneets Equation

R. K. Gazizov1, N. Kh. Ibragimov1,2
1Ufa State Aviation Technical University, Ufa, 450000 Russia
2Blekinge Institute of Technology, Karlskrona, SE-37179 Sweden
Keywords: Kompaneets equation, Compton effect, approximate symmetries approximate solutions

Abstract >>
Different approximations of the Kompaneets equation are studied using approximate symmetries, which allows consideration of the contributions of all terms of this equation previously neglected in the analysis of the limiting cases.

On the Action of Internal Heat Sources on Convective Motion in a Porous Medium Heated from Below

V. N. Govorukhin
Southern Federal University, Rostov-on-Don, 344090 Russia
Keywords: filtration convection, cosymmetry, convection, steady regimes

Abstract >>
The effect of internal heat sources on the flow pattern in the filtration convection problem with cosymmetry is studied numerically. Low-intensity heat sources, whose presence leads to violation of cosymmetry and breakdown of the one-parameter family of steady regimes are considered. Theoretically predicted scenarios for the breakdown of the family into a finite number of steady regimes and the occurrence of slow periodic motions are confirmed. The existence of relaxation oscillations is established for large Rayleigh numbers.

Unsteady Flows with a Constant Total Pressure, Described by the Equations of Ideal Magnetohydrodynamics

S. V. Golovin1,2, M. N. Dudnik3
1Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
2Novosibirsk State University
3Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: magnetohydrodynamics, exact solutions, total constant pressure

Abstract >>
Exact solutions of the equations of ideal magnetohydrodynamics describing the class of unsteady flows of an electrically conducting fluid with a constant total pressure are constructed. The solutions are written in the Lagrange coordinate system; arbitrariness in its choice was used to parameterize magnetic field lines. The wide functional arbitrariness the solutions provide a significant variation in the picture of the described fluid motions. An example of unsteady flow of an ideal electrically conducting fluid in a cylindrical channel with fixed magnetic tubes is given.

Example of an Exact Solution of the Stationary Problem of Two-Layer Flows with Evaporation at the Interface

O. N. Goncharova1,2, E. V. Rezanova1
1Altai State University, Barnaul, 656049 Russia
2Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: convection, thermocapillary interface, two-layer flow, evaporation, exact solution

Abstract >>
Stationary two-layer liquid and gas flows with fluid evaporation at the interface are studied. On the solid impermeable boundaries of the channel, no-slip conditions are satisfied and a linear temperature distribution along the longitudinal coordinate and a condition for the vapor concentration at the upper boundary are specified. On the thermocapillary interface, remaining undeformed, the following conditions are specified: kinematic and dynamic conditions, a condition for thermal flows with mass transfer, continuity conditions for the velocity, temperature, and mass balance, and a relation for the saturated vapor concentration. An exact solution of the stationary problem for a given gas flow rate is obtained. Examples of velocity profiles are given for stationary flows of the ethanol-nitrogen system under normal and reduced gravity are given. The effect of longitudinal temperature gradients specified at the boundaries of the channel on the flow pattern is investigated.

Linear Stability of the Couette Flow of a Vibrationally Excited Gas. 1. Inviscid Problem

Yu. N. Grigor'ev1, I. V. Ershov2
1Institute of Computational Technologies, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
2Novosibirsk State University of Architecture and Civil Engineering, Novosibirsk, 630008 Russia
Keywords: linear stability theory, vibrational relaxation, equations of two-temperature aerodynamics, inviscid modes of disturbances

Abstract >>
Stability of the Couette flow of a vibrationally excited diatomic gas with a parabolic profile of static temperature is studied within the framework of the linear theory. A set of explicit asymptotic estimates are obtained for inviscid disturbances described by a system of linearized equations of two-temperature gas dynamics. It is shown that the first Rayleigh condition (theorem) is satisfied for unstable modes, and the classification of inviscid modes into even and odd modes is valid. A generalized condition of the presence of an inflection point on the velocity profile, which is necessary for disturbances to evolve, is obtained. The sufficient condition in Howard's semi-circle theorem is refined. Complex phase velocities of two-dimensional even and odd inviscid modes are numerically calculated as functions of the Mach number, degree of excitation of vibrational levels of energy, and characteristic relaxation time. In the Couette flow problem, in contrast to the case of a free shear layer, the growth rate of the most unstable second mode increases with increasing Mach number and tends to a certain limit for which an asymptotic expression in the form of an ordinary differential equation is obtained. The calculated results show that the effect of reduction of the growth rate on the background of the relaxation process is clearly expressed in the range of flow parameters considered.

Gas-Jet Synthesis of Diamond-Like Films from an H 2+CH 4 Gas Mixture Glow

A. A. Emel'yanov, A. K. Rebrov, I. B. Yudin
Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: diamond-like films, gas-jet deposition, DSMC method

Abstract >>
Synthesis of diamond-like coatings from a high-velocity flow of gas mixtures in flow regimes from free-molecular to continuum with flow velocities from hundreds to thousands meters per second at different specific flow rates and temperatures in the case of activation of gases on hot surfaces is studied experimentally. Deposition of carbon films at low (less than 0.15 Pa) and high (2600 Pa) pressures from a mixture of hydrogen and methane is considered. The hydrogen flow is computed by the Direct Simulation Monte Carlo (DSMC) method in accordance with test conditions with given surface temperatures and chemical transformations on the surfaces. It is found that coatings obtained at the high pressure contain particles typical for diamonds and unusual inclusions shaped as prisms with a hexagonal cross section.

Role of Nuclei Density as a ``Hidden" Parameter in the Formation of Anomalous Zones in a Heavy Cavitating Magma

V. K. Kedrinskii
Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: decompression wave, diffusion, saturation zone, viscosity, nucleation, concentration of the gas phase, microcrystallites

Abstract >>
The formation of zones with anomalously high values of the basic flow characteristics in decompression waves in a heavy cavitating magma with an intense increase in the density of cavitation nuclei is numerically studied within the framework of a mathematical model of multiphase media with a system of kinetic equations. The basic effects leading to the formation of the anomalous zone are identified.

Axisymmetric-Conical and Locally Conical Flows without Swirling

A. N. Kraiko, N. I. Tillyaeva
Baranov Central Institute of Aviation Motors, Moscow, 111116 Russia
Keywords: local conicity condition, diffuser, trail part, cone, detonation wave, characteristics, streamlines

Abstract >>
Axisymmetric steady conical and locally conical non-swirled flows of an ideal (inviscid and non-heat-conducting) gas are considered. Like two-dimensional conical flows, the examined one-dimensional (axisymmetric) flows can be conically subsonic and supersonic. If the uniform flow is not considered as a conical flow, then the type of one-dimensional conical flows can change only on the shock wave, except for the junction of two one-dimensional conical flows of different types on the C+ characteristic. C± characteristics and streamlines are constructed for a number of locally conical flows and some already known and new conical flows.

Mixing Layer under a Free Surface

V. Yu. Liapidevskii1,2, A. A. Chesnokov1,2
1Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Novosibirsk, 630090 Russia
2Novosibirsk State University
Keywords: shallow water equations, mixing layer, turbulent bore, breakdown of waves

Abstract >>
In the long-wave approximation, the flow of a homogeneous fluid with a free surface in the gravity field is considered. Mathematical models of the surface turbulent layer in shear flows are derived. Steady solutions of the problem of evolution of the mixing layer under the free surface and formation of a surface turbulent jet are constructed. In particular, the problem of the structure of a turbulent bore in a supercritical flow is solved, and the conditions for the formation of a local subcritical zone ahead of the obstacle are studied.

Steady Waves in a Stratified Flow over a Combined Obstacle

N. I. Makarenko1,2, J. L. Maltseva1,2
1Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
2Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: stratified flows, internal waves

Abstract >>
A problem of steady internal waves in subcritical flows of a stratified liquid over a finite sequence of bottom elevations is considered. An asymptotic solution of the second order of accuracy, which takes into account nonlinear effects, is constructed by the method of perturbations in terms of the small parameter of the obstacle height. Near-field interference wave structures are studied.

Mechanisms of Elastic Energy Dissipation in the Transition Layer between a Coating and a Substrate under Contact Interaction

V. E. Panin1,2, D. D. Moiseenko1, S. V. Panin1,2, P. V. Maksimov1, I. G. Goryacheva3, C. H. Cheng4
1Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, 634021 Russia
2Tomsk Polytechnic University, Tomsk, 634028 Russia
3Ishlinskii Institute of Problems of Mechanics, Russian Academy of Sciences, Moscow, 119526 Russia
4National Cheng Kung University, Tainan, 70701 China
Keywords: contact interaction, interface, discrete simulation, materials with coatings, cellular automata

Abstract >>
A theoretical and experimental analysis of the influence of the dispersed transition layer between a coating and a substrate on the development of deformation structures near the interface has been performed as part of an interdisciplinary study of the deformation and fracture of coating-substrate compositions under contact interaction. Elastic energy transfer from an indenter was simulated using excitable cellular automata taking into account the self-organization of translations and rotations of the structure near the interface. The effect of the transition layer between the coating and the substrate on the development of deformation structures during contact interaction with the indenter in three-point bending was studied experimentally using a TOMSC television-optical measuring complex.

Steady Vortex Flows of a Self-Gravitating Gas

D. V. Parshin1, A. A. Cherevko1,2, A. P. Chupakhin1,2
1Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Scinces, Novosibirsk. 630090 Russia
2Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: self-gravitating gas, Ovsyannikov vortex, partly invariant solution, group of rotations, vortex flow of the gas, spherical source

Abstract >>
An exact solution of equations of steady motion of a self-gravitating gas is found. This solution describes a vortex flow of the gas from the surface of a spherical source and is partly invariant with respect to the group of rotations (Ovsyannikov vortex, singular vortex). The factor-system of the solution is reduced to finite formulas and one ordinary differential equation of the third order. Various regimes of gas motion described by this solution are determined: unlimited spreading of the gas with swirling from the surface of a spherical source and gas exhaustion with formation of a sphere with elevated density at a finite distance from this source.

Motion of a Load Over a Floating Sheet in a Variable-Depth Pool

A. V. Pogorelova1, V. M. Kozin2,3
1Institute of Metallurgy and Mechanical Engineering, Far-East Branch, Russian Academy of Sciences, Komsomol'sk-on-Amur, 681005 Russia
2Komsomol'sk-on-Amur State Technical University, Komsomol'sk-on-Amur, 681013 Russia
3Amur Humanitarian Pedagogical State University, Komsomol'sk-on-Amur, 681000 Russia
Keywords: incompressible liquid, elastic sheet, unsteady motion

Abstract >>
A straightline unsteady motion of a load over a floating elastic sheet in a pool whose depth changes in the direction of load motion is studied. The influence of the pool depth, sheet thickness, load size and intensity, and velocity of uniform motion on the amplitude and maximum deflection of the sheet is analyzed.

Point Vortex in a Viscous Incompressible Fluid

V. V. Pukhnachev1,2
1Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
2Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: Navier-Stokes equations, no-slip condition, point vortex

Abstract >>
A plane steady problem of a point vortex in a domain filled by a viscous incompressible fluid and bounded by a solid wall is considered. The existence of the solution of Navier-Stokes equations, which describe such a flow, is proved in the case where the vortex circulation Γ and viscosity ν satisfy the condition | Γ | < 2πν. The velocity field of the resultant solution has an infinite Dirichlet integral. It is shown that this solution can be approximated by the solution of the problem of rotation of a disk of radius γ with an angular velocity ω under the condition 2πγ2ω → Γ as γ → 0 and ω → ∞.

Waves on Down-Flowing Fluid Films: Calculation of Resistance to Arbitrary Two-Dimensional Perturbations and Optimal Downflow Conditions

Yu. Ya. Trifonov
Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: viscous film flow, nonlinear waves, stability

Abstract >>
Wavy downflow of viscous fluid films is studied. The full Navier-Stokes equations are used to calculate the hydrodynamic characteristics of the flow. The stability of calculated nonlinear waves to arbitrary two-dimensional perturbations is considered within the framework of the Floquet theory. It is shown that, for small values of the Kapitza number, the waves are stable over a wide range of wavelengths and values of the Reynolds number. It is found that, as the Kapitza number increases, the parameter range where nonlinear waves are calculated is divided into a series of alternating zones of stable and unstable solutions. A large number of narrow zones where the solutions are stable are revealed on the wavelength-Reynolds number parameter plane for large values of the Kapitza number. Optimal regimes of film downflow that correspond to the minimum value of average film thickness for nonlinear waves with different wavelengths are determined. The basic characteristics of these waves are calculated in a wide range of Reynolds and Kapitza numbers.

Simple Waves of a Seven-Dimensional Subalgebra of All Translations in Gas Dynamics

S. V. Khabirov
Mavlyutov Institute of Mechanics, Ufa Scientific Center, Russian Academy of Sciences, Ufa, 450054 Russia
Keywords: gas dynamics, subalgebra of translations, simple wave

Abstract >>
A differentially invariant solution of rank 1 for gas dynamics equations is considered in a seven-dimensional subalgebra of all translations including Galilean translations. Exact solutions with a uniformly accelerated plane of the level of invariant functions are obtained. A nonisentropic wave depends on two arbitrary functions and constants. A general solution of an isentropic simple wave depends on the constants only.

Oil Flow to a Slit-Like Well in a Reservoir Containing other Fluids

V. N. Emikh
Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: horizontal well, flow velocity hodograph, conformal mapping parameters, double critical regime, boundary line

Abstract >>
An exact solution is obtained for the boundary-value problem of oil flow to a slit-like well in a reservoir containing other fluids adjacent to its top and base. This solution is used to compare the maximum possible oil production from slit-like and tubular wells in the double critical regime.