

2014 year, number 5
E. A. Demekhin^{1,2}, M. G. Baryshev^{1}, G. S. Ganchenko^{1}, E. V. Gorbacheva^{1}
^{1}Kuban' State University, Krasnodar, 350040 Russia ^{2}Lomonosov Moscow State University, Moscow, 117192 Russia
Keywords: nano– and microflows, Nernst—Planck—Poisson equations, concentration polarization, electrolyte, ionic double layer, nano– and microdiodes, quasispectral Galerkin method, τvariant of the Galerkin method, stiff systems, Gear method
Abstract >>
A method for rectifying electric current in micro and nanoscale devices is proposed based on the asymmetric concentration of polarization in an electrolyte solution in the case where the current in a microdiode successively passes through two mutually undissolvable fluids with different dielectric constants and diffusion coefficients in tubes with different dimensions. It is assumed that both fluids contain the ions of a completely dissociated substance which provide electric charge transfer upon application of a potential difference to the walls of the device, and the interface between the two fluids has a charge. The process is described by a one–dimensional nonstationary Nernst—Planck—Poisson system. The boundary conditions on the electrodes are the impermeability condition for anions and the Arrhenius equation which defines the flow of cations. The system of equations was solved numerically: the unknowns were decomposed into a complete system of orthogonal functions of the spatial variable, and the resulting dynamical system for the Galerkin coefficients was integrated over time by the Gear method because of its stiffness. The parameters of the system that have the most significant effect on the degree of rectification are determined, and their optimum values are evaluated.

S. V. Stankevich and G. A. Shvetsov
Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: electromagnetic rail launchers, external magnetic field, metal armature, numerical simulation, current density, armature heating
Abstract >>
This paper presents the results of numerical simulation of the Joule heating of the armature and rails in rail launchers of metal bodies with one or two augmenting rails generating an external magnetic field. The ultimate projectile velocity is calculated under the assumption that the Joule heating of the armature and rails during acceleration does not exceed the melting temperature of the materials of which they are made. It is shown that, with an appropriate choice of the position of the coils generating the augmenting magnetic field with respect to the launcher channel and the current magnitude in them, the heat load on the rails and armature in electromagnetic launchers of solids can be substantially reduced and the ultimate kinematic characteristics of these launchers in crisisfree regimes can be considerably increased.

S. P. Kiselev, V. P. Kiselev, and V. N. Zaikovskii
Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: overexpanded supersonic jet, obstacle, selfoscillations, shock wave
Abstract >>
Results of the numerical solution of the problem of impingement of an overexpanded supersonic jet onto an obstacle are reported. The massflowrate mechanism of selfoscillations is revealed. This mechanism consists of periodic changes in the regimes of gas inflow and outflow from the separation region to the jet around this region. It is shown that the shockwave structure of the impinging supersonic jet exerts a significant effect on the amplitude of selfoscillations.

Yu. V. Pivovarov
Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia pivov@hydro.nsc.ru
Keywords: regularization, nonNewtonian fluid, conformal mapping, Stokes equation, vortex, stream function
Abstract >>
The experimentally observed process of approach of two identical oil droplets in an alcoholwater solution (matrix) which have identical density and move cyclically in the rest  acceleration  deceleration mode with a cycle time of 0.01 s is mathematically modeled. The imbalance of forces on the boundaries of the droplets at rest is due to the fact that the shear stresses on them cannot exceed the yield strength of the matrix and the normal stresses are determined from the solution of the elastic problem because the intermolecular bonds in the resting matrix make it similar to a solid. The calculation and experimental results are compared, and it is noted that they are in good agreement.

Yu. G. Yermolaev, A. D. Kosinov, and N. V. Semionov
Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: supersonic velocities, boundary layer, swept wing, transition, stability
Abstract >>
Results of experiments aimed at studying the linear and nonlinear stages of the development of natural disturbances in the boundary layer on a swept wing at supersonic velocities are presented. The experiments are performed on a swept wing model with a lensshaped airfoil, leadingedge sweep angle of 45°, and relative thickness of 3%. The disturbances in the flow are recorded by a constanttemperature hotwire anemometer. For determining the nonlinear interaction of disturbances, the kurtosis and skewness are estimated for experimentally obtained distributions of the oscillating signal over the streamwise coordinate or along the normal to the surface. The disturbances are found to increase in the frequency range from 8 to 35 kHz in the region of their linear development, whereas enhancement of highfrequency disturbances is observed in the region of their nonlinear evolution. It is demonstrated that the growth of disturbances in the highfrequency spectral range ( f > 35 kHz) is caused by the secondary instability.

A. A. Ivanova, V. G. Kozlov, and V. D. Shchipitsyn
Perm' State Humanitarian Pedagogical University, Perm' 614990 Russia
Keywords: vibrations, vibrational mechanics, cylindrical body, hydrodynamic interaction, lift force
Abstract >>
The averaged lift force acting on a cylindrical body near the boundary of a cavity with a fluid performing translational vibrations was studied. Experiments were performed with variation the viscosity of the fluid, the size and relative density of the body, and vibration parameters were varied. The lift force was measured by the method of dynamic suspension of a body in a gravitational field in the case where the body performed inertial vibrations without touching the walls. It was found that the vibrations generated a repulsion force which held the heavy body over the bottom of the cavity, and the light body at a certain distance from the top wall. It was shown that the effect of the repulsion forces manifested itself at a distance comparable to the thickness of the Stokes layer and increased with approach to the wall. A description of the mechanism of generation of the lift force is given. It is shown that in the case of high dimensionless frequencies, the experimental and theoretical results are in agreement.

P. S. Farrugia and A. Micallef
University of Malta, Msida, MSD2080, Malta
Keywords: thermal plumes, power series, recurrence relations, natural convection
Abstract >>
Power series solutions to the boundary layer equations for laminar point and line thermal plumes in natural convection have been derived in terms of recurrent relations. These together with the initial conditions constitute closedform solutions for any Prandtl number in the region where the series converge. The starting conditions are related to the maximum values of the temperate and velocity profiles. Their values together with those for the radius of convergence of the series have been obtained numerically for different Prandtl numbers, and bestfitting functions have been proposed for the variation observed. The validity of the approach has been tested against the known closedform solutions giving identical results in the region of convergence. While the utility of the equations does not extend to infinity, the tests conducted indicate that the range of convergence can be potentially extended by using the Euler transform. This is especially true for results involving point heat sources, where it has been shown that, for all Prandtl numbers, the nearest singularity is found in the complex plane and, hence, has no physical significance.

H. A. Attia^{1}, M. A. M. Abdeen^{2}, M. T. M. M. Elbarawy^{1}
^{1}ElFayoum University, ElFayoum63514, Egypt ^{2}Cairo University, Giza 12211, Egypt
Keywords: circular pipe flow, nonNewtonian fluid, unsteady state, conducting fluid, ion slip, Hall current
Abstract >>
In this paper, a steady magnetohydrodynamic (MHD) flow of a dusty incompressible electrically conducting Oldroyd 8constant fluid through a circular pipe is examined with considering the ion slip effect. A constant pressure gradient in the axial direction and an external uniform magnetic field in the perpendicular direction are applied. A numerical solution is obtained for the governing nonlinear momentum equations by using finite differences. The effect of the ion slip, the nonNewtonian fluid characteristics, and the particlephase viscosity on the velocity, volumetric flow rates, and skin friction coefficients of both the fluid and particle phases is reported.

A. V. Bolesta^{1,2} and V. M. Fomin^{1,2}
^{1}Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia ^{2}Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: molecular dynamics, polycrystal, copper, uniaxial deformation, shock wave, phase transition
Abstract >>
An approach for molecular dynamics simulation of the formation of polycrystalline materials from a melt during its cooling is proposed. Atomic configurations of copper corresponding to polycrystals with the mean grain size from 2 to 16 nm are obtained. Isothermal uniaxial tension and compression of these polycrystals is studied by the molecular dynamics method. For the mean grain size of polycrystalline copper being smaller than 10 nm, it is shown that Young's modulus and yield stress decrease as the grain size decreases. Shock adiabats for polycrystalline copper are constructed. For a material with the grain size approximately equal to 2 nm, the temperature behind the shock wave front is demonstrated to be 10% higher than that in a polycrystal with the grain size greater than 10 nm. Molecular dynamics calculations predict the presence of copper with a bodycentered cubic lattice behind the shock wave front at pressures ranging from 100 to 200 GPa.

A. G. Petrov^{1} and I. I. Potapov^{2}
^{1}Ishlinskii Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526 Russia ^{2}Computing Center, Far East Branch of Russian Academy of Sciences, Khabarovsk, 680000 Russia
Keywords: channel processes, sediment transport, bottom layer, bottom waves, channel resistance
Abstract >>
It is shown that the sediment transport rate is uniquely determined by the normal and tangential stresses and the slope of the bottom surface. The dependence of the mass flow rate of sediments transported over an uneven eroded bottom on these three characteristics was obtained analytically. A formula for the sediment transport rate that generalizes a number of wellknown formulas of sediment transport was derived within the framework of a twovelocity model.

V. I. Pen'kovskii^{1}, N. K. Korsakova^{1}, and G. V. Nesterova^{2}
^{1}Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia ^{2}Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: invaded zone, flow of immiscible fluids, thermal front, resistivity
Abstract >>
A mathematical model of the axisymmetric distribution of the phases in the zone of invasion of the waterbased drilling mud into the productive stratum whose porous space can simultaneously contain three immiscible fluids (oil, gas, and natural water) is constructed; the model takes into account the high rate of heat transfer between the fluids and the rock matrix. It is shown that the resistivity of the invaded zone depends not only on saturation of the latter by the fluids and the concentration of salts in the water phase, but also on the drilling mud filtrate temperature. It is also shown that there is a jump in the function of stratum saturation by oil on the thermal front.

A. A. Bochkarev^{†} and V. I. Polyakova
Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: Langmuir adsorption model, sorption of water vapor, emission of dimers, multilayer adsorption, condensation, film boiling
Abstract >>
The emission rate of water dimers from a free surface and a wetted solid surface in various cases was calculated by a simplified Monte Carlo method with the use of the binding energy of water molecules. The binding energy of water molecules obtained numerically assuming equilibrium between the free surface of water and vapor in the temperature range of 298–438 K corresponds to the coordination number for liquid water equal to 4.956 and is close to the reference value. The calculation results show that as the water temperature increases, the free surface of water and the wetted solid surface become sources of free water dimers. At a temperature of 438 K, the proportion of dimers in the total flow of water molecules on its surface reaches 1%. It is found that in the film boiling mode, the emission rate of dimers decreases with decreasing saturation vapor. Two mechanisms of the emission are described.

E. B. Sibiryakov
Trofimuk Institute of Petroleum Geology and Geophysics, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: boundary integral equation method in elasticity theory, cavernous medium, microstructure, pore structure, contrast media, block media
Abstract >>
The dependence of the effective parameters of microheterogeneous media on the frequency and structure of the pore space is obtained using the boundary integral equation method. The potential method is first used to solve dynamic threedimensional elastic problems in multiply connected domains in the case of stationary oscillations. It is shown that if the wavelength corresponds to a finite number of blocks, the effective elastic moduli decrease.

V. N. Paimushin^{1,2}
^{1}Kazan' National Research Technical University named after Tupolev, Kazan', 420111 Russia ^{2}Kazan' Federal University, Kazan', 420008 Russia
Keywords: thin shell, Timoshenko model, nonlinear theory, finite displacements, finite strains, true stresses, true strains, spherical shell, internal pressure, static instability, elastomer
Abstract >>
A new modified version of the Timoshenko theory of thin shells is proposed to describe the process of deformation of thin shells with arbitrary displacements and strains. The new version is based on introducing an unknown function in the form of a rotation vector whose components in the basis fitted to the deformed midsurface of the shell are the components of the transverse shear vector and the extensibility in the transverse direction according to Chernykh. For the case with the shell midsurface fitted to an arbitrary nonorthogonal system of curvilinear coordinates, relationships based on the use of true stresses and true strains in accordance with Novozhilov are obtained for internal forces and moments. Based on these relationships, a problem of static instability of an isotropic spherical shell experiencing internal pressure is solved. The shell is considered to be made either of a linear elastic material or of an elastomer (rubber), which is described by Chernykh's relationships.

W. Tebboune, M. Merdjah, K. H. Benrahou, and A. Tounsi
Université de Sidi Bel Abbes, BP 89 Cité Ben M'hidi 22000 Sidi Bel Abbes, Algérie
Keywords: thermoelastic buckling, functionally graded material, shear deformation theory, neutral surface position
Abstract >>
The thermoelastic buckling behavior of a thick plate made of a functionally graded material is investigated in this paper by using an exponential shear deformation plate theory. A simple power law based on the rule of mixtures is used to estimate the effective material properties as functions of the plate thickness. The neutral surface position for such functionally graded plates is determined on the basis of the nonlinear straindisplacement relations. Uniform, linear, and nonlinear temperature distributions across the plate are considered. An analytical approach is presented to find the critical buckling temperature, which can be used in engineering calculations. A numerical solution of the problem with the use of an exponential dependence for shear strains is presented. The results obtained are compared with available data.

N. P. Adamov^{1}, L. N. Puzyrev^{1}, A. M. Kharitonov^{1}, E. A. Chasovnikov^{1}, A. A. Dyad'kin^{2}, A. N. Krylov^{2}
^{1}Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia ^{2}Korolev Rocket and Space Corporation Energia, Korolev, 141071 Russia
Keywords: reentry vehicle model, free oscillations, frequency of oscillations, damping decrement, longitudinal damping
Abstract >>
The experimental equipment, model, test conditions, and methods used for determining the streamwise damping on a setup with free oscillations on rolling bearings are described. Characteristics of aerodynamic damping of the model with two positions of the rotation axis and Mach numbers M_{∞} = 2, 4, and 6 are measured. Irregular oscillations of the model with a greater displacement of the rotation axis with respect to the longitudinal axis are found to arise at M_{∞} = 2.

V. M. Tikhomirov
Siberian State University of Railway Transport, Novosibirsk, 630049 Russia
Keywords: threedimensional body with a notch, stress intensity factors, numerical analysis of stress strain states
Abstract >>
Dependences of displacements of the surface of a notch on the corresponding stress intensity factors were obtained for axisymmetric bodies with internal and external notches under different deformations (tensile, shear, bending, and torsion). An algorithm is proposed to determine the stress intensity factors of three types (opening mode, longitudinal shear, and transverse shear) from displacements of the notch surface near its tip. The effectiveness of the algorithm is shown, as an example, for numerical analysis of various threedimensional problems of fracture mechanics.

L. V. Stepanova and E. M. Adylina
Samara State University, Samara, 443011 Russia
Keywords: mixed deformation, selfsimilar representation of solution, selfsimilarity of the second kind, intermediate asymptotic form
Abstract >>
A method is proposed to calculate the eigenvalues of the class of nonlinear eigenvalue problems resulting from the problem of determining the stress  strain state in the vicinity of a crack tip in powerlaw materials over the entire range of mixed modes of deformation, from the opening mode to pure shear. The proposed approach was used to found eigenvalues of the problem that differ from the wellknown eigenvalue corresponding to the HutchinsonRiceRosengren solution. The resulting asymptotic form of the stress field is a selfsimilar intermediate asymptotic solution of the problem of a crack in a damaged medium under mixed loading. Using the new asymptotic form of the stress field and introducing a selfsimilar variable, we obtained an asymptotic solution of the problem of a crack in a damaged medium and constructed the regions of dispersed material near the crack.

G. G. Savenkov^{1}, Yu. I. Meshcheryakov^{2}, B. K. Barakhtin^{1}, and N. V. Lebedeva^{1}
^{1}St. Petersburg State Marine Technical University, St. Petersburg, 190008 Russia ^{2}Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, 199178 Russia
Keywords: spall strength, selfsimilarity, fractal dimension
Abstract >>
Results of experiments on shockwave deformation of M2 copper under uniaxial loading are presented. Light, scanning, and transmission electron microscopy methods are used to reveal specific features of mechanisms of deformation and fracture of copper during the formation of a main spall crack. The parameters of spall strength, damage, and selfsimilarity of the spall crack contour are determined.

