Home – Home – Jornals – Journal of Applied Mechanics and Technical Physics 2014 number 2

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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 → ∞.

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.

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.

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.