Home – Home – Jornals – Journal of Applied Mechanics and Technical Physics 2019 number 2

The exact solution of the Euler equations of relativistic hydrodynamics of an compressible fluid - the relativistic analog of the Ovsyannikov vortex (a special vortex) in the classical gas dynamics. The theorem on the representation of the factor system in the form of a union of a noninvariant subsystem for the function defining the deviation of the velocity vector from the meridian and an invariant subsystem for the function defining thermodynamic parameters, the Lorentz factor, and the radial component of the velocity vector. Compatibility conditions of the overdetermined noninvariant system are obtained. The stationary solution is studied in detail. It is proved that the invariant subsystem reduces to an implicit differential equation. The branching manifold of the solution of this equations was studied, and many singular points were found. The existence of two flow regimes, i.e., solutions describing the vortex source relativistic gas, was proved. One of these solutions is defined only at a finite distance from the source, and the other is an analog of supersonic gas flow from the surface of a sphere.

Several problems of motion of a viscous incompressible fluid with a point source in the flow region are considered. The corresponding initial-boundary-value problems for the Navier-Stokes equations have no solutions in the standard class of functions because the flow velocity field contains an infinite Dirichlet integral. Problem regularization allows one to prove its solvability under certain constraints on the initial data.

The development of perturbations of a tangential discontinuity surface separating two stationary flows of an ideal incompressible fluid slowly varying in space is studied taking into account surface tension. Perturbations are described using the complex Hamilton equations. The dependences of the amplitude of the perturbations on the coordinate and time are obtained.

A presentation of the general solution of the equations of dynamics of a transversely isotropic thermoelastic medium is obtained in the case where the Carrier-Gassmann condition is satisfied with due allowance for the additional expression relating the temperature stress coefficients to the elasticity moduli. The displacements are expressed via three resolving potentials satisfying three inhomogeneous quasi-wave equations. The potentials are related by the heat conduction equation. A presentation of the solution with the use of the stress and displacement functions is provided. Two displacement functions are determined by solving the system of two homogeneous equations, which do not involve the temperature. After these displacement functions are determined, the temperature can be found from the third equation. The resultant presentation of the solution also yields the solution of the static equations of thermoelasticity.

Equations of ideal magnetohydrodynamics that describe stationary flows of an inviscid ideally electroconducting fluid are considered. Classes of exact solutions of these equations are described. With the use of the natural curvilinear coordinate system, where the streamlines and magnetic force lines are coordinate curves, the model equations are partially integrated and converted to the form that is more convenient for the description of the magnetic lines and streamlines of particles. As the coordinate system used is related to the initial coordinate system by a nonlocal transformation, the group admitted by the system can change. An infinite-dimensional (containing three arbitrary functions of time) group of symmetries is calculated for the system in the natural coordinates. An optimal system of subgroups of dimensions 1 and 2 is constructed for this group. For one of the optimal system subgroups, an invariant exact solution is found, which describes the electroconducting fluid flow of the vortex source type with swirling magnetic lines and streamlines.

A long-wave approximation that describes running solitary waves is considered within the framework of the model of a weakly stratified two-layer fluid. It is demonstrated that wave regimes occur near the boundary of the parametric domain of shear instability in the stratified flow. This fact offers an explanation for the mechanism of intense mixing in deep near-bottom layers.

A possibility of using the 4D Qflow protocol, which is commonly used for medical diagnostics in magnetic resonance imaging, for determining the structure of the three-dimensional fluid flow in the human blood circulation system is considered. Specialized software is developed for processing DICOM images obtained by the magnetic resonance scanner, and the retrieved unsteady three-dimensional velocity field is analyzed. It is demonstrated that magnetic resonance measurements allow one to detect the existence of the flow in blood vessel models and also to study the degree of its swirling (helicity) both qualitatively and quantitatively.

This paper describes an inverse problem for determining the right side of a parabolic equation with a nonstandard growth condition and integral overdetermination condition. The Galerkin method is used to prove the existence of two solutions of the inverse problem and their uniqueness, one of them being local and the other one being global in time. Sufficient blow-up conditions for the local condition for a finite time in a limited region with a homogeneous Dirichlet condition on its boundary. The blow-up of the solution is proven using the Kaplan method. The asymptotic behavior of the inverse problem solutions for large time values is investigated. Sufficient conditions for vanishing of the solution for a finite time are obtained. Boundary conditions ensuring the corresponding behavior of the solutions are considered.

A problem of changing of the orientation of a solid in a space by means of motion of the internal mass is under consideration. It is shown that it is possible for a solid to be arbitrarily reoriented due to special motions of the internal mass. Approaches to controlling the internal motions ensuring this reorientation are proposed.

This paper presents a review of theoretical, experimental, and numerical studies of geometric attractors of internal and/or inertial waves in a stratified and/or rotating fluid. The dispersion relation for such waves defines the relationship between the frequency and direction of their propagation, but does not contain a length scale. A consequence of the dispersion relation is energy focusing during wave reflection from inclined walls. In a limited volume of fluid, focusing leads to the concentration of wave energy near closed geometrical configurations called wave attractors. The evolution of the ideas of wave attractors attractors from ray-theory predictions to observations of wave turbulence in physical and numerical experiments is described.

The nonstationary motion of a spherical layer of an ideal fluid is investigated taking into account the adiabatic distribution of gas pressure in the internal cavity. The existence of nonlinear oscillations of the layer is established, and their period is determined. It is shown that there is only one equilibrium state of the layer. Amplitude equations taking into account the action of capillary forces on the surfaces of the layer in a linear approximation are obtained and used to study the stability of nonlinear oscillations of the layer. The limiting cases of a spherical bubble and soap film are considered.

Different approximate crack opening models in a porous stratum are derived, based on a priori representations of the crack size, accurate solutions of motion equations of viscous fluid, approximate expressions of the filtering model in the stratum, and approximate expressions of the theory of elasticity of the stratum. The law of conservation of momentum of the fluid in the crack is used to obtain quasilinear parabolic equations, describing the crack opening.

A generalized phenomenological Leith model of wave turbulence in a medium with nonstationary viscosity is under study. Group analysis methods are used to obtain the main models possessing nontrivial symmetries. All invariant submodels are determined for each model. Invariant solutions describing these submodels are either determined in explicit form or satisfy the integral equations obtained. The main models are used to study turbulent processes. At an initial instance and with a fixed value of the wave number modulus, either turbulence energy spectrum and its gradient or turbulence energy spectrum and the rate of its variation are specified for the above-mentioned models. It is determined that solutions of the problems describing these processes exist and are unique under certain conditions.

Nonlocal equations of the coagulation theory are studied by methods of group analysis. In addition to the integrodifferential Smoluchowski equation, equivalent models are also considered, including the equation for the Laplace transform of the original equation, an infinite system of equations for the power moments of its solution, and the equation for the generating function of the power moments. Admissible Lie groups for the considered equations are found, their relationships are studied, and the corresponding invariant solutions are analyzed.

Horizontal shear motion of a homogeneous fluid in an open channel is considered in the approximation of the shallow water theory. The main attention is paid to studying the mixing process induced by the development of the Kelvin-Helmholtz instability and by the action of bottom friction. Based on a three-layer flow pattern, an averaged one-dimensional model of formation and evolution of the horizontal mixing layer is derived with allowance for friction. Steady solutions of the equations of motion are constructed, and the problem of the mixing layer structure is solved. If bottom friction is taken into account, the mixing process becomes slower and the width of the intermediate mixing layer does not increase. Verification of the proposed one-dimensional model is performed through comparisons with available experimental data and with the numerical solution of the two-dimensional equations of the shallow water theory.

The classical Boussinesq equation describing gravity waves in shallow waters is under consideration. Hirota's bilinear representation is used to construct exact solutions describing wave packets, waves on solitons, and «dancing» waves. The principle of multiplying the solutions of the Hirota equation is formulated, which helps constructing more complex structures made of solitons, wave packets, and other types of waves.

A fundamentally new unsaturated technique for the numerical solution of the Dirichlet-Neumann problem for the Laplace equation was designed. This technique makes it possible, due to the smoothness of the sought solution of the problem, to take into account the axisymmetric specificity of the problem, which prevents the use of any saturated numerical methods, i. e., methods with a leading error term.

Variational approach and sensitivity theory methods are used to construct algorithms for solving the problems of environmental forecast and design. The behavior of the model in a parameter space is studied by calculating sensitivity functions as partial derivatives of target functions with respect to the model, used to investigate the properties of mathematical models and solutions of inverse problems. Not only the proposed approach implies the atmospheric quality model in a Novosibirsk agglomeration and an algorithm based on an ensemble of sensitivity functions, but is also used to solve the inverse problem of position estimation and pollution source intensity.