Vol 60, No 2 (2019)
- Year: 2019
- Articles: 20
- URL: https://journals.rcsi.science/0021-8944/issue/view/9802
Article
Ovsyannikov Vortex in Relativistic Hydrodynamics
Abstract
The exact solution of the Euler equations of relativistic hydrodynamics of an compressible fluid—a relativistic analog of the Ovsyannikov vortex (singular vortex) in the classical gas dynamics—was found and investigated. A theorem was proved which shows that the factor system can be represented as 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 were obtained. The stationary solution was studied in detail. It was proved that the invariant subsystem reduces to an implicit differential equation. The branching manifold of the solutions of this equations was studied, and many singular points were found. It is proved that there exist two flow regimes, i.e., the solutions describing the vortex source of a 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.
Problem of a Point Source
Abstract
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.
On Perturbations of a Tangential Discontinuity Surface between Two Non-Uniform Flows of an Ideal Incompressible Fluid
Abstract
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 complex Hamiltonian equations. The dependences of the amplitude of perturbations on the coordinate and time are obtained.
Presentation of the General Solution of Three-Dimensional Dynamic Equations of a Transversely Isotropic Thermoelastic Medium
Abstract
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.
Exact Solutions of Stationary Equations of Ideal Magnetohydrodynamics in the Natural Coordinate System
Abstract
Equations of ideal magnetohydrodynamics that describe stationary flows of an inviscid ideally electrically conducting 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 play the role of the 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 electrically conducting fluid flow of the vortex source type with swirling magnetic lines and streamlines.
Steady Internal Waves in Deep Stratified Flows
Abstract
A long-wave approximation that describes travelling solitary waves within the framework of the model of a weakly stratified two-layer fluid is considered. It is demonstrated that wave regimes occur near the boundary of the parametric domain of shear instability in the stratified flow. This property offers an explanation for the mechanism of intense mixing in deep bottom layers.
Application of Magnetic Resonance Imaging for Studying the Three-Dimensional Flow Structure in Blood Vessel Models
Abstract
A possibility of using the 4D Qflow protocol, which is commonly applied for medical diagnostics by 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 taken by a 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 swirling flow in blood vessel models and also to study the degree of its swirling (helicity) both qualitatively and quantitatively.
Inverse Problem for an Equation with A Nonstandard Growth Condition
Abstract
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 solution for a finite time in a limited region with a homogeneous Dirichlet condition on its boundary are obtained. 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.
Controlling the Orientation of a Solid Using the Internal Mass
Abstract
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.
Internal and Inertial Wave Attractors: A Review
Abstract
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 due to wave reflection from sloping 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 concept of wave attractors from ray-theory predictions to observations of wave turbulence in physical and numerical experiments is described.
Stability of Nonlinear Oscillations of A Spherical Layer of an Ideal Fluid
Abstract
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.
Modeling of Aircraft Flight Through the Wake Vortex
Abstract
A mathematical model is proposed for calculating the forces and moments acting on an aircraft in the region affected by the wake vortex generated by another aircraft. This model takes into account various random factors and determines the wake vortex characteristics, as well as its shape, which depends on integral and distributed characteristics of atmospheric turbulence. The computational scheme, which involves the use of artificial neural networks, allows real-time calculation of the aerodynamic characteristics of the aircraft and to ensure training of pilots on flight simulators.
Simulation and Optimization in the Problems of Design of Spherical Layered Thermal Shells
Abstract
This paper describes the inverse problems of heat transfer, arising in designing multilayered spherical cloaking shells and other functional devices for controlling thermal fields. It is assumed that shells are comprised of a finite number of layers, each filled by a homogeneous isotropic or anisotropic medium. An exact solution for a partial case of a single-layer homogeneous anisotropic shell is presented and analyzed. An optimization approach used to reduce the inverse problems under consideration to control problems. A numerical algorithm for solving them is proposed, which is based on a particle swarm optimization, and the results of numerical experiments are discussed.
Crack Opening Models Based on the Exact Solutions of the Navier-Stokes Equations
Abstract
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 mass of the fluid in the crack is used to obtain quasilinear parabolic equations, describing the crack opening.
Invariant Submodels of the Generalized Leith Model of Wave Turbulence in a Medium with Nonstationary Viscosity
Abstract
A generalized phenomenological Leith model of wave turbulence in a medium with unsteady 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.
Group Properties of Equations of the Kinetic Theory of Coagulation
Abstract
Nonlocal equations of the coagulation theory are studied by the group analysis methods. In addition to the integro-differential 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. Admitted Lie groups for the considered equations are found, their relationships are studied, and the corresponding invariant solutions are analyzed.
Evolution of the Horizontal Mixing Layer In Shallow Water
Abstract
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. The bottom friction produces a stabilizing effect and reduces the growth of the mixing layer. 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.
Waves and Structures in the Boussinesq Equations
Abstract
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.
Numerical Implementation of Nonstationary Axisymmetric Problems of an Ideal Incompressible Fluid with a Free Surface
Abstract
A fundamentally new unsaturated technique for the numerical solution of the Dirichlet–Neumann problem for the Laplace equation was developed. 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 is an insurmountable obstacle to any saturated numerical methods, i.e., methods with a leading error term.
Methods for Studying the Sensitivity of Air Quality Models and Inverse Problems of Geophysical Hydrothermodynamics
Abstract
Variational approach and sensitivity theory methods are used to construct algorithms for solving the problems of environmental forecast and design. When studying the behavior of the model in the parameter space, sensitivity functions are calculated as partial derivatives of the target functionals with respect to the model parameters. The sensitivity functions are used to investigate the properties of mathematical models and solve inverse problems. Using the proposed approach, which involves a model of air quality of the Novosibirsk agglomeration and an algorithm based on an ensemble of sensitivity functions, the inverse problem of estimating the position and intensity of pollution sources is solved.