Vol 300, No 1 (2018)

An application of the symmetries of fundamental solutions in continuum mechanics is presented. It is shown that the Riemann function of a second-order linear hyperbolic equation in two independent variables is invariant with respect to the symmetries of fundamental solutions, and a method is proposed for constructing such a function. A fourth-order linear elliptic partial differential equation is considered that describes the displacements of a transversely isotropic linear elastic medium. The symmetries of this equation and the symmetries of the fundamental solutions are found. The symmetries of the fundamental solutions are used to construct an invariant fundamental solution in terms of elementary functions.

The work is devoted to the stability analysis of the flow of a non-Newtonian powerlaw fluid in an elastic tube. Integrating the equations of motion over the cross section, we obtain a one-dimensional equation that describes long-wave low-frequency motions of the system in which the rheology of the flowing fluid is taken into account. In the first part of the paper, we find a stability criterion for an infinite uniform tube and an absolute instability criterion. We show that instability under which the axial symmetry of motion of the tube is preserved is possible only for a power-law index of n < 0.611, and absolute instability is possible only for n < 1/3; thus, after the loss of stability of a linear viscous medium, the flow cannot preserve the axial symmetry, which agrees with the available results. In the second part of the paper, applying the WKB method, we analyze the stability of a tube whose stiffness varies slowly in space in such a way that there is a “weakened” region of finite length in which the “fluid–tube” system is locally unstable. We prove that the tube is globally unstable if the local instability is absolute; otherwise, the local instability is suppressed by the surrounding locally stable regions. Solving numerically the eigenvalue problem, we demonstrate the high accuracy of the result obtained by the WKB method even for a sufficiently fast variation of stiffness along the tube axis.

We derive a quasi-one-dimensional energy equation that corresponds to the flow of a compressible viscous fluid in a deformable pipeline. To describe the flow of such a fluid in a pipeline, we couple this equation with the previously derived continuity and momentum equations as well as with the equations of state for the internal energies of the fluid, the pipe deformations, pressure, and the cross-sectional area of the pipe. The derivation of the equations is based on averaging over the pipeline cross section. The equations obtained are designed for numerical simulations of long-distance transportation of a compressible fluid.

The following self-similar problem is considered. At the initial instant of time, a phase transformation front starts moving at constant velocity from a certain plane (which will be called a wall or a piston, depending on whether it is assumed to be fixed or movable); at this front, an elastic medium is formed as a result of solidification from a medium without tangential stresses. On the wall, boundary conditions are defined for the components of velocity, stress, or strain. Behind the solidification front, plane nonlinear elastic waves can propagate in the medium formed, provided that the velocities of these waves are less than the velocity of the front. The medium formed is assumed to be incompressible, weakly nonlinear, and with low anisotropy. Under these assumptions, the solution of the self-similar problem is described qualitatively for arbitrary parameters appearing in the statement of the problem. The study is based on the authors’ previous investigation of solidification fronts whose structure is described by the Kelvin–Voigt model of a viscoelastic medium.

Within the framework of a detailed kinetic mechanism of chemical interaction, we study detonation combustion of a stoichiometric hydrogen–air mixture flowing at a supersonic velocity into a plane symmetric channel with narrowing cross section (constriction). The goal of the study is to determine conditions that guarantee the formation of a thrust-producing flow with a stabilized detonation wave in the channel. With a view to increasing the efficiency of detonation combustion of the gas mixture, we analyze the effect of variations of the inflow Mach number, the concentration of inert particles added to the combustible mixture flowing into the channel, and the geometric parameters of the channel on the position of a stabilized detonation wave in the flow. We also study the possibility of formation without energy expenditure of a thrust-producing flow with a stabilized detonation wave in a channel with narrowing cross section.

A Couette flow of a viscoelastic medium is considered that is described by the Johnson–Segalman–Oldroyd model with two relaxation times. The development of singularities related to the appearance of internal discontinuities is studied both analytically and numerically within one-dimensional nonstationary hyperbolic models of viscoelastic Maxwell-type media. A numerical model for calculating nonstationary one-dimensional discontinuous solutions is constructed, discontinuous solutions are studied, and the hysteresis phenomenon, i.e., the dependence of the structure of a steady Couette flow on the prehistory of its formation, is analyzed.

Nonlinear spatial oscillations of a material point on a weightless elastic suspension are considered. The frequency of vertical oscillations is assumed to be equal to the doubled swinging frequency (the 1 : 1 : 2 resonance). In this case, vertical oscillations are unstable, which leads to the transfer of the energy of vertical oscillations to the swinging energy of the pendulum. Vertical oscillations of the material point cease, and, after a certain period of time, the pendulum starts swinging in a vertical plane. This swinging is also unstable, which leads to the back transfer of energy to the vertical oscillation mode, and again vertical oscillations occur. However, after the second transfer of the energy of vertical oscillations to the pendulum swinging energy, the apparent plane of swinging is rotated through a certain angle. These phenomena are described analytically: the period of energy transfer, the time variations of the amplitudes of both modes, and the change of the angle of the apparent plane of oscillations are determined. The analytic dependence of the semiaxes of the ellipse and the angle of precession on time agrees with high degree of accuracy with numerical calculations and is confirmed experimentally. In addition, the problem of forced oscillations of a spring pendulum in the presence of friction is considered, for which an asymptotic solution is constructed by the averaging method. An analogy is established between the nonlinear problems for free and forced oscillations of a pendulum and for deformation oscillations of a gas bubble. The transfer of the energy of radial oscillations to a resonance deformation mode leads to an anomalous increase in its amplitude and, as a consequence, to the break-up of a bubble.

We construct a one-dimensional nonstationary model of explosive degassing for a volcano with plugged conduit. The motion of the plug is assumed to be due to the presence of a gas cavity under it into which a gas flows from the underlying magma, which increases the pressure. We show that periodic explosions of gas occur in the system that are separated by long quiescent stages, which agrees with the data of field observations.

We consider a flow of a fluid in a long vertical tube with elastic walls and show that, for certain parameters of the flow, small perturbations of the flow at the inlet section of the tube give rise to roll waves. Depending on the properties of the closing relation, either regular or anomalous roll waves are formed. In the latter case, a roll wave is characterized by two strong discontinuities that connect regions of continuous flow. We present the results of numerical simulations of the development of a pulsatile flow mode for convex and nonconvex closing relations that demonstrate the formation of regular and anomalous roll waves. We also construct a two-parameter class of exact periodic solutions and obtain existence diagrams for roll waves.

The paper is devoted to the mathematical modeling of the dynamics of geophysical flows on mountain slopes, e.g., rapid landslides, debris flows, avalanches, lava flows, etc. Such flows can be very dangerous for people and various objects. A brief description is given of models that have been used so far, as well as of new, more sophisticated, models, including those developed by the authors. In these new models, nonlinear rheological properties of the moving medium, entrainment of the underlying material, and the turbulence are taken into account. The results of test simulations of flows down long homogeneous slopes are presented, which demonstrate the influence of rheological properties, as well as of turbulence and mass entrainment, on the behavior of the flow.