A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

Study of three-dimensional wave structure of nonstationary gas outflow from a planar sonic nozzle

Results are reported of an experimental study of the three-dimensional structure of nonstationary gas outflow from a planar nozzle. Outflow of a heated shock wave in a nitrogen tube at different moments of time from the start of outflow (0–1 msec) in two mutually perpendicular directions is considered. A scheme for reconstructing the flow at different outflow stages is proposed. The dimensions of the Riemann wave are found to oscillate.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 41–45, January–February, 1976.     

Properties of shock adiabats in the neighborhood of jouguet points

Two well-known properties of shock adiabats in a gas [1] are proved for shock adiabats corresponding to discontinuous solutions of hyperbolic systems of equations expressing conservation laws. If the state on one side of a discontinuity is fixed, then at the point of extremum of the discontinuity velocity on the shock adiabat the velocity of the discontinuity is equal to one of the velocities of the characteristics on the other side of the discontinuity and vice versa. If for the systems there is defined an entropy flux or mass density of entropy, then at the points of extremum of the velocity there is an extremum of the entropy production at the discontinuity and the entropy mass density. If the system is a symmetric hyperbolic system [2, 3], then the extrema of the entropy production at the discontinuity correspond to extrema of the velocity. These properties may be helpful in the study of discontinuities in complex media, since the sections of a shock adiabat whose points can correspond to actually existing discontinuities are frequently bounded by points corresponding to discontinuities whose velocity is equal to the velocity of a characteristic on one of the sides of the discontinuity (see, for example, [1, 4, 5]).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 184–186, March–April, 1979.     

Development of perturbations at the interface between two liquids

The asymptotic behavior is studied in the case of large times of initially localized, one-dimensional, small perturbations of the interface between two liquids in the presence of a tangential velocity discontinuity, taking account of surface tension and the force of gravity. The asymptotic behavior of the perturbed region is found; i.e., on the plane x, t a sector is shown with vertex at the origin of the coordinates, inside of which the perturbations tend to infinity with increase of t, and outside of which the perturbations tend to zero, and the velocities of motion of the boundaries of the perturbed region are calculated. The conditions are shown for which the instability of the tangential discontinuity will not be absolute; i.e., when they are fulfilled, flows with a tangential velocity discontinuity can occur. For the case where the effect of the force of gravity can be neglected, these conditions are independent of the magnitude of the surface tension.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 46–49, September–October, 1977.     

Concerning the description of long nonlinear waves in channels

A method of deriving the equations that describe long nonlinear waves in channels of arbitrary cross section, taking the transverse acceleration of fluid particles into account (the Boussinesq approximation), is proposed. For channels of certain cross sections the equations are written in explicit form. In the case of narrow channels the Boussinesq equations and those of the next approximation are written in explicit form for arbitrary cross sections.     

Occurrence of periodic regimes in steady supersonic mid flows due to loss of electrical conductivity of medium

A study is made of the features of supersonic magnetohydrodynamic (MHD) flows due to the vanishing of the electrical conductivity of the gas as a result of its cooling. The study is based on the example of the exhausting from an expanding nozzle of gas into which a magnetic field (Re_m 1) perpendicular to the plane of the flow is initially frozen. It is demonstrated analytically on the basis of a qualitative model [1] and by numerical experiment that besides the steady flow there is also a periodic regime in which a layer of heated gas of electric arc type periodically separates from the conducting region in the upper part of the nozzle. A gas-dynamic flow zone with homogeneous magnetic field different from that at the exit from the nozzle forms between this layer and the conducting gas in the initial section. After the layer has left the nozzle, the process is repeated. It is established that the occurrence of such layers is due to the development of overheating instability in the regions with low electrical conductivity, in which the temperature is approximately constant due to the competition of the processes of Joule heating and cooling as a result of expansion. The periodic regimes occur for magnetic fields at the exit from the nozzle both greater and smaller than the initial field when the above-mentioned Isothermal zones exist in the steady flow. The formation of periodic regimes in steady MHD flows in a Laval nozzle when the conductivity of the gas grows from a small quantity at the entrance due to Joule heating has been observed in numerical experiments [2, 3]. It appears that the oscillations which occur here are due to the boundary condition. The occurrence of narrow highly-conductive layers of plasma due to an initial perturbation of the temperature in the nonconducting gas has previously been observed in numerical studies of one-dimensional flows in a pulsed accelerator [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 138–149, July–August, 1985.     

Asymptotic laws of damping of weak continuous and shock waves in dust-laden gas

Analytic investigations into the damping of perturbations in dust-laden gas have been restricted to self-similar flows [1, 2] and flows with a symmetry plane, it being assumed in the latter case that thermal and velocity equilibrium of the phases is established instantaneously [3–6], i.e., the relaxation time of the medium is short. In the present paper, asymptotic laws of damping are obtained for plane, cylindrical, and spherical shock and continuous waves whose amplitude and width are such that the acceleration of the particles and the change in their temperature can be ignored. It is assumed that between the phases there is heat transfer proportional to the temperature difference and frictional momentum transfer proportional to the difference between the velocities of the phases. The obtained laws of damping of plane waves are found to be entirely analogous to the laws of damping of magnetohydrodynamic waves in a medium with finite conductivity, when the presence of Joule dissipation and the additional ponderomotive force in the traveling wave or in the gas flow behind the shock wave leads to exponential damping of the wave amplitude [7–9].     

On behavior of the small perturbations of one-dimensional steady transonic flows

Behavior of unsteady perturbations of steady solutions of quasilinear hyperbolic or parabolic degenerate systems of differential equations in partial derivatives is considered in the critical point neighborhood. The sought functions of analyzed equations are assumed dependent on two arguments, viz. coordinate x and time t., with an arbitrary number of sought functions. The point at which one of the system characteristic velocities vanishes, is called critical.