Riemannian Submersions with Discrete Spectrum

We prove some estimates on the spectrum of the Laplacian of the total space of a Riemannian submersion in terms of the spectrum of the Laplacian of the base and the geometry of the fibers. When the fibers of the submersions are compact and minimal, we prove that the spectrum of the Laplacian of the total space is discrete if and only if the spectrum of the Laplacian of the base is discrete. When the fibers are not minimal, we prove a discreteness criterion for the total space in terms of the relative growth of the mean curvature of the fibers and the mean curvature of the geodesic spheres in the base. We discuss in particular the case of warped products.     

Nonlinear Electromagnetic Delay of Electromagnetic Signals Propagating in the Magnetic Meridian Plane of Pulsars and Magnetars

We analyze the propagation of electromagnetic waves in the magnetic meridian planes of neutron stars with a strong magnetic field in the framework of the parameterized post-Maxwellian electrodynamics of the vacuum. The origin of these electromagnetic waves is the curvature emission of X-rays and gamma rays from high-energy electrons in the vicinity of the magnetic poles of neutron stars. We show that in the case of a slowly varying intensity of X-ray and gamma-ray emission, the delay of the slow normal mode of electromagnetic waves relative to the fast mode results in a shift of the time dependence of the intensity of the detected radiation with one polarization relative to that of the radiation with the orthogonal polarization. In the case of single X-ray or gamma-ray pulses, the delay effect results in the polarization of the detected pulse varying during the pulse length, the leading edge of all pulses being polarized normally to the magnetic equator plane of the neutron star. We note that the modern level of the experimental technique, in principle, allows observing the manifestations of the delay effect for signals of different polarizations.     

Seepage of a gas condensate mixture when using the geoloosening method

The problem of the seepage of a two-phase multicomponent hydrocarbon mixture for evaluating the efficiency of the use of the geoloosening method in gas condensate deposits is considered. The geoloosening method is a technology for increasing the productivity of wells, developed at the Institute of Problems in Mechanics of the Russian Academy of Sciences, and it ensures an increase in the permeability of the critical zone of a well because of directed relief of the stratum. The initiation of the geoloosening process requires the creation of deep depressions at the well bottom and, as a result, there is an accumulation of retrograde condensate in the neighbourhood of the well, which leads to a decrease in the phase permeability with respect to the gas. It is necessary to take account of the existence of these two processes, which are mutually counter directed from the point of view of the change in permeability, when this method is used for gas condensate deposits. Due to the change in the chemical composition of the mixture in the condensation process and the action of capillary forces, the gas content at each point and each instant is not the equilibrium content and, consequently, cannot be directly determined from the phase diagram of the substance. A differential scheme is used to describe the seepage of the mixture, according to which, unlike an integral scheme, the relation for the transition into the liquid phase is specified for increments and not for the pressure and volume values themselves. Numerical calculations of the steady seepage of a hydrocarbon mixture are carried out for the necessary depression levels for the conditions in the Astrakhan gas condensate deposit and the effectiveness of the use of the geoloosening method there is demonstrated.     

Prediction of the deformation properties of a polymeric composite with a granular-fibrous filler

Conclusion A variant of calculation of the characteristics of the deformation properties of a hybrid composite containing a complex disperse filler in the form of granular particles and short fibers was proposed. The effect of aggregation of the granular filler, the statistical distributions of the fibers by lengths and orientation in the material, and the anisotropy of the fibers are taken into consideration in the calculation. The statistical distribution of the orientation of the fibers is given by a function proportional to the distance from the center to the surface of a triaxial ellipsoid in the corresponding direction. The uniform random distribution of the fibers in bulk and in the plane is a special case of this distribution. The results of the analysis of the effect of the parameters of the statistical distributions of the length and orientation of short fibers on the elasticity characteristics of a composite are reported. The dependence of the creep of the composite on the ratio of the concentration of the components of the complex filler was determined, and the efficiency of partial replacement of a granular filler by a short-fiber filler to inhibit creep of the composite was demonstrated. The possibilities of predicting the long-term creep were experimentally confirmed on the example of LDPE filled with ground limestone and short glass fibers.Presented at the Sixth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, November, 1986).Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 898–909, September–October, 1987.     

The topology of the zero-level lines of the components of the acceleration vector in subsonic flows

The steady subsonic flow past bodies of finite dimensions, when the stream is unbounded and uniform at infinity is considered. The structure formed by the stationary points (points where both components of the acceleration vector vanishes), by the zero-level of the components of the acceleration vector emerging from them and the body past which the flow occurs is studied. It is shown that each of the above-mentioned lines must reach the surface of the body past which the flow takes place. This fact, in particular, enables one to estimate the overall number of streamlines with zero curvature emerging from the stationary points in terms of the number of zeros of the curvature of the streamlines on the body around which the flow takes place, including the branch points of a dividing streamline. With a view to refining the above mentioned number of zeros, the known solution for the neighbourhoods of the branch points of a streamline is considered and the singularity of the flow in the neighbourhoods of points of discontinuity of the curvature of the wall around which the flow occurs is investigated. In order to illustrate the above, certain properties of the flow past convex bodies are refined and a fairly broad class of so-called convex-concave bodies with zero angle of tapering of the trailing edge is constructed and considered. It is shown that, for this body, there are not more than four zeros of the curvature of the streamline and, as a consequence, there are no branch points of the isobars and isoclines in the flow field, including at infinity, an infinitely distant point is the sole stationary point and, most important of all, in the case of the flow past the given bodies the values of the circulation and the lifting force cannot vanish. The mathematical apparatus employed is based on the equations of gas dynamics constructed earlier for certain combinations of the components of the acceleration vector.     

The thermodynamically consistent equations of a thermoelastic saturated porous medium

A phenomenological model of a porous medium saturated with fluid is considered with in the framework of the hypothesis of interpenetrating continua. Assuming that there are no phase transitions, that the contribution of pulsations to the stress tensor and kinetic energy is small, and the components of the medium are in thermal equilibrium, mass, momentum and energy equations and a law of conservation of compatibility of the deformations and velocities are formulated. Using a representation of the force of interaction of the components in the form of the sum of equilibrium and dissipative components, a new form of inequality is obtained for the rate of entropy production. A definition of a thermoelastic saturated porous medium is given. The symmetry group of such a medium is considered as a set of two groups, corresponding to the symmetry of the skeleton and the fluid. It is shown that, in the class of thermoelastic porous media with an arbitrary type of symmetry of the skeleton, the saturating fluid can only be an ideal fluid, while the thermodynamic potentials and the porosity, stresses and entropies determined by them do not depend on the temperature gradient and the relative fluid velocity. It is found that the condition of incompressibility of only one of the components of the medium leads to the elimination of the porosity from the governing relations, rather than to kinematic limitations. The limitations imposed on the governing relations by the principle of thermodynamic consistency and the requirement of independence of the choice of the frame of reference are investigated. A form of the governing relations, necessary and sufficient to satisfy these principles, is obtained. It is shown that the Biot equations are one of the forms of thermodynamically consistent governing relations. A thermodynamic validation of the effective-stress tensor is given.     

An investigation on the influence of slot injection/suction on the spreading of a thin film under gravity on a rotating disk

We investigate the influence of slot injection/suction on the axisymmetric spreading of a thin film under the influence of gravity and rotation. The effects of surface tension are ignored. We allow a very thin film to precede the bulk of the fluid to overcome the singularity which arises as a consequence of applying the no-slip boundary condition. We show how the width of the slot and magnitude of the injection/suction influences the height of ridges and depth of cavities on the profile of the free surface of the thin film. Rotation increases the depth of the cavities and the height of the ridges as compared to the effects of gravity alone. The presence of rotation also results in the formation of a breaking wave.     

Identification and orbit determination of a tethered satellite system

Orbital motion of a tethered satellite system, composed of two satellites and an inextensible tether, is considered by using a perturbed two-body model. This approach is adopted so that the determination of the orbit of one of the satellites can be attempted without using observations of the motion of the other satellite in the system. The identification of the tethered condition of the system using observations of only one of the satellites in the tethered satellite system is considered. The characteristics of the `tether perturbed' motion of the observed satellite are investigated. Estimation of the state of the system using near perfect data is also illustrated. Observations of one satellite provide the entire state of the system and a parameter involving the ratio of the masses of satellites and the tether length.     

Rates of convergence for continued fractions of irrational numbers

The dynamics of the Gauss Map suggests a way to compare the convergence to a real number ζ ε(0,l) of a continued fraction and the divergence of the orbit of ζ Of particular interest is the comparison of the rate of convergence to ζ of its simple continued fraction and the rate of divergence by the Gauss Map of the orbit of ζ for all irrational numbers in (0,l). We state and prove sharp inequalities for the convergence of the sequence of rational convergents of an irrational number ζ. We show that the product of the rate of convergence of the continued fraction of ζ and the rate of divergence by the Gauss Map of the orbit of ζ equals 1.     

The increasingly complex structure of the bifurcation tree of a piecewise-smooth system

A new approach to the study of the dynamics of a piecewise-smooth system is proposed, which uses the a priori known possible bifurcation structures of the parameter space. In Section 1 the synthesis of the structures of the bifurcation tree of the system is considered, namely, the local structures, bifurcation bands, sources and nodes. It is shown that a node corresponding to a doubling bifurcation with reorientation of the domain of existence can generate a sequence of increasingly complex structures. Then the increasing number of unstable orbits serves as one of the mechanisms giving rise to the chaotic behaviour of the dynamical system. In Section 2 the procedure for synthesizing the structures of the bifurcation tree of a piecewise-smooth system proposed in the first part of the paper is applied to the problem of the forced vibrations of a linear oscillator with impacts against a stopping device. Period-doubling cascades are discovered, which are accompanied by the reorientation of the domain of existence of a solution relative to some bifurcation surface, namely, the trunk of the tree. A set of frequency intervals is distinguished on the bifurcation trunk, each containing an infinite sequence of increasingly complex local structures appearing and disappearing at the nodes. This specific mechanism, giving rise to the chaotic motion of the oscillator, is realized in neighbourhoods of the limiting nodal bifurcation points.     

Harmonic distrotions under nonlinear periodic deformation of fluid polymers

Conclusions 1. It has been shown for a number of viscoelastic fluid systems that under nonlinear periodic deformation, the contribution of the third harmonic of the stress to the fundamental does not exceed 20% of the amplitude.2. In the case of clay soil and melt of filled polyethylene, the shape of the stress waves is essentially definable by the relative phase angle of the third harmonic of the stress and is practically independent of the deformation amplitude in a growing nonlinear range of deformation.3. In the case of the polyethylene melt, the amplitude dependence of the phase angles of the stress harmonics is in satisfactory agreement with the analysis of model I. With increasing deformation amplitude, the modulus vector of the first harmonic rotates counterclockwise and remains in the first trigonometric quadrant; the modulus vector of the third harmonic passes from the second to the third quadrant, and the modulus vector of the fifth harmonic passes from the second to the fourth quadrant via the third.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 893–898, September–October, 1978.     

基于综合重要度的动车组多部件系统机会维修策略研究

针对我国动车组列车现行维修方式,提出基于综合重要度序列的动车组多部件系统机会维修策略,对提高系统可靠度贡献大的关键部件进行准时优先维修。建立部件综合重要度指数计算模型,并依据其对部件维修优先级进行排序。以维修总成本最低为目标计算单部件最优维修周期及时刻,以系统维修总成本最低为目标,以关键部件的维修时刻为系统停机时刻建立考虑重要度的多部件系统机会维修模型。算例选取某型动车组四级修时更换的四部件系统为研究对象,讨论机会维修里程窗的大小及其偏移量对维修效果的影响,对比结果表明,考虑综合重要度的机会维修策略能够在维修费用基本持平的条件下,保证对系统可靠性贡献大的关键部件的可靠性,进而保证系统的整体可靠性。     

Flow of thin liquid film over a rough rotating disk in the presence of a transverse magnetic field

The development of a flow of a viscous conducting fluid over a rough spinning disk in the presence of a transverse magnetic field has been analysed for different patterns of surface roughness of the disk and different initial distributions of the height of the liquid lubricant. The numerical solution of the governing equation of motion subject to initial and boundary conditions has been obtained by a finite-difference method. The temporal evolution of the free surface of the fluid and the rate of retention of the liquid lubricant on the spinning disk have been obtained for different values of the two parameters M , the Hartmann number and N_ratio, the ratio of the surface tension effect to the centrifugation effect. In the absence of the magnetic field, the results have been observed to agree with those of [6]. It has been observed that the effect of surface roughness is to enhance the relative volume of the fluid retained on the spinning disk and this is further enhanced by the presence of the magnetic field.     

Longitudinal waves in partially saturated porous media: the effect of gas bubbles

The problem of the propagation of longitudinal waves in a liquid-saturated porous medium when there are gas bubbles present is considered. The decay factor and the phase velocity of Frenkel–Biot waves of the first and second kind are found as a function of the frequency in the linear approximation. It is shown that, in the neighbourhood of the resonance frequency of the bubbles, longitudinal Frenkel–Biot waves change their form. A wave of the first kind is transformed from a fast wave at low frequencies into a slow wave at high frequencies. The dispersion curve of a wave of the second kind consists of two branches – a “low-frequency” branch, the oscillations of which possess the classical properties, and a “high-frequency” branch, which is a weakly decaying high-velocity mode. The frequency dependences of the ratio of the mass velocities of a gas-liquid mixture and of a porous matrix, and also of the perturbations of the stress in the matrix and the pressure in the mixture, are constructed. It is shown that the “high-frequency” branch of a wave of the second kind is characterized by the in phase motion of the gas-liquid mixture and of the porous matrix, while their mass velocities are close, which explains the weak decay of this mode of oscillations. An analytical expression is obtained for the “boundary frequency”, which determines the offset of the “high-frequency” branch of the dispersion curve of the wave of the second kind.     

Influence of the choice of a rheological law on computer simulation results of slab subduction

The influence of the choice of the type of yield surface for elastoplastic materials and material constants for the slab and the mantle on the scenario of mathematical modeling of the slab collision is investigated. Computer simulation is performed by the FEM numerical solution of nonlinear equations for deformable solid mechanics, using the MSC.Marc 2005 code. The simulation results essentially depend on the choice of material constants for the slab and the mantle, as well as on the type of the yield surface for the elastoplastic material of the subduction slab. The numerical simulations demonstrate that the primary driving mechanism of subduction can be a geometrical inhomogeneity of the subduction slab near the zone of the slab collision with simultaneous consideration of consolidation of the slabmaterial as the slab descends into the mantle.     

An axisymmetric model of the Mush-Chimney system

We develop a model of the mush-chimney system based on the assumption that the order of magnitude of the vertical velocity component in the chimney is much greater than the speed of upward advance of the mush, and making use of the fact that the radius of chimney is much less than the (dimensionless) thickness of the mush. We determine the fluid flow structure in the chimney by utilizing the knowledge of the mass fraction of light constituent, the vertical velocity component, and the pressure that are obtained from the mush. We find a relation between a parameter measuring the ratio of viscous and buoyancy forces in the chimney and the vertical velocity component on the top of the mush, and estimate numerically the value of this velocity.     

On the boundary integral formulation of the plane theory of elasticity with applications (analytical aspects)

A formulation of the plane strain problem of the theory of elasticity in stresses, for simply connected domains, is carried out in terms of real functions within the frame of what is known as the boundary integral method. Special attention is devoted to the problem of determination of the arbitrary constants appearing in the solution, in view of work in progress where numerical techniques are used. Relying on some mathematical results formulated in the appendix, simple applications concerning the first and the second fundamental problems for the circle and for the ellipse are given, which show the correctness of the formulation and the necessity of recurring to numerical techniques, once the geometry of the problem or the type of boundary conditions deviates from being simple. Following parts of the present work are devoted to the numerical treatment of the obtained system of equations, as well as to the theories of thermoelasticity and thermo-electromagneto-elasticity.