Nonlinear Electrohydrodynamic Stability of a Poiseuille Two–Layer Flow

The long–wave stability of the Poiseuille two–layer flow of homogeneous viscous dielectrics between plate electrodes under a constant potential difference is studied in an electrohydrodynamic approximation. A linear asymptotic stability analysis shows that surface polarization forces are a destabilizing factor, in addition to viscous stratification. The method of many scales is used to obtain the Kuramoto—Sivashinsky equation governing the weakly nonlinear evolution of the interface between the dielectrics. Within the framework of the approaches used, it is shown that nonlinear interactions limit perturbation growth and the interface does not fail even for a rather large potential difference.     

Molecular gas dynamics applied to phase change processes at a vapor–liquid interface: shock-tube experiment and MGD computation for methanol

This paper deals with a molecular gas-dynamics method applied to the accurate determination of the condensation coefficient of methanol vapor. The method consisted of an experiment using a shock tube and computations using a molecular gas-dynamics equation. The experiments were performed in such situations where the shift from a vapor–liquid equilibrium state to a nonequilibrium one is realized by a shock wave in a scale of molecular mean free time of vapor molecules. The temporal evolution in thickness of a liquid film formed on the shock-tube endwall behind a reflected shock wave is measured by an optical interferometer. By comparing the measured liquid-film thickness with numerical solutions for a polyatomic version of the Gaussian–BGK model of the Boltzmann equation, the condensation coefficient of methanol vapor is accurately determined in vapor–liquid nonequilibrium states. As a result, it is clear that the condensation coefficient is just unity very near to an equilibrium state, but is smaller far from the equilibrium state.     

Non-linear waves in a fluid-filled inhomogeneous elastic tube with variable radius

In the present work, by employing the non-linear equations of motion of an incompressible, inhomogeneous, isotropic and prestressed thin elastic tube with variable radius and the approximate equations of an inviscid fluid, which is assumed to be a model for blood, we studied the propagation of non-linear waves in such a medium, in the longwave approximation. Utilizing the reductive perturbation method we obtained the variable coefficient Korteweg–de Vries (KdV) equation as the evolution equation. By seeking a progressive wave type of solution to this evolution equation, we observed that the wave speed decreases for increasing radius and shear modulus, while it increases for decreasing inner radius and the shear modulus.     

Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk

The effect of the Coriolis force on the evolution of a thin film of Newtonian fluid on a rotating disk is investigated. The thin-film approximation is made in which inertia terms in the Navier–Stokes equation are neglected. This requires that the thickness of the thin film be less than the thickness of the Ekman boundary layer in a rotating fluid of the same kinematic viscosity. A new first-order quasi-linear partial differential equation for the thickness of the thin film, which describes viscous, centrifugal and Coriolis-force effects, is derived. It extends an equation due to Emslie et al. [J. Appl. Phys. 29, 858 (1958)] which was obtained neglecting the Coriolis force. The problem is formulated as a Cauchy initial-value problem. As time increases the surface profile flattens and, if the initial profile is sufficiently negative, it develops a breaking wave. Numerical solutions of the new equation, obtained by integrating along its characteristic curves, are compared with analytical solutions of the equation of Emslie et al. to determine the effect of the Coriolis force on the surface flattening, the wave breaking and the streamlines when inertia terms are neglected.     

Rock vibration and finite oil recovery

Basic information concerning the possibility of mechanical stimulation of an oil reservoir is presented. The positive effect of vibration on the oil fraction in the output of flooded wells is demonstrated. The effect is attributed to the restoration of permeability for dispersed oil as a result of drop clusterization or breakdown. A mathematical model illustrating the special role of dominant vibration frequencies is proposed. This model is based on the nonlinear effects associated with internal viscoelastic resonance. The corresponding evolution equation of the seismic waves emitted by the vibrator is a generalization of the Burgers-Korteweg-de Vries equation. For this equation the existence of an asymptotic regular wave structure is proved. Taking the microparticle rotation effect into account leads to bimodal wave vibrations, and under conditions of long-short-wave resonance the nonlinear generation of high ultrasonic frequencies by seismic waves is possible. The ultrasonic vibrations created enable the oil drops to recover their mobility.Based on paper presented to the fluid mechanics section of the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 110–119, September–October, 1992.     

Evolution of quasiplane weakly nonlinear internal waves in a viscous fluid

The method of multiscale asymptotic expansions is used to derive a model equation describing the evolution of internal waves in a viscous stratified fluid with allowance for nonlinearity, dispersion, and diffraction in the diffusion approximation. The approximate analytic solution of the obtained equation in the case of weak nonlinearity is analyzed. The possibility of using the Boussinesq approximation is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 158–162, October–December, 1981.     

Waves in an inhomogeneous plasma with Hall dispersion

The evolution of steady-state periodic solutions of the Korteweg-de Vries equation (the socalled cnoidal waves), propagating along the direction of the gravitational force with an arbitrary orientation of the magnetic field, is studied for plasma characterized by Hall dispersion and Joule dissipation, using the magnetohydrodynamic approximation. The wavelength is regarded as much shorter than the characteristic scale of the inhomogeneity. The dependence of the wave amplitude on the distance to the source of the wave is considered for various limiting cases. The behavior of the wave depends on the temperature distribution in the medium. In the particular case of an isothermal atmosphere, the problem is solved analytically for a cold plasma in the absence of dissipation. The amplitude of both fast and slow waves increases when the wave travels upward and diminishes when the wave travels downward. The nonlinearity of the wave (i.e., the parameter characterizing the deviation of the wave from sinusoidal form) diminishes in the case of fast magnetoacoustic waves when the wave travels upward and increases when the wave travels downward. The situation is reversed for slow magnetoacoustic waves.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 139–144, September–October, 1976.The author is grateful to V. B. Baranov for constant interest in the work and valuable comments.     

Effect of nonlinearity on the development of oscillatory magnetohydrodynamic Kelvin-Helmholtz instability in the weakly supercritical regime

The nonlinear stage of development of perturbations at a tangential magnetohydrodynamic discontinuity is investigated in the weakly subcritical and supercritical regimes. It is assumed that the fluid is incompressible and that the density and magnetic field, as well as the velocity, suffer a discontinuity. An equation describing the evolution of low-amplitude nonlinear perturbations is obtained. For periodic perturbations this equation reduces to an infinite system of ordinary differential equations for the amplitudes of the Fourier harmonics. The system is reduced to finite form by truncation and then integrated numerically. Calculations show that the evolution of an initially sinusoidal perturbation always ends with the appearance in the wave profile of an infinite derivative. This can take the form of either an infinitely sharp peak (knife-edge) or wave breaking.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 30–39, May–June, 1988.     

Quasilongitudinal propagation of narrow beams of nonlinear magnetohydrodynamic waves

An equation, analogous to the Khokhlov-Zabolotskaya equation, is derived for narrow beams of quasitransverse waves propagating at small angles to a magnetic field. The effect of diffraction on wave propagation is investigated in the linear approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 21–28, July–August, 1986.     

Set of steady-state traveling waves on the surface of a liquid film flowing down an inclined plane

The nonlinear evolution equation often encountered in modeling the behavior of perturbations in various nonconservative media, for example, in problems of the hydrodynamics of film flow, is examined. Steady-state traveling periodic solutions of this equation are found numerically. The stability of the solutions is investigated and a bifurcation analysis is carried out. It is shown how as the wave number decreases ever new families of steady-state traveling solutions are generated. In the limit as the wave number tends to zero a denumerable set of these solutions is formed. It is noted that solutions which also oscillate in time may be generated from the steadystate solutions as a result of a bifurcation of the Landau-Hopf type.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–125, November–December, 1989.     

A level set-based Eulerian approach for anisotropic wave propagation

The geometric optics approximation to high frequency anisotropic wave propagation reduces the anisotropic wave equation to a static Hamilton–Jacobi equation. This equation is known as the anisotropic eikonal equation and has three different coupled wave modes as solutions. We introduce here a level set-based Eulerian approach that captures all three of these wave propagations. In particular, our method is able to accurately reproduce the quasi-transverse, or quasi-S, waves with cusps, which form a class of multi-valued solutions. The level set formulation we use is borrowed from one for moving curves in three spatial dimensions, with the velocity fields for evolution following from the method of characteristics on the anisotropic eikonal equation. We present here our derivation of the algorithm and numerical results to illustrate its accuracy in different cases of anisotropic wave propagations related to seismic imaging.     

Diffraction of a sound wave by a nonmoving plate

The problem of determining the velocity field excited by a sound wave impinging on a plate at rest is analyzed as an initial- and boundary-value problem with a movable boundary for the two-dimensional wave equation. The latter problem is solved by the formulation and inversion of integral equations of the Volterra type. The solution is obtained in closed form for any angle of inclination of the incident wave relative to the plate surface and is represented by recursion relations allowing for the influence of any number of diffracted waves generated in succession at the plate boundary.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 123–130, March–April, 1972.     

Propagation of surface gravity waves in a layer of electrically conducting liquid

The wave motion of a weakly conducting incompressible liquid in a transverse magnetic field is investigated within the framework of the nonlinear theory of magnetohydrodynamics. The influence of MHD interaction effects on harmonic perturbations of infinitesimal amplitude is analyzed and a long-wave equation of the Kortewegde Vries-Burgers type describing the evolution of weakly nonlinear perturbations of the free surface is derived. It is shown that the influence of the electrical conductivity leads to a change in both the dissipative and the dispersive properties of the system.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 173–175, July–August, 1989.     

Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.     

Evolution of cylindrical waves of finite amplitude diverging in heated gas

The behavior of weak cylindrical and spherical waves of finite amplitude in a dissipative gas close to the wave front is described by a generalized Burgers equation [1]. The construction of various types of solution of this equation for large Reynolds numbers is known [1–3]. For the evolution of diverging perturbations in heated gas, a study of this equation in the region Re < 1, where Re is the effective Reynolds number at the initial time, is of interest. The direct application of the method of successive approximations to this problem is restricted by the condition Re 1, and becomes more and more difficult as the Reynolds number grows and the form of the initial wave becomes more complex. This paper describes in explicit form the construction of an approximate solution of the Cauchy problem for the generalized Burgers equation in the case of cylindrical symmetry in the region Re < 1. The initial wave selected is the arbitrary perturbation represented by a function which is absolutely integrable on the real axis. An integral estimate of the error as a function of Re is given. The question of how the structure of the solution corresponds to the Cole-Hopf transformation is discussed. All the treatment can easily be extended to the spherically symmetric case.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 150–153, July–August, 1985.     

Distribution of radiant thermal flux over the surface of a sphere for hypersonic flow of a nonviscous radiating gas past the sphere

In the present study using the Newtonian approximation [1] we obtain an analytical solution to the problem of flow of a steady, uniform, hypersonic, nonviscous, radiating gas past a sphere. The three-dimensional radiative-loss approximation is used. A distribution is found for the gasdynamic parameters in the shock layer, the withdrawal of the shock wave and the radiant thermal flux to the surface of the sphere. The Newtonian approximation was used earlier in [2, 3] to analyze a gas flow with radiation near the critical line. In [2] the radiation field was considered in the differential approximation, with the optical absorption coefficient being assumed constant. In [3] the integrodifferential energy equation with account of radiation was solved numerically for a gray gas. In [4–7] the problem of the flow of a nonviscous, nonheat-conducting gas behind a shock wave with account of radiation was solved numerically. To calculate the radiation field in [4, 7] the three-dimensional radiative-loss approximation was used; in [5, 6] the self-absorption of the gas was taken into account. A comparison of the equations obtained in the present study for radiant flow from radiating air to a sphere with the numerical calculations [4–7] shows them to have satisfactory accuracy.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 44–49, November–December, 1972.In conclusion the author thanks G. A. Tirskii and É. A. Gershbein for discussion and valuable remarks.     

Numerical simulation of plastic-flow localization for simple shear

An algorithm was developed to numerically simulate plastic-flow localization for simple shear of a thermally plastic and viscoplastic material. The algorithm is based on solving the partial differential equations describing continuum flow. The closing equation is the constitutive relation known in the literature as the power law linking the plastic-strain rate to the flow stress, temperature, and accumulated plastic strain. Calculated relations for the time evolution of the shear-band width and the temperature and plastic strains localized in it agree satisfactorily with experimental relations. Good agreement with experimental results is also obtained for the sample temperature distribution at the developed stage of the localization process.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 173–180, January–February, 2005     

Nonlinear waves in a viscous compressible fluid with relaxation

The propagation and stability of nonlinear waves in a viscous compressible fluid with relaxation that satisfies a Theological equation of state of Oldroyd type are investigated. An equation that describes the structure of the wave perturbations and its evolution is derived subject to the condition of balance of the nonlinear dissipative and relaxation effects, and its solutions of the solitary wave type are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 31–35, May–June, 1993.     

Generation of Nonlinear Waves on a Viscoelastic Coating in a Turbulent Boundary Layer

Self–induced excitation of periodic nonlinear waves on a viscoelastic coating interacting with a turbulent boundary layer of an incompressible flow is studied. The response of the flow to multiwave excitation of the coating surface is determined in the approximation of small slopes. A system of equations is obtained for complex amplitudes of multiple harmonics of a slow (divergent) wave resulting from the development of hydroelastic instability on a coating with large losses. It is shown that three–wave resonant relations between the harmonics lead to the development of explosive instability, which is stabilized due to the deformation of the mean (Sover the wave period) shear flow in the boundary layer. Conditions of soft and hard excitation of divergent waves are determined. Based on the calculations performed, qualitative features of excitation of divergent waves in known experiments are explained.