Nonlinear waves on the surface of a charged liquid. Instability,bifurcation, and nonequilibrium shapes of the charged surface

Plane nonlinear waves in shallow water are described by the Kortewegde Vries equation [1–3]. The present paper contains theoretical investigations of nonlinear waves and nonlinear equilibrium shapes on the surface of a charged liquid. The influence of the field on the velocity and shape of a hydrodynamic soliton is considered. The bifurcation of the equilibrium shapes is investigated. Problems of the equilibrium shapes of a charged liquid are solved in the nonlinear formulation of the dynamics of nonlinear solitary forms (lunes, trenches) on the surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–102, May–June, 1984.     

Propagation of modulated nonlinear waves in a relaxing gas-liquid mixture

The propagation of nonstationary weak shock waves in a chemically active medium is essentially dispersive and dissipative. The equations for short-wavelength waves for such media were obtained and investigated in [1–4]. It is of interest to study quasimonochromatic waves with slowly varying amplitude and phase. A general method for obtaining the equations for modulated oscillations in nonlinear dispersive media without dissipation was proposed in [5–8]. In the present paper, for a dispersive, weakly nonlinear and weakly dissipative medium we derive in the three-dimensional formulation equations for waves of short wavelength and a Schrödinger equation, which describes slow modulations of the amplitude and phase of an arbitrary wave. The coefficients of the equations are particularized for the considered gas-liquid mixture. Solutions are obtained for narrow beams in a given defocusing medium as well as linear and nonlinear solutions in the neighborhood of a diffraction beam. A solution near a caustic for quasimonochromatic waves was found in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–143, January–February, 1980.     

Sphere scattering of nonlinearly interacting acoustic waves

Sphere scattering of the field of nonlinearly interacting plane acoustic waves when the sphere is located in the region of nonlinear interaction between the primary pumping waves of a parametric antenna is considered. An analytic expression for the secondary field pressure at the difference frequency is obtained. This expression describes the process of nonlinear interaction of the incident and scattered waves. The secondary-field total pressure components, which characterize the interaction between the incident plane waves and scattered spherical waves are analyzed. The numerical results and experimental data are given.Taganrog. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 4–12, March–April, 1995.     

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.     

Determination of the turbulent Prandtl number in an asymptotic boundary layer with suction

The problem of heat transfer in a turbulent asymptotic boundary layer with suction is solved in the framework of the monoharmonic model. The flow is one dimensional on the average, which is why it is chosen for investigation. The theoretically determined mean and pulsation characteristics of the flow, in particular the turbulent Prandtl number, agree with the experimental results for a boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 74–79, January–February, 1981.     

Nonlinear evolution of waves in forced decaying capillary jets

The investigation of nonlinear waves in decaying capillary jets is of great interest both from the point of view of nonlinear wave processes in media and for practical applications associated with the generation and propagation of flows of monodisperse droplets [1–4]. The formation and dynamics of satellite droplets are particularly important in the study of the decay of thin capillary jets [5–8]. Investigation of the conditions of formation of satellites open up important prospects for the preparation of monodisperse microscopic granules with diameters appreciably less than the diameter of the original jet. This is of great importance in modern technologies based on the use of materials in disperse form [9–13]. The present paper is devoted to the investigation of nonlinear waves in decaying capillary jets.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 54–60, May–June, 1993.     

Interaction of wall turbulence with a compliant surface

The effect of various parameters of the compliant surface on the interaction with wall turbulence is analyzed using the monoharmonic approximation. It is shown that the interaction is resonant in character and that for certain values of the parameters a considerable reduction in turbulent skin friction is possible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 67–72, July–August, 1990.     

New Solitary Waves in Elastic Materials: Existence and Nonlinear Interaction

Waves mentioned in the title were revealed in composite materials that are described by the microstructural theory of the second order — the theory of two-phase mixtures. For harmonic periodic waves, a mixture is always a dispersive medium. This medium admits existence of other waves — waves with profiles described by functions of mathematical physics (the Chebyshov–Hermite, Whittaker, Mathieu, and Lamé functions). If the initial profile of a plane wave is chosen in the form of the Chebyshev–Hermite or Whittaker function, then the wave may be regarded as an aperiodic solitary wave. The dispersivity of a mixture as a nonlinear frequency dependence of phase velocities transforms for nonperiodic solitary waves into a nonlinear phase-dependence of wave velocities. This and some other properties of such waves permit us to state that these waves fall into a new class of waves in materials, which is intermediate between the classical simple waves and the classical dispersion traveling waves. The existence of these new waves is proved in a computer analysis of phase-velocity-versus-phase plots. One of the main results of the interaction study is proof of the existence of this interaction itself. Some features of the wave interaction — triplets and the concept of synchronization — are commented on     

Reflection of a Dam–Break Wave at a Vertical Wall. Numerical Modeling and Experiment

Results of the experimental study and numerical modeling of the reflection of a dam–break wave at the vertical end wall of a channel are given. A wave forms with distance from a partition creating the initial level difference of the liquid. It is shown that a numerical calculation based on the Zheleznyak—Pelinovskii nonlinear dispersion model satisfactorily describes the height of the splash–up, the amplitude of reflected waves, and the wave velocity in front of the wall for smooth and dam–break waves. It is also shown that, for smooth and weakly breaking (without significant entrainment of air) incoming waves, the experimental values of the height of the splash–up at the wall agree well with relevant experimental and calculated data for solitary waves.     

The propagation of nonlinear waves in a bidisperse fluidized bed

A study is made of the propagation of nonlinear kinematic waves of concentrations of solid particles in a fluidized bed of particles of two different sizes. A hyperbolic system of quasilinear equations is obtained which describes the propagation of the waves. A dependence of the characteristic velocities on the concentrations of the phases and the ratio of the sizes of the particles is found. The influence of an admixture of fine particles on the propagation of porosity waves in the fluidized bed is analyzed. The nature of the formation of jumps in the porosity depending on the concentration of the admixture is studied, as is the process of the transfer of the admixture of fine particles in the bed. The nature of the propagation of nonlinear waves in a fluidized bed of identical particles is clarified. A characteristic velocity is found and conditions are determined for the formation of discontinuities of concentration of the dispserse phase in rarefaction compression waves.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 49–58, January–February, 1985.     

Existence and uniqueness of a solution to a problem of steady composite waves on the surface of a liquid flowing over a wavy bottom

Composite waves on the surface of the stationary flow of a heavy ideal incompressible liquid are steady forced waves of finite amplitude which do not disappear when the pressure on the free surface becomes constant but rather are transformed into free nonlinear waves [1]. It will be shown that such waves correspond to the case of nonlinear resonance, and mathematically to the bifurcation of the solution of the fundamental integral equation describing these waves. In [2], a study is made of the problem of composite waves in a flow of finite depth generated by a variable pressure with periodic distribution along the surface of the flow. In [3], such waves are considered for a flow with a wavy bottom. In this case, composite waves are defined as steady forced waves of finite amplitude that, when the pressure becomes constant and the bottom is straightened, do not disappear but are transformed into free nonlinear waves over a flat horizontal bottom. However, an existence and uniqueness theorem was not proved for this case. The aim of the present paper is to fill this gap and investigate the conditions under which such waves can arise.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–98, July–August, 1980.     

Nonlinear waves in short-term injection of a non-Newtonian fluid into a stratum

The one-dimensional nonlinear pressure waves excited in short-term (instantaneous) injection of finite mass of power-law non-Newtonian fluid into a stratum (or instantaneous extraction from the stratum) are analyzed. The propagation laws for plane, cylindrical, and spherical waves in closed (pressurized) and open (unpressurized) strata are obtained.Baku. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 89–96, May–June, 1996.     

Complete model of the process of propagation of long waves and their interaction with a vertical wall

Asymptotic analysis of the nonlinear three-dimensional boundary-value problem of potential theory is carried out and a complete system of equations describing the process of propagation of long surface waves is obtained. Approximate solutions of the problems for both traveling and standing waves are constructed to the third approximation.Translated by Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 173–176, May–June, 1985.     

Nonresonance parametric interactions of surface waves in isotropic solid bodies

A solution in parametric approximation is given of the nonlinear interaction problem of Rayleigh surface waves propagating in a solid body with given elastic fields. Shortened equations that govern the modulation effect of the surface waves, and also expressions for the modulation index in terms of the third-order elasticity constant, are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 163–172, July–August, 1973.     

Generation of higher electron cyclotron frequency harmonics in plasma with a beam

We examine nonlinear excitation of the higher electron-cyclotron frequency harmonics for waves propagating perpendicular to an external uniform magnetic field in a Maxwell plasma for the case of low-density electron beam passage through the plasma. It is shown that the nonlinear excitation mechanism leads to the possibility of generating cyclotron harmonics for plasma parameters for which generation does not occur from the linear theory viewpoint. The nonlinear cyclotron harmonic generation increments are calculated for nonlinear scattering by the beam and plasma electrons of the high frequency longitudinal waves excited in the plasma by the beam.Translated from Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 40–51, November–December, 1969.The author wishes to thank V. N. Tsytovich for posing the problem and for many discussions of the questions touched upon in the article.     

Nonlinear electric field oscillations in continuous media

Strong nonlinear potential and drift-velocity waves in a nonequilibrium medium, in which the current-carrier collision frequency depends on the electric field, are considered in the hydrodynamic approximation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 55–57, May–June, 1969.     

The initialization of nonlinear waves using an adjustment scheme

A procedure for initializing nonlinear free-surface simulations is developed and validated. Numerical simulations of nonlinear progressive waves are prone to developing spurious high-frequency standing waves unless the flow field is given sufficient time to adjust. An adjustment procedure is developed that allows nonlinear free-surface simulations to be initialized with linear solutions. The adjustment scheme allows the natural development of nonlinear self-wave (locked modes) and wave–wave (free modes) interactions. The implementation of the adjustment procedure is illustrated using a high-order spectral method. Comparisons are made to fully-nonlinear Stokes waves and Schrodinger theory.     

Nonlinear steady-state oscillations of an elastic plate floating on the surface of a liquid of finite depth

In addition to obtaining solutions by the perturbation method it is shown that in the case of nonlinear wave interaction given a certain relationship between the parameters of the interacting waves steady-state compound waves may exist.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 146–154, May–June, 1989.     

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.