Nonlinear waves in a liquid film on a near-horizontal surface

Nonlinear waves in a liquid film on a slightly inclined rigid plane are studied. A mathematical model is reduced to a system of two evolutionary equations for the layer thickness and the local fluid mass flow. In addition to viscous forces, gravity, and surface tension, the pressure difference over the layer thickness, induced by the gravity force projection on the normal to the underlying surface, is also taken into account. Spatially periodic solutions developing with time from small initial disturbances into regular nonlinear waves are considered. A spectral representation of the solution, the Galerkin method with respect to the uniform coordinate, and subsequent numerical calculation of the corresponding dynamic system on large time intervals are employed. Different variants in the space of the three governing parameters are calculated and some basic mechanisms of nonlinear dynamics of the two-dimensional waves are detected. The calculation results are compared with the existing experimental data. It is shown that the theoretical conclusions can be used to interpret and predict experiments.     

Instability of monochromatic waves on the surface of a liquid of arbitrary depth

A study is made of the stability, with respect to the spontaneous appearance of modulation, of steady periodic waves of small amplitude on the surface of an ideal liquid depth. The well-known instability of waves on the surface of a liquid of infinite depth disappears on making the transition to small depths.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 45–49, September–October, 1970.     

Nonlinear resonance interaction between three capillary-gravity waves on the plane charged fluid surface

Three-wave interaction between capillary-gravity waves on a uniformly charged free fluid surface is analyzed using second-order analytic calculations. The time evolution of the wave amplitudes in the state of nonlinear resonance is studied. It is shown that the number of three-wave resonances is infinite and their exact locations for waves of finite amplitude depend on the initial conditions.     

Nonlinear waves on the free surface of an electrically conducting liquid

Gravity waves of small but finite amplitude on the free surface of an electrically conducting liquid are examined. For the waves whose propagation is described by the Korteweg-de Vries equation (in the absence of a magnetic field), equations are derived. In addition to nonlinearity and dispersion, these equations include the influence of applied magnetic fields. As an example, the effect of magnetic damping on a solitary wave is presented.     

Nonlinear standing waves on the surface of a viscous fluid

Standing surface waves in a viscous infinite-depth fluid are studied. The solution of the problem is obtained in the linear and quadratic approximations. The case of long, as compared with the boundary layer thickness, waves is analyzed in detail. The trajectories of fluid particles are determined and an expression for the vorticity is derived.     

Instability of a liquid surface with laser heating

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 28–34, May–June, 1991.     

Nonlinear surface waves on cubic materials

The effect of nonlinearity on the propagation of surface Rayleigh waves on an elastic medium of cubic symmetry is examined by applying the results of an earlier analysis for general anisotropic materials. Explicit calculations are given of the parameters relating to the generation of higher harmonics in an initially sinusoidal wave for a series of thirty-five materials for which experimental measurements of the third-order moduli are available.     

Nonlinear capillary waves on a viscous fluid jet surface

Capillary instability of a fluid jet is one of the classical problems of hydrodynamics [1]. Studying it is of practical interest, particularly for the optimization of the ignition of a liquid propellant and the development of granulating apparatus in the chemical industry [2]. Until recently, the main attention has been paid to analyzing linear problems. Dispersion equations have been obtained for small perturbations of a jet surface with the viscosity of the external medium taken into account [3]. The construction of a theory of finite-amplitude waves on an ideal fluid jet surface was started in [4, 5]. Up to now this theory has achieved substantial results, as can be assessed by the successful numerical modeling of the dissociation of an inviscid fluid jet into drops [6] (see [7, 8] also). This paper is devoted to a discussion of the nonlinear development stage of viscous fluid jet instability under conditions allowing the influence of the surrounding medium and the gravity field to be neglected.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 179–182, March–April, 1977.The author is grateful to B. M. Konyukhov and G. D. Kuvatov for suggesting this problem and performing the experiment and to M. I. Rabinovich for useful discussions.     

Progressive surface waves on a liquid of non-uniform depth

The propagation of harmonic progressive waves on the surface of a homogeneous liquid of non-uniform depth is considered. Particular attention is devoted to the action of the Reynolds stresses in generating a mean or time-independent component in the flow.     

Nonlinear surface waves on a ferromagnetic fluid of variable depth

A nonlinear ray method is used to study surface waves on a ferromagnetic fluid of variable depth subject to a horizontal magnetic field, and an equation of the KdV type with variable coefficients is derived. An approximate solution of the equation representing a three-dimensional soliton with varying amplitude and phase is constructed and numerical results are presented.     

Mutual effect of long waves and turbulence on the surface of a liquid

The interaction of a long coherent wave with the turbulence on the surface of a liquid is investigated within the framework of the theory of weak turbulence. A closed system of equations is obtained which consists of the dynamic equation for the coherent wave and equations of kinetic type describing the turbulent subsystem. It is shown that because of the interaction with the turbulent subsystem, coherent waves with wave vectors identical in magnitude but opposite in direction are coupled. The additional attenuation of the coherent wave because of the interaction is estimated; this attenuation may be considerably greater than that caused by molecular viscosity. A change in the spectrum of height correlators of the liquid surface is seen in the presence of a coherent wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 100–109, January–February, 1973.     

Nonlinear analysis of the finite-amplitude oscillations of a charged liquid layer on a rigid spherical core for the multimodal initial deformation of the free surface

Within the framework of the analytical asymptotic procedure, in the quadratic approximation in the small dimensionless amplitude of the capillary oscillation of a charged electroconductive fluid layer on the surface of a rigid spherical core, the problem of calculation of the free surface shape at an arbitrary moment is solved for a multimodal initial deformation. The conditions under which the internal nonlinear resonant interaction between the liquid layer oscillation modes and the intermodal energy exchange are realized are analyzed for both the degenerate and the secondary combinational resonances.     

Symmetric shapes of contact of a jet with the surface of a heavy liquid

The plane linearized problem of oblique impingement of a weightless jet of an ideal incompressible fluid on the surface of a heavy fluid is considered. Flows are sought with symmetric forms of the contact region. Mathematically we arrive at the problem of the eigenvalues and eigenfunctions of an integral equation; solving this problem we obtain various contact forms. The fundamental result for the infinitesimally thin jet of finite intensity is derived by passing to the limit, yielding a result analagous with the forms of free vibrations of a string. Some results are presented for the problem under consideration in the nonlinear formulation.The two-dimensional problem on (vertical) impingement of a jet on a liquid was solved by Olmstead and Raynor [1]. Some results for oblique impingement of a sufficiently thin, slightly curved jet are presented by Frolov [2], Information on other studies, primarily experimental, is presented in [3].This problem is related to the model of a jet curtain of an air-cushion vehicle; in this regard we note the study of Stepanov [4] in which, in particular, a result is obtained for an infinitesimally thin jet curtain.     

Instability in the oscillations of a moving oscillator while it radiates surface and internal waves

In the study of wave radiation by oscillators moving in a fluid considerable interest attaches to the question of the effect which the radiation has on the nature of the motion of the source. The main task here is to clarify the conditions under which the component of the radiation reaction force connected with the radiation of anomalous Doppler harmonics prevails over the component of the reaction force in the region in which normal Doppler frequencies are radiated. The latter results in a buildup in the oscillations of an oscillator moving in a dispersing medium [1, 2]. In the present study the authors obtain in the dipole approximation expressions for the reaction force of radiation (wave resistance) of a mass source of the oscillator type [2] when it excites surface waves in an incompressible fluid of finite depth, and internal waves in a smoothly stratified fluid. The thresholds are also found for the occurrence of radiation instability with respect to gravitational waves, and is found that with respect to internal waves radiation instability is nonthreshold in character in the Boussinesq approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 168–171, January–February, 1986.The authors are grateful to, E. V. Teodorovich for discussing the results and for his useful comments, and to A. I. Notik for performing numerical calculations.     

Nonlinear magnetoacoustic waves in a conducting liquid containing gas bubbles

The propagation of long waves in an incompressible conducting liquid saturated with nonconducting gas bubbles is considered on the basis of the equations of magnetohydrodynamics of a homogeneous gas-liquid medium. It is shown that the propagation of weakly nonlinear MHD waves in such a medium is described by the Burgers-Korteweg-de Vries (BKdV) equation. The influence of MHD interaction effects on the parameters of fast and slow weak magnetoacoustic shock waves is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 142–147, March–April, 1991.