Solitary stokes waves in heavy-fluid slit flow

Flow with a solitary Stokes wave is obtained as a result of solving the plane nonlinear problem of the steady flow of an ideal heavy incompressible fluid into a slit at the bottom of a vessel for a narrow range of Froude numbers on which there exists a solution with a jet surface descending monotonically to the slit [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 173–176, November–December, 1991.The authors are grateful to A. N. Ivanov and G. Yu. Stepanov for their assistance and useful discussions.     

Two-dimensional instabilities of generalized Korteweg-de Vries waves

Instabilities of the generalized Korteweg-de Vries ((u_t+₁u^mu_x+₂uⁿu_x+u_xxx)_x+₃u_yy = 0) waves wi th respect to two-dimensional infinitesimal longitudinal disturbances are investigated using the Infeld-Rowlands method. A linear dispersion relation expressed as a cubic equation in w₁ is derived and instabilities of waves are discussed.     

Numerical calculation for the coefficients of stokes harmonic waves of high order

This paper has reformulated the mathematical model and boundary conditions in the semi-physical plan (x,ψ)by using W.H.Hui’s method and suggested two new ways of numerical calculation for the coefficients of Stokes harmonic waves of high order. By transforming the perturbation parameter? into a new one we we rejind Cokelet’s results (1977) of phase speed and semi-waveheight expressions.     

Two-dimensional reactive flow dynamics in cellular detonation waves

This investigation deals with the two-dimensional unsteady detonation characterized by the cellular structure resulting from trajectories of triple-shock configurations formed by the transverse waves and the leading shock front. The time-dependent reactive shock problem considered here is governed by a system of nonlinear hyperbolic conservation laws coupled to a polytropic equation of state and a one-step Arrhenius chemical reaction rate with heat release. The numerical solution obtained allowed us to follow the dynamics of the cellular detonation front involving the triple points, transverse waves and unreacted pockets. The calculations show that the weak tracks observed inside the detonation cells around the points of collision of the triple-shock configurations arise from interactions between the transverse shocks and compression waves generated by the collision. The unreacted pockets of gas formed during the collisions of triple points change form when the activation energy increases. For the self-sustained detonation considered here, the unreacted pockets burn inside the region independent of the downstream rarefaction, and thus the energy released supports the detonation propagation. The length of the region independent of the downstream is approximately the size of one or two detonation cell. Received 13 February 1998 / Accepted 13 August 1998     

Two-dimensional stokes flow of a viscous fluid with a free boundary under the effect of capillary forces

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 5, pp. 47–51, September–October, 1992.     

Fourth order evolution equations and stability analysis for stokes waves on arbitrary water depth

Fourth order evolution equations have been derived for three-dimensional Stokes waves on arbitrary water depth. In deep water the equations reduce to those of Dysthe, and on finite depth the third order terms agree with those of Benney and Roskes, Hasimoto and Ono and Davey and Stewartson. The results of the stability analysis for uniform waves based on the new arbitrary depth expressions are superior to those based on the finite depth approximation and they agree fairly well with the exact calculations of McLean. It is demonstrated that dimensionless water depth as well as wave steepness influences the applicability of the deep water stability expressions.     

Averaged equations and wave jumps describing the propagation of stokes waves with slowly varying parameters

The equations describing the stationary envelope of periodic waves on the surface of a liquid of constant or variable depth are investigated. Methods previously used for investigating the propagation of solitons [1–5] are extended to the case of periodic waves. The equations considered are derived from the cubic Schrödinger equation assuming slow variation of the wave parameters. In using these equations it is sometimes necessary to introduce wave jumps. By analogy with the soliton case a wave jump theory in accordance with which the jumps are interpreted as three-wave resonant interactions is considered. The problems of Mach reflection from a vertical wall and the decay of an arbitrary wave jump are solved. In order to provide a basis for the theory solutions describing the interaction of two waves over a horizontal bottom are investigated. The averaging method [6] is used to derive systems of equations describing the propagation of one or two interacting wave's on the surface of a liquid of constant or variable depth. These systems have steady-state solutions and can be written in divergence form.The author wishes to thank A. G. Kulikovskii and A. A. Barmin for useful discussions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 113–121, September–October, 1989.     

Two-dimensional problem for internal waves generated by moving singular sources

When bodies move in a liquid with inhomogeneous density in a gravitational field waves are excited even at low velocities and in the absence of boundaries. They are the so-called internal waves (buoyancy waves), which play an important part in geophysical processes in the ocean and the atmosphere [1–4]. A method based on the replacement of the bodies by systems of point sources is now commonly used to calculate the fields of internal waves generated by moving bodies. However, even so the problems of the generation of waves by a point source and dipole are usually solved approximately or numerically [5–11]. In the present paper, we obtain exact results on the spectral distribution of the emitted waves and the total radiation energy per unit time for some of the simplest sources in the two-dimensional case for an incompressible fluid with exponential density stratification. The wave resistance is obtained simply by dividing the energy loss per unit time by the velocity of the source. In the final section, some results for the three-dimensional case are briefly formulated for comparison.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 77–83, March–April, 1981.     

Two-dimensional equations for electromagnetic waves in multi-layered thin dielectric films

Two-dimensional equations for electromagnetic fields in a multi-layered thin dielectric film are derived from the three-dimensional equations of electrodynamics by expanding the vector potential of the electromagnetic fields into trigonometric series expansions of the film thickness coordinate. The lower order equations are examined. It is shown that they can describe certain long waves in the film. The equations are useful for modeling thin film devices.     

Fluid particle dynamics and stokes drift in gravity and capillary waves generated by the faraday instability

In this paper we study particle motions in nearly square containers due to gravity and capillary waves generated by vertical, periodic oscillation of the container. The method of second order partial averaging is used to decompose the particle motions into periodic oscillations and a slow Stokes drift. In the case of gravity waves, it is shown that long distance (several wavelengths) particle transport is possible. In the case of capillary waves, it is shown that, in agreement with experimental observations of Ramshankaret al., particle trajectories can be chaotic even when the wave pattern is regular so long as the pattern is spatially modulated.Dedicated to Professor P. R. Sethna on the Occasion of His 70th BirthdayThis research was partially supported by an NSF Presidential Young Investigator Award and an ONR Grant No. N00014-89-J-3023.     

Quasiperiodic and exponential transient phase waves and their bifurcations in a ring of unidirectionally coupled parametric oscillators

Phase waves rotating in a ring of unidirectionally coupled parametric oscillators are studied. The system has a pair of spatially uniform stable periodic solutions with a phase difference and an unstable quasiperiodic traveling phase wave solution. They are generated from the origin through a period doubling bifurcation and the Neimark?CSacker bifurcation, respectively. In transient states, phase waves rotating in a ring are generated, the duration of which increases exponentially with the number of oscillators (exponential transients). A power law distribution of the duration of randomly generated phase waves and the noise-sustained propagation of phase waves are also shown. These properties of transient phase waves are well described with a kinematical equation for the propagation of wave fronts. Further, the traveling phase wave is stabilized through a pitchfork bifurcation and changes into a standing wave through pinning. These bifurcations and exponential transient rotating waves are also shown in an autonomous system with averaging and a coupled map model, and they agree with each other.     

Structure and bifurcations of plane shock waves in a vibrationally excited gas with an external pumping source

The structure of the density profiles in stationary plane shock waves in a vibrationally excited gas is investigated. For self-similar solutions a bifurcation diagram is plotted in the parametric “traveling wave velocity—degree of nonequilibrium” plane. The bifurcation boundaries of the domains of existence of the structures of different types are analytically derived. It is shown that weak plane shock waves are unstable, accelerate, and break down into a sequence of pulses or-at a fairly high pumping rate-waves with nonzero asymptotics, whose amplitude and propagation velocity are independent of the initial disturbance and are determined by the parameters of the medium itself.     

CODIMENSION TWO BIFURCATIONS AND HOPF BIFURCATIONS OF AN IMPACTING VIBRATING

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Some properties of multi-fluid stokes flows

The solution of Stokes' equations for a rotating axisymmetric body which possesses reflection symmetry about a planar interface between two infinite immiscible quiescent viscous fluids is shown to be independent of the viscosities of the fluids and identical with the solution when the fluids have the same viscosity. The result is generalized to a rotating axisymmetric system of bodies which possesses reflection symmetry about each interface of a plane stratified system of fluids. An analogous result for two-fluid systems with a nonplanar static interface is also derived. The effect on torque reduction produced by the presence of a second fluid layer adjacent to a rotating axisymmetric body is considered and explicit calculations are given for the case of a sphere. A proof of uniqueness for unbounded multi-fluid Stokes' flow is given and the asymptotic far field structure of the velocity field is determined for axisymmetric flow caused by the rotation of axisymmetric bodies.     

Two-dimensional modulation and instabilities of flexural waves of a thin plate on nonlinear elastic foundation

In the present paper we intend to examine in detail the formation of localized modes and waves mediated by modulational instability in an elastic structure. The elastic composite structure consists of a nonlinear foundation coated with an elastic thin plate. The problem deals with flexural waves traveling on the plate. The attention is devoted to the behavior of nonlinear waves in the small-amplitude limit in view of deducing criteria of instability which produce localized waves. It is shown that, in the small-amplitude limit, the basic equation which governs the plate deflection is approximated by a two-dimensional nonlinear Schrödinger equation. The latter equation allows us to study the modulational instability conditions leading to different zones of instability. The examination of the instability provides useful information about the possible selection mechanism of the modulus of the carrier wave vector and growth rate of the instabilities taking place in both (longitudinal and transverse) directions of the plate. The mechanism of the self-generated nonlinear waves on the plate beyond the birth of modulational instability is numerically investigated. The numerics show that an initial plane wave is then transformed, through the instability process, into nonlinear localized waves which turn out to be particularly stable. In addition, the influence of the prestress on the nature of localized structures is also examined. At length, in the conclusion some other wave problems and extensions of the work are evoked.     

Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension

Nonlinear Dynamics - We derive two new two-dimensional third- and fifth-order nonlinear evolution equations that model a unidirectional wave motion in shallow water waves with surface tension....