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.     

Three-dimensional waves on the surface of a viscous fluid

The study of the effect of viscosity on the propagation of surface waves has traditionally been confined to the consideration of plane (two-dimensional) waves [1, 2]. So far the effect associated with taking the transverse modulation of the wave profile into account have not been studied. In what follows, a solution is constructed and analyzed for linear three-dimensional periodic waves in an infinitely deep fluid. Their distinguishing property is the presence of a vorticity in the direction of propagation. An expression for the average (over the wavelength) velocity of horizontal particle drift is found in the quadratic approximation in the small wave steepness.     

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.     

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.     

Capillary waves in a relaxing conducting viscous fluid with a surface charge

The influence of weak viscosity and dynamical surface tension relaxation effect on the structure of the wave motion spectrum in a conducting viscous fluid with a surface charge is investigated. Taking these phenomena into account leads to the appearance of additional branches of both wave and aperiodic motions associated with fluid elasticity effects and the finite time taken by the fluid surface layer to respond to fast external excitation.Yaroslavl. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 98–105, January–February, 1996.     

Stability of stationary traveling waves on the surface of a vertical film of viscous fluid

The stability of stationary traveling waves of the first and second families with respect to infinitesimal perturbations of arbitrary wavelength is subjected to a detailed numerical investigation. The existence of a unique region of stability of the first family is established for wave numbers (₁, ₁) corresponding to the optimal wave regime. There are several regions of stability of the second family ( _k , _k),k=2,3,..., lying close to the local flow rate maxima. In the regions of instability the growth rates of perturbations of the first family are several times greater than for the second family. This difference increases with increase in the Reynolds number. The calculations make it possible to explain a number of experimental observations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–41, May–June, 1989.The authors are grateful to V. Ya. Shkadov for his constant interest, and to A. G. Kulikovskii, A. A. Barmin and their seminar participants for useful discussions and suggestions.     

Vortex on the surface of a viscous fluid

The system of Navier-Stokes equations is solved for boundary conditions corresponding to the case when an axisymmetric tangential transversal load acts at the surface of a gravity viscous incompressible fluid of infinite depth. An integral representation is obtained for the shape of the free surface under the prolonged effect of a stationary vortex load. The example of a tangential load, similar to a concentrated vortex, is examined. In this case a column is squeezed out of the fluid, the height of the column being directly proportional to the square of the moment of the transverse tangential forces and inversely proportional to the square of the product of the dynamic fluid viscosity and the area of the tangential stress distribution. The depth of the annular funnel being formed in front of the column is determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 127–132, July–August, 1978.     

Influence of rotation on the damping of surface waves in a cylindrical layer of viscous fluid

A study is made of the oscillations of the free surface of a cylindrical layer of viscous incompressible fluid attracted by gravity to a solid, uniformly rotating cylinder. Logarithmic decay rates are found for the damping of the surface gravitational waves when large Reynolds numbers are assumed. It is shown that rotation introduces an asymmetry into the damping of the waves traveling in and against the direction of rotation of the fluid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 171–173, November–December, 1984.     

Structure of the drift flow initiated by periodic waves traveling over a viscous fluid surface

An analytical procedure used in calculating the Stokes drift velocity (the drift motion initiated by the propagation of a capillary-gravity wave over an ideal fluid surface) is applied to the problem of the calculation of an analogous drift flow in a viscous fluid. An expression for the velocity of the Stokes drift modified with allowance for viscosity is constructed. The properties and the role of the modified Stokes drift in the general pattern of the drift in a viscous fluid are analyzed.     

VELOCITY FIELD IN SHIP WAVES ON THE VISCOUS FLUID

ＩｎｔｒｏｄｕｃｔｉｏｎＷｈｅｎａｍｏｖｉｎｇｂｏｄｙ (ｆｏｒｅｘａｍｐｌｅ:ｓｈｉｐ)ｉｓｉｎｍｏｔｉｏｎｏｎｔｈｅｏｃｅａｎ ,ａＶ＿ｓｈａｐｅｄｓｈｉｐｗａｖｅａｐｐｅａｒｓｂｅｈｉｎｄｔｈｅｂｏｄｙ .Ｉｎｔｈｉｓａｒｔｉｃｌｅ,ｗｅｗｉｌｌｄｉｓｃｕｓｓｔｈｅｓｈａｐｅｏｆｔｈｅｖｅｌｏｃｉｔｙｆｉｅｌｄａｎｄｔｈｅＶ＿ｓｈａｐｅｄｓｈｉｐｗａｖｅ .Ａｔｆｉｒｓｔ,Ｒｅｆ.[1]ｓｔｕｄｉｅｄｔｈｅｑｕｅｓｔｉｏｎｓｏｆｓｈｉｐｗａｖｅｓｏｎａｎｉｎｃｏｍｐｒｅｓｓｉｂｌｅｉｎｖｉｓｃｉｄｆｌｕｉ…     

Unsteady waves in a viscous fluid of finite depth

Here we study the plane and three-dimensional problems of unsteady waves which arise on the surface of a viscous fluid of finite depth under the influence of a velocity pulse applied on the bottom of the basin.The problem is considered as the simplest scheme for studying, with account for the effect of viscosity, the propagation of waves of the tsunami type which result from an underwater shock.Similar problems on the propagation of waves which arise from initial surface disturbances are considered in [1–9].     

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.     

The role of external influences in the formation of waves on the surface of a flowing-down film of a viscous fluid

The example of two non-stationary forces is used to study the impact of external influences leading to the occurrence of additional ponderomotive forces on the wave regimes of the film freely flowing down a vertical surface. The first case describes a ferromagnetic fluid film affected by the magnetic field, and the second case touches upon a dielectric fluid film affected by the electric field. For the given forces, in the case of small flow rates, the problem is reduced to the solution of a model equation for the perturbation of the film thickness. The numerical solutions of the problem are obtained, and several characteristic scenarios of evolution of periodical perturbations are considered. It is shown that changes in the boundaries of the region of linear stability of the unperturbed flow with a flat free surface under the influence of ponderomotive forces have a great impact on the flow.     

Nonlinear dynamic characteristics of a micro-vibration fluid viscous damper

Nonlinear Dynamics - This study is concerned with the nonlinear dynamic characteristics of a micro-vibration fluid viscous damper used in a satellite. When a control moment gyroscope is working, it...     

Nonlinear waves propagating over a conducting ideal fluid surface in an electric field

For wave perturbations of a heavy conducting fluid in an electric field orthogonal to the undisturbed surface evolutionary equations quadratically nonlinear in amplitude are obtained. Equations for the long-wave approximation are derived. A method of deriving the nonlinear and simple-wave equations is proposed. Solutions for solitary waves are considered. It is shown that even a weak electric field significantly affects the form of the soliton solution, which is related with fundamental changes in the spectrum of the linear waves.     

Internal and inertial waves in a viscous rotating stratified fluid

The effects of viscosity on the propagation of a St. Andrew's cross wave which is generated by a simple-harmonic localized disturbance in a rotating stratified fluid are considered. A similarity solution of the linearised equations shows that the velocities decay and that the wave width increases away from the disturbance. Previous solutions in a stratified non-rotating fluid are recovered by letting the rotation tend to zero. The solutions are also valid in the limit of a homogeneous rotating fluid. Further solutions for waves in a realistic ocean and in an isothermal atmosphere on a rotating Earth are also included.