Asymptotic analysis of the unsteady problem of waves in a two-layer flow

The problem investigated is the unsteady problem of the internal waves generated in a two-layer flow by a certain periodic perturbation which leads to small deviations from the basic flow. A method of constructing an approximate solution uniformly valid throughout the region of variation of the variables and the parameters of the problem is indicated. It is confirmed that for large times and near-resonance parameters the motion of the fluid is described by the mixed problem for a cubic Schrödinger equation. Certain qualitative properties of the solution of this nonlinear problem are noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 82–90, November–December, 1987.The author is grateful to V. I. Bukreev and to I. V. Sturova for their interest in his work.     

Permanent Waves in Slow Free-Surface Flow of a Herschel–Bulkley fluid

Unsteady flow of a viscoplastic fluid on an inclined plane is examined. The fluid is described by the three-parameter Herschel–Bulkley constitutive equation. The set of equations governing the flow is presented, recovering earlier results for a Bingham fluid and steady uniform motion. A permanent wave solution is then derived, and the relation between wave speed and flow depth is discussed. It is shown that more types of gravity currents are possible than in a Newtonian fluid; these include some cases of flows propagating up a slope. The speed of permanent waves is derived and the possible surface profiles are illustrated as functions of the flow behavior index.     

A cylindrical analog of trochoidal gerstner waves

This paper investigates isobaric motions for which the values of the pressure are conserved in fluid particles. In it, a new analytic exact particular solution of nonlinear multidimensional hydrodynamic equations is obtained; it describes a trochoidal wave in cylindrical geometry. It is also proved that trochoidal waves in cylindrical and plane geometry exhaust the class of nonlinear isobaric motions. Here and below by a wave in plane geometry we mean a wave in a uniform gravitational field which is characterized by the wave vector k. It is obvious that waves in both plane and cylindrical geometry are two-dimensional motions, since the fluid particles in motion are fixed in the plane and the motions in parallel planes are the same. The trochoidal wave in cylindrical geometry is of interest, since it describes a nonlinear wave on the surface of a cavity in a rotating fluid, a situation which is frequently encountered in applications.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–150, September–October, 1985.     

Effect of internal linear waves on the hydrodynamic characteristics of a vortex source

The results of an investigation to estimate the effect of surface and internal waves on the hydrodynamic characteristics are presented for the problem of the uniform motion of a vortex source in a three-layer fluid. The behavior of the lift force and wave drag is studied in the neighborhood of the critical Froude number. Some results of the numerical experiments are presented. An analogous investigation is also carried out for the motion in a two-layer fluid beneath a rigid top and in the presence of a bottom.Omsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 146–153, September–October, 1996.     

Effect of an impermeable inclusion in the underlying highly permeable pressurized horizon on the conditions of ground water in an irrigated layer of soil

The problem of plane, nonpressurized, steady-state filtration through a layer of soil into an underlying pressurized horizon, which contains an impermeable section at the top, with uniform infiltration on the free surface is solved in a hydro-dynamic formation. A constructive solution of the problem is given with the help of the method of P. Ya. Polubarinova-Kochina; representations are obtained for the characteristic dimensions of the flow scheme and the depression. The case of limiting flow — no head in the bottom, highly permeable layer — studied in [1] is noted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 3–5, May–June, 1986.The author thanks V. N. Emikh for useful remarks and discussions.     

Theory of convective combustion of an aerosuspension of fuel which contains an oxidant

In earlier papers [1, 2], the author and Nigmatulin developed a theory of the unsteady propagation of combustion in aerosuspensions of a unitary fuel (a fuel containing an oxidant) at low subsonic velocities of the gas. The assumption of low velocities permits a number of simplifications which reduce the equations describing the mechanics of multiphase reacting media [3, 5] to the equations of unsteady homobaric (with uniform pressure [1, 3, 4, 6]) motion of gas with distributed blowing and heat release. At a constant mass rate of combustion of the particles, these equations have a class of analytic solutions describing the initial stage of convective combustion of aerosuspensions in open and closed regions with allowance for the possible formation of weak shock waves. In the present paper, these solutions are generalized to the case of a more realistic law of combustion of a particle of a unitary fuel [7]. The results are given of a numerical solution to the problems, and these results make it possible, in particular, to analyze the conditions of applicability of the model with a constant mass rate of combustion of the particles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 152–155, October–December, 1981.     

Quasi-three-dimensional calculation of flow in blade passages of turbines

The flow in turbomachines is currently calculated either on the basis of a single successive solution of an axisymmetric problem (see, for example, [1-A]) and the problem of flow past cascades of blades in a layer of variable thickness [1, 5], or by solution of a quasi-three-dimensional problem [6–8], or on the basis of three-dimensional models of the motion [9–11]. In this paper, we derive equations of a three-dimensional model of the flow of an ideal incompressible fluid for an arbitrary curvilinear system of coordinates based on averaging the equations of motion in the Gromek–Lamb form in the azimuthal direction; the pulsation terms are taken into account in the equations of the quasi-three-dimensional motion. An algorithm for numerical solution of the problem is described. The results of calculations are given and compared with experimental data for flows in the blade passages of an axial pump and a rotating-blade turbine. The obtained results are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 69–76, March–April, 1991.I thank A. I. Kuzin and A. V. Gol'din for supplying the results of the experimental investigations.     

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.     

Propagation of three-dimensional waves generated by a traveling region of time-periodic pressure in a rotating layer of uniform fluid

The effect of rotation of a uniform fluid on the kinematic characteristics of the waves generated by the motion of a variable-intensity pressure region over its surface is investigated. All the possible types of wave fields for gyroscopic modes are analyzed, and the dependence of the shape of the leading fronts and the angular width of the wave zones of the wave systems excited on the model parameters is investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 110–116, May–June, 1992.     

Thermocapillary instability of a deformable liquid film

The instability of a plane liquid film with a uniform transverse temperature gradient under conditions of weightlessness is considered. The surface tension is assumed to depend linearly on the temperature. On the basis of an exact solution of the neutral perturbation problem for a layer with deformable boundaries, the instability domains, the dispersion curves, and the shape of the perturbations are determined. It is shown that on the interval of low Prandtl numbers both thermocapillary waves with predominantly longitudinal flow and capillary waves, supported by the thermocapillary effect, with intense transverse liquid flow can develop on the film.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 30–36, September–October, 1996.     

Nonlinear steady-state motion of an elastic plate floating on the surface of a liquid of infinite depth

The nonlinear problem of steady-state waves in an ideal fluid of infinite depth with a thin elastic plate floating on its surface is considered. The solution is found by a perturbation method. Three approximations are obtained. A case of branching of the solution is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 119–123, March–April, 1987.     

Unsteady flow over a rough bottom

The unsteady motion of an ideal incompressible fluid with a free surface, developing from a state of rest, is considered. The flow is assumed to be irrotational, continuous and two-dimensional; it may be the result either of an initial disturbance of the free boundary or of a given boundary pressure distribution. The rigid boundaries of the flow region are fixed, and the free surface does not cross them at any time during the motion. The fluid is located in a uniform gravity force field and there is no surface tension. A method which in the case of localized roughness of the bottom makes it possible to find the shape of the free surface at any moment of time with predetermined accuracy is proposed. The method involves reducing the initial linear problem to a Volterra integral equation of the second kind, the kernel of this equation being a nonlocal operator. This operator has a smoothing effect, which makes it possible to reduce the solution of the initial problem to the solution of an infinite, perfect lyregular system of Volterra integral equations for a denumerable set of auxiliary functions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–119, November–December, 1989.The author is grateful to I. V. Sturova and B. E. Protopopov for useful discussions and criticism.     

Simple stationary waves in an inclined magnetic field

One component of the solution to the problem of flow around a corner within the scope of magnetohydrodynamics, with the interception or stationary reflection of magnetohydrodynamic shock waves, and also steady-state problems comprising an ionizing shock wave, is the steady-state solution of the equations of magnetohydrodynamics, independent of length but depending on a combination of space variables, for example, on the angle. The flows described by these solutions are called stationary simple waves; they were considered for the first time in [1], where the behavior of the flow was investigated in stationary rotary simple waves, in which no change of density occurs. For a magnetic wave, of parallel velocity, the first integrals were found and the solution was reduced to a quadrature. The investigations and the applications of the solutions obtained for a qualitative construction of the problems of streamline flow were continued in [2–8]. In particular, problems were solved concerning flow around thin bodies of a conducting ideal gas. The general solution of the problem of streamline flow or the intersection of shock waves was not found because stationary simple waves with the magnetic field not parallel to the flow velocity were not investigated. The necessity for the calculation of such a flow may arise during the interpretation of the experimental results [9] in relation to the flow of an ionized gas. In the present paper, we consider stationary simple waves with the magnetic field not parallel to the flow velocity. A system of three nonlinear differential equations, describing fast and slow simple waves, is investigated qualitatively. On the basis of the pattern constructed of the behavior of the integral curves, the change of density, magnetic field, and velocity are found and a classification of the waves is undertaken, according to the nature of the change in their physical quantities. The relation between waves with outgoing and incoming characteristics is explained. A qualitative difference is discovered for the flow investigated from the flow in a magnetic field parallel to the flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1976.The author thanks A. A. Barmin and A. G. Kulikovskii for constant interest in the work and for valuable advice.     

Internal waves arising in the presence of the unsteady motion of a source in a continuously stratified fluid

A general approach to the investigation of linear internal and surface waves in a stably stratified fluid, arising from different types of perturbations, is presented in [1]. The methods of calculation of the internal waves generated by the motion of a mass source are developed for particular cases of steady horizontal motion in a continuously stratified fluid in [2] and arbitrary motion in an unbounded exponentially stratified fluid in [3]. Internal waves generated by other types of perturbations have also been investigated but only for particular cases of motion (see, for example, [4]). This paper presents a method for the calculation of unsteady, linear, internal gravity waves arising in an inviscid incompressible fluid with continuous stable stratification.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 122–130, July–August, 1985.     

Propagation of solitary waves over underwater ridges

The propagation of solitary waves is investigated on the basis of a nonlinear system of equations of hyperbolic type describing the motion of the crest of a solitary wave over the surface of a liquid of variable depth [1]. The existence of solutions with discontinuities, the boundary conditions at which are introduced on the basis of [2, 3], is assumed. In the case of an infinite cylindrical ridge both solitary and periodic captured waves are found. Depending upon the height of the ridge and the parameters of the wave, the encounter between a uniform wave and a semi-infinite ridge yields qualitatively different solutions — continuous and discontinuous, where the primary wave is broken down by the ridge into several solitary waves. The amplitude of the wave may either increase or decrease over the ridge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–93, January–February, 1985.The author is grateful to A. G. Kulikovskii and A. A. Barmin for their interest in his work, useful discussions and valuable comments offered during the preparation of the article for the press.     

Head-on collision of two coaxial vortex rings in an ideal fluid

The opposing motion of two coaxial vortex rings is classified in relation to the initial parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 60–64, July–August, 1989.The authors are grateful to V. V. Meleshko for discussing their results.     

Structure of internal waves in a channel

One method of describing wave motion in a fluid with continuous stratification is to use normal waves (modes). The propagation of internal gravity waves in closed rectangular regions whose boundaries coincide with planes through which there is no normal motion is essentially different from wave motion in an unbounded medium [1, 2]. This paper describes a theoretical and experimental investigation of the propagation of internal waves in an exponentially stratified fluid in a horizontal channel of finite height.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–110, January–February, 1935.In conclusion the authors wish to express their gratitude to A. T. Onufriev for his interest in their work.     

The stability of the motion of a liquid,due to thermocapillary forces

A study is made of the stability of the plane-parallel flow of a viscous liquid in a layer with a free boundary, under weightless conditions. The motion of the liquid is due to the dependence of the surface tension on the temperature. An exact solution for an unperturbed boundary is obtained by the same method used in [1], but with a more general boundary condition for the temperature. A study of the stability was carried out by the method of small vibrations, taking account of the perturbation of the free boundary. The article discusses the asymptotic behavior of long waves at small Reynolds numbers, and the conditions for instability are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 94–98, November–December, 1971.     

Solution to the problem of a swirling jet from an annular orifice

The problem of the propagation of a laminar immersed fan jet with swirling was considered in [1–3]. In [1], the jet source scheme was used to find a self-similar solution for a weakly swirling jet. An attempt to solve by an integral method the analogous problem for a jet emanating from a slit of finite size was made in [2]. In [3], the equations of motion for a jet with arbitrary swirling were reduced under a number of assumptions to the equations that describe the flow of a flat immersed jet. This paper gives the numerical solution to the problem of the propagation of a radial jet emanating with arbitrary swirling from a slit of finite size and an analytic solution for the main section of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 49–54, March–April, 1991.     

Motion of shock waves in an inhomogeneous gas

Whitham [1] has proposed a method of approximate calculation of the motion of shock fronts giving results in fairly good agreement with the experiments when one considers various nonstationary problems of the interaction of shock waves with one another and solid surfaces. However, the method gives rise to fundmental difficulties when one considers the propagation of shock waves in a moving medium [1, 2]. In the present paper, the method is generalized to the case of a gas which is moving nonuniformly in front of the shock wave, no restrictions being made on the nature of the motion or the variation of its parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 175–178, January–February, 1981.I thank M. D. Gerasimov for interest and assistance in the work.