Similarity autooscillations in a plane jet

Experimental investigations into the stability of a plane jet [1, 2] show that after the stationary flow has lost its stability a stable autooscillatory regime arises. In the present paper, an autooscillatory flow in a jet is studied theoretically on the basis of a plane-parallel flow in a fairly wide channel in the presence of a field of external forces. The external forces are such that at zero amplitude of the autooscillations they produce a Bickley—Schlichting velocity profile. The excitation of the secondary regimes is studied by the methods of bifurcation theory [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 26–32, May–June, 1979.We thank M. A. Gol'dshtik and V. N. Shtern for discussing the formulation of the problem and the results.     

Excitation of waves on the surface of helium II

The stability of the free surface of a superfluid in a variable gravitational field is considered in the framework of two-fluid hydrodynamics. The problem is posed in the linear formulation by the Laplace transformation method. For the displacement of the surface from the equilibrium position an integro-differential equation is obtained with periodic coefficients; its solution is sought by the method of averaging [6]. It is shown that the excitation has a threshold nature. Estimates are obtained for the excitation threshold.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti Gaza, No. 4, pp. 148–151, July–August, 1979.I should like to thank N. A. Leontovich, who drew my attention to this problem, G. Z. Gershuni and E. M. Zhukhovitskii for discussing the results, and V. A. Briskman for constant interest and support.     

Parametric excitation of waves on a liquid surface in the presence of an insoluble film

The presence of a film of surface-active agents leads to a change of the force acting on the surface of a liquid. This change does not lead to a simple decrease of the surface tension , and it is connected with the appearance of tangential forces acting on the free surface of the fluid [1]. The stability of the free surface of a liquid with a film of a surface-active agent in a variable gravitational field is examined. The linear formulation of the problem is solved. A solution is sought in the form of a series in powers of the small viscosity by the method of Laplace transforms in time and Fourier transforms in the x and y variables (the xy-plane coincides with the undisturbed liquid surface). An integrodifferential equation of the second-order with periodic coefficients is derived for the displacement of the surface from the equilibrium position. The solution is found by the method of averaging [2]. It is shown that the excitation energy should not be less than the energy dissipated in the system. It is shown that the presence of the film substantially increases the threshold of the instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 159–162, September–October, 1976.The authors thank G. Z. Gershuna for calling their attention to this problem.     

Cylindrical equilibrium surfaces of a capillary liquid and their stability

We consider the plane problem of the equilibrium of a capillary surface. We study the stability of a two-dimensional surface with respect to plane and spatial disturbances. We give data which can be used for deciding the question of the stability of any symmetric equilibrium surface in a field of gravitational forces and in conditions of weightlessness. We solve the problems of the stability of a liquid in a rectangular and a sectorial channel and also the problem of the separation of a plane drop from a horizontal wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 57–68, January–February, 1976.The authors are grateful to A. D. Myshkis and A. D. Tyuptsov for their evaluation and their useful comments.     

Effect of surface tension on the onset of convection in a layer of liquid with a free surface

The problem of stability of a plane horizontal layer of liquid heated from below is considered with surface tension at the upper surface taken into account. The problem is stated in section 1, proof of the existence of stability threshold is given in section 2, while section 3 concerns the construction of neutral curves by numerical methods and with the stabilizing effect of surface tension on the state of equilibrium.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 89–92, May–June, 1969.     

Characteristics of vibrosource in a liquid layer

Attention is given to the axisymmetric problem of the excitation of steady oscillations in a layer of ideal incompressible liquid in a gravitational field by the vibration of two circular plates fixed at the edges and immersed in the liquid at some depth. The liquid layer, with a stress-free upper surface, is supported on an absolutely rigid base. The corresponding linear boundary-value problem reduces to the solution of an integral equation of the first kind. The method proposed for the solution of this equation is to reduce it to a Fredholm equation of the second kind by the factorization of functions. Using the numerical results obtained, the velocity-potential distribution over the surface of the vibrosource may be estimated, and the associated mass of the source may be determined. The results may be used for the calculation and optimization of the parameters of marine vibroseismosources.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–8, March–April, 1978.It remains to thank V. A. Babeshko for his interest in the work and useful advice.     

Surface waves and stability of free surface of a magnetizable liquid

Linear theory of surface waves in a magnetizable liquid is examined. A stability criterion for a plane-free surface of a magnetizable liquid is studied for an arbitrary magnetization law.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 35–40, July–August, 1974.     

The stability of steady regimes in an isothermal chemical flow reactor

One of the fundamental problems in the theory of chemical reactors is the determination of the number of steady regimes and their stability. The problem of the number of steady regimes has been considered in many studies, for example, in [1–4]. The stability of a steady regime is usually established from an analysis of the behavior of small perturbations. The corresponding linear boundary-value problem for perturbations has been studied mainly in the limiting cases of ideal mixing and ideal displacement. When account was taken of longitudinal mixing, the only criteria obtained were ones which imposed fairly severe restrictions on the parameters [5]. In the present study numerical analysis is used in order to investigate the stability of steady concentration distributions in an isothermal chemical flow reactor with longitudinal mixing in the case of a single chemical reaction. The eigenvalues were obtained for the Sturm-Liouville problem, which fully characterize the stability for several laws of variation of the chemical reaction rate as a function of the concentration. A knowledge of the eigenvalues is essential, for example, in order to construct the stabilization system proposed in [6] for the unsteady regime.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 179–182, March–April, 1985.     

Evolution of a Small Distortion of the Spherical Shape of a Gas Bubble under Strong Expansion-Compression

The evolution of a small distortion of the spherical shape of a gas bubble which undergoes strong radial expansion-compression upon a single oscillation of the ambient liquid pressure under a harmonic law are analyzed by numerical experiments. It is assumed that the distortions of the spherical bubble shape are axisymmetric and have the form of individual spherical surface harmonics with numbers of 2–5. Bubble-shape oscillations prior to the beginning of expansion are taken into account. Generally, the distortion value during bubble expansion-compression depends on the phase of bubble-shape oscillation at the beginning of the expansion (initial phase). Emphasis is placed on the dependence of the maximum distortions in the initial phase at certain characteristic times of bubble expansion-compression on the amplitude of the external excitation, liquid viscosity, and distortion mode (harmonic number). The parameters of the problem are typical of the stable periodic sonolumiescence of an individual air bubble in water at room temperature. An exception is the liquid pressure oscillation amplitude, which is varied up to values that are five times the static pressure. That large excitation amplitudes are beyond the stability threshold of periodic oscillations of spherical bubbles. Their consideration is of interest from the point of view of increasing the compression ratio of the bubble gas, i. e., increasing the maximum temperature and density achievable in the final compression stage.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 17–28, July–August, 2005.     

Thermocapillary instability of the interface between two immiscible liquids in the presence of volume heat sources

The problem of the stability of the interface between two infinite layers of different immiscible liquids is considered. It is assumed that within the liquid a distributed volume heat source, simulating Joule heating, is given. The stability of the rest state with respect to small unsteady disturbances is investigated. The investigation is carried out using the real boundary conditions at the interface between the two liquids rather than the model boundary conditions usually employed in such problems [5]. The problem considered is related to the practical question of the stability of electrolyzer processes. In the present case a possible threshold mechanism of development of oscillations of the electrolyte-aluminum interface is examined. A numerical example with liquid parameters that coincide with those of the electrolyte and aluminum shows that the thermocapillary instability mechanism can, in fact, be the source of surface waves at the electrolyte-aluminum interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 156–160, September–October, 1990.     

Regime change in a planar jet with a stepped velocity profile

Flows with a nearly stepped velocity profile are seen on the initial section of jets in experiments. The present article theoretically analyzes the stability and character of branching of secondary regimes in a planar submerged jet with a stepped velocity profile. The piecewise-constant form of the profile makes it possible to perform many of the calculations analytically. A mild excitation regime is discovered and the structure of the secondary regime is calculated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 43–47, November–December, 1984.The author expresses thanks to V. N. Shtern for his constant attention to the work and useful consultations.     

Stability of Filtration of a Gas—Liquid Mixture

The stability of steady regimes of filtration of a gas—liquid mixture at pressure lower than the saturation pressure is studied for the case of a nonmonotonic dependence of the relative phase permeability of the liquid on the gas saturation. It is shown that periodic self–oscillations can appear, and their evolution leads to deterministic chaos due to the appearance and destruction of quasiperiodic motions.     

Threshold excitation of photoabsorption convection

The article considers questions of the stability of the equilibrium states of a liquid which absorbs light. Threshold values are found for the intensity of the light in the problem of the stability of the equilibrium of a liquid in a square cavity with three thermally insulated walls. A steady-state integro-interpolation scheme is presented for the numerical calculation of problems of photoabsorption convection. The propagation of light waves in absorbing media is accompanied by the dissipation of radiant energy. In heavy liquids, absorption heating of a substance in the field of a wave may be the reason for the appearance of convection [1–3]. It is important to study the conditions for the appearance and the special characteristics of this type of convection, and its inverse effect on the structure of the light field. The first problem is important when the light beams are regarded only as a source of convection [4], and the second in questions of the directed propagation of light [5] and of self-focusing phenomena [2, 3, 6–10]. For high-energy heat fluxes and a liquid with a strong temperature dependence of its dielectric permeability, the convective self-stress will be very considerable; in this case, both problems are mutually interconnected. The excitation of convection by the absorption of light, without taking account of the inverse effect on the structure of the light beam, was studied numerically in [1, 4]. Equations for photoabsorption convection, taking account of convective self-stress in the Boussinesq approximation and of the geometry of the optics, were formulated in [11]. Several economical finite-difference schemes for solving problems of photoabsorption convection problems in rectangular cavities are discussed in [12]. The present article is devoted to an investigation of the threshold intensities of light for the excitation of photoabsorption convection. The existence of critical intensities of light, above which the mechanically equilibrium states of the liquids absorbing the light become unstable, was demonstrated in [1, 4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 128–135, September–October, 1971.The authors thank A. V. Lykov for his continuing interest and aid, and G. I. Petrov and V. I. Polezhaev for their useful evaluation of the work.     

Self-similar thinning of a liquid film separating liquid phases or a liquid and a solid phase

The article discusses the thinning of a thin flat film of liquid between coalescing bubbles or coalescing drops due to motion of the liquid under the action of capillary forces, under drag conditions of the surfaces of the film. A study was also made of the outflow of a liquid from a film separating a bubble (drop) from a solid surface. The problem of the decrease in thickness and subsequent breakaway of thin liquid films is of interest in investigations of the kinetics of coalescence and coagulation and in the theory of the stability of foams and lyophobic colloids in the presence of surfactants or electrolytes, as well as in the theory of heterogeneous boiling. Information on some work and on special characteristics of the statement of the problem may be found in [1].Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp, 39–48, March–April, 1973.     

The Nonlinear Vibrations and Stability of Orthotropic Cylindrical Liquid-Carrying Shells Under Periodic Actions

The nonlinear resonance vibrations and stability of nonlinear orthotropic shells partially filled with a liquid and subjected to longitudinal and transverse periodic loading are studied. The equations of motion are derived with regard for the existence and nonlinear interaction of conjugate flexural modes of the shell. Stationary regimes for forced and parametric vibrations are found and analyzed for stability under the conditions of fundamental and subharmonic resonances. The amplitude–frequency characteristics of these regimes are constructed for various parameters of the system     

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.     

Stability of flow of a viscous liquid down an inclined plane

Using the Navier-Stokes equation the stability of a layer of viscous liquid flowing down a solid surface under gravity is studied in the linear formulation. The effect of surface tension and the inclination of the solid surface on the limits of stability are examined also. Curves are calculated for the neutral stability with respect to two types of perturbations — surface waves and shear waves.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskol Fiziki, No. 2, pp. 172–176, March–April, 1975.     

Hypersonic flow past a wing at large angles of attack with detached shock wave

The thin shock layer method [1–3] has been used to solve the problem of hypersonic flow past the windward surface of a delta wing at large angles of attack, when the shock wave is detached from the leading edge (but attached to the apex of the wing) and the velocity of the gas in the shock layer is of the same order as the speed of sound. A classification of the regimes of flow past a delta wing at large angles of attack has been made. A general solution has been obtained for the problem of three-dimensional hypersonic flow past the wing allowing for nonequilibrium physicochemical processes of thermal radiation of the gas at high temperatures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 149–157, May–June, 1985.     

Nonlinear wave motions of a liquid in a vibrating axisymmetric cavity

The problem of the nonlinear wave deformation of the free surface of a liquid due to the translational motions of the containing vessel is examined. Bogolyubov's averaging method is used to investigate the characteristics of the wave motions of the liquid in the resonance zones in the case of a cylindrical vessel. Relations are obtained characterizing the variation of the amplitude of the circular wave with the frequencies of the external perturbations in the steady-state wave process; the conditions of occurrence and stability of such processes are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 120–125, May–June, 1989.     

Linear stability in the flow of a liquid film concurrent with a gas flow

The two-phase flow of liquid films are often encountered in practice, but the number of theoretical papers devoted to this problem is limited. The problem of the linear stability of a viscous liquid film subjected to a gas flow has been formulated in [1] and, in somewhat different form, in [2]. The linear stability of plane-parallel motion in films has been studied analytically in [1–8] for some limiting cases. The range of validity of the analytic approaches remains an open question. Therefore, an exact numerical analysis of flow stability over a fairly broad range is required. In the present paper a separate solution of the problem for the gas and the liquid is shown to be possible. The Orr-Sommerfeld equation has been integrated numerically, and the results are compared to the results of analytic calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–146, January–February, 1976.The author is grateful to É. É. Markovich for directing the work and to V. Ya. Shkadov for his interest in the work and many useful comments.