Flexural perturbations of free jets of maxwell and Doi-Edwards liquids

Flexural perturbations of high-velocity free jets of drop liquids moving in air are reinforced by the fact that the air pressure on the concave sections of the jet surface is greater than on the convex sections. The linear and nonlinear stages of development of flexural perturbations were studied in [1–5] for viscous Newtonian fluids. The effect of elastic stresses in the fluid on the growth of flexural perturbations of jets was first examined in [6], where it was assumed in an analysis of the growth of small disturbances that surface tension was constant along the jet, i.e., the investigators actually studied a tensed string. The studies [7, 8] examined the linear stage of growth of flexural perturbations of jets of Maxwell liquids. Our goal here is to analyze the dynamics of long-wave flexural perturbations of jets of viscoelastic fluids in both the linear and nonlinear stages of development. The rheological behavior of the fluid is described by two models — the phenomenological (Maxwell) model and the physical-molecular (Doi-Edwards) model. It is shown that the disturbances are oscillatory in character in the nonlinear stage of development. Meanwhile, the results of calculations performed with the Maxwell (M) and Doi-Edwards (DE) rheological models in the given problem agree with each other quantitatively as well as qualitatively.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 43–53, November–December, 1986.     

Nonlinear evolution of waves in forced decaying capillary jets

The investigation of nonlinear waves in decaying capillary jets is of great interest both from the point of view of nonlinear wave processes in media and for practical applications associated with the generation and propagation of flows of monodisperse droplets [1–4]. The formation and dynamics of satellite droplets are particularly important in the study of the decay of thin capillary jets [5–8]. Investigation of the conditions of formation of satellites open up important prospects for the preparation of monodisperse microscopic granules with diameters appreciably less than the diameter of the original jet. This is of great importance in modern technologies based on the use of materials in disperse form [9–13]. The present paper is devoted to the investigation of nonlinear waves in decaying capillary jets.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 54–60, May–June, 1993.     

Ultrasonic jet in an external acoustic field

The interaction of supersonic jets with external acoustic waves is investigated in connection with the emission of sound of discrete frequency by the jets. A plausible physical scheme explaining the appearance and maintenance of the oscillations of supersonic jets with discrete frequency was proposed in [1]. A model problem of the effect of pressure perturbations of a given frequency, traveling along the surface of a two-dimensional jet is also investigated there. The results of the solution of this problem (in particular, the presence of critical frequencies at which the perturbations in the jet grow indefinitely in the direction of motion of the flow) substantiate the hypothesis that by virtue of its periodic (cellular) structure a supersonic jet has the properties of a resonator. In [1] the more general problem of interaction of a supersonic jet with an external acoustic field is also formulated, which is in complete correspondence with the physical scheme of the phenomena developed in that article. In the present work this problem is solved in its complete form for plane and cylindrical jets for symmetric and antisymmetric perturbations in an external acoustic field, and also in the presence of subsonic accompanying flow in the outer medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 105–113, March–April 1974.     

The nonlinear evolution of two-dimensional and three-dimensional waves in mixing layers

The authors consider problems connected with stability [1–3] and the nonlinear development of perturbations in a plane mixing layer [4–7]. Attention is principally given to the problem of the nonlinear interaction of two-dimensional and three-dimensional perturbations [6, 7], and also to developing the corresponding method of numerical analysis based on the application to problems in the theory of hydrodynamic stability of the Bubnov—Galerkin method [8–14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza, No. 1, pp. 10–18, January–February, 1985.     

Collision of two jets of perfect fluid with different Bernoulli constants

We consider the problem of the collision between two plane jets of perfect fluid with different Bernoulli constants in jets flowing into a mediumfilled space out of channels with parallel walls, converging at an angle. In [1–3] the problem is reduced to a system of nonlinear equations, whose solution is obtained in the form of a formal series in powers of the small quantity , equal to the ratio of the total dynamic heads of the colliding jets. The zeroth and first approximations of the unknown, and also the second approximation for the angle of deflection of the jets, are calculated. Here the nonlinear problem of the collision of two jets is solved in an exact mathematical formulation [4]. The results of the calculations are given for different geometric parameters of the problem in the entire range of variation of the Bernoulli number Be equal to the ratio of the difference between the Bernoulli constants of the jets to the dynamic head of one of them.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 38–42, March–April, 1987.     

Solution of three-dimensional nonlinear problems of the theory of jets in ideal fluids

A method of successive approximations is proposed for solving three-dimensional nonlinear problems of the theory of jets in ideal fluids (see, for example, [1–3]). Each approximation includes the calculation of the flow over a known surface, i.e., the solution of the exterior Neumann problem for the Laplace equation in the velocity potential and the correction of part of that surface for the purpose of reducing the discrepancy in the constant-pressure condition at the surface of the jets. The correction takes the form of small deformations found from a system of integral equations; the shape of the cavity in plan is also refined. The results of calculating the flow past triaxial ellipsoids, obtained using the generalized Zhukovskii-Roshko method for closing the jets, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 175–179, March–April, 1989.The authors are grateful to V. P. Karlikov for useful comments.     

Long-wavelength stability of the jet flows of a perfect fluid

The spatial stability of jets whose velocity profiles depend linearly on one coordinate is considered. The influence of the velocity profile and the presence of solid walls on the development of perturbations in the jets is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkostl I Gaza, No. 5, pp. 170–175, September–October, 1979.We thank A. A. Pavel'ev for making available experimental material and for a helpful discussion of the results.     

Propagation of surface gravity waves in a layer of electrically conducting liquid

The wave motion of a weakly conducting incompressible liquid in a transverse magnetic field is investigated within the framework of the nonlinear theory of magnetohydrodynamics. The influence of MHD interaction effects on harmonic perturbations of infinitesimal amplitude is analyzed and a long-wave equation of the Kortewegde Vries-Burgers type describing the evolution of weakly nonlinear perturbations of the free surface is derived. It is shown that the influence of the electrical conductivity leads to a change in both the dissipative and the dispersive properties of the system.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 173–175, July–August, 1989.     

Nonmonodisperse breakup of capillary jets in a variable electric field

The electrical charging of capillary jets has a strong influence on their stability [1–10]. Well-known theoretical studies have been devoted to the linear [1–6], weakly linear [7], or finite-amplitude [10] stability of such jets in a constant electric field. In the present paper, an investigation is made in the framework of the full nonlinear equations. The main attention is devoted to effects associated with allowance for a time-variable electric field. It is shown that a sharp decrease of the surface charge may lead to an appreciable decrease in the size of the satellite droplets; allowance for the long-wavelength background also leads to a decrease in the size of the satellite droplets. In contrast, a sharp increase of the surface charge increases the relative contribution of the satellite droplets. At the same time, introduction of small-scale background perturbations can lead to a decrease in the contribution of the fine satellite droplets and to a weakening of their reaction to a rapidly increasing electric field. It is shown that the degree of monodisperseness can be increased by a relatively slowly varying electric field. An averaged effect of an electric field that varies rapidly in time is found. Appreciable increase of the initial perturbation amplitude in the case of a periodically varying electric field can lead to an appreciable increase in the degree of monodisperseness. The introduction of short-wavelength perturbations in a periodic electric field with large amplitude of the pulsations can lead to disappearance of the satellite droplets.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 55–62, March–April, 1991.     

Nonlinear stability of two-dimensional steady flows of an ideal fluid with constant vorticity and their axisymmetric analogs

Flows with constant vorticity are widely used as local models of more complicated flows [1]. In many cases, such flows are stable against finite two-dimensional perturbations. In particular, the inviscid plane-parallel Couette flow has the property of nonlinear stability. Similar treatment of a class of axisymmetric flows yields nonlinear stability of a spherical Hill vortex and inviscid Poiseuille flow in a circular tube with respect to axisymmetric perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 16–21, October–December, 1981.     

Effect of nonlinearity on the development of oscillatory magnetohydrodynamic Kelvin-Helmholtz instability in the weakly supercritical regime

The nonlinear stage of development of perturbations at a tangential magnetohydrodynamic discontinuity is investigated in the weakly subcritical and supercritical regimes. It is assumed that the fluid is incompressible and that the density and magnetic field, as well as the velocity, suffer a discontinuity. An equation describing the evolution of low-amplitude nonlinear perturbations is obtained. For periodic perturbations this equation reduces to an infinite system of ordinary differential equations for the amplitudes of the Fourier harmonics. The system is reduced to finite form by truncation and then integrated numerically. Calculations show that the evolution of an initially sinusoidal perturbation always ends with the appearance in the wave profile of an infinite derivative. This can take the form of either an infinitely sharp peak (knife-edge) or wave breaking.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 30–39, May–June, 1988.     

Instability and breakup of capillary liquid jets in a parallel airstream

The problem of the development and interaction of nonlinear two-dimensional perturbations in a rotating capillary jet is solved. The main attention is devoted to the study of the nonuniform breakup of the jet with allowance for the influence of the parallel airstream and the rotation. The solution is found by Galerkin's method [1–3]. The nonlinear development and interaction of a large number of perturbations is considered. A significant influence of long-wavelength modulation on the nature of drop formation is established. It is shown that an increase in the velocity of the parallel stream leads to a decrease in the relative size of the satellite (for the characteristic wavelengths). It is also shown that the rotation extends the region of unstable wave numbers in the complete range of flow velocities and air densities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 124–128, May–June, 1981.I am sincerely grateful to G. I. Petrov, V. Ya. Shkadov, and S. Ya. Gertsenshtein for constant interest in the work.     

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.     

Experimental investigation of the interaction of a supersonic annular jet and an obstacle

There has been much interest in recent years in gas-dynamic problems involving the interaction of gas jets with obstacles, and there have been studies of combinations of individual jets, systems of jets, and also annular jets. Various papers have been published with the results of theoretical and experimental investigations of the interaction of axisymmetric continuous jets with obstacles [1–3]. However, there have been only a few experiments on the fluctuations of an annular system of jets that encounter an obstacle [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 109–117, May–June, 1983.     

Finite-amplitude self-oscillations in a rotating Hagen-poiseuille flow

The nonlinear stability of a viscous incompressible flow in a circular pipe rotating about its own axis is investigated. A solution of the initial—boundary value problem for the unsteady three-dimensional Navier—Stokes equations is found by means of the Bubnov—Galerkin method [1–5]. A series of methodological investigations were made. The nonlinear evolution of the periodic self-oscillating regimes is studied, and their characteristic stabilization times, amplitudes, and other integral and fluctuational characteristics are found. The secondary instability of these finite-amplitude wave motions is examined. It is established that the secondary instability is initially weak and linear in character; the corresponding growth times are approximately an order greater than for the primary perturbations. There is the possibility of a sharp, explosive restructuring of the motion when the secondary perturbations reach a certain critical amplitude. A survival curve [5] is constructed, which makes it possible to determine the preferred perturbation, distinguishable from the rest if the initial perturbation amplitudes are equal, and the critical amplitude values starting from which the other perturbations may prevail even over the preferred one. The range of these surviving perturbations is obtained. It is shown that as a result of the non-linear interaction of several perturbations at low levels of supercritlcality periodic motion in the form of a single traveling wave is generated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 22–28, July–August, 1985.     

The stability of vortex filaments

The article discusses the problem of the nonlinear oscillations of concentrated vortexes, For a vortical annulus and a spiral vortical filament, the article demonstrates the character of the change in the form and the frequency of the oscillations with an increase in the amplitude. In the case of standing waves, the solution is obtained in the form of a series with respect to the amplitude of the perturbations, with an accuracy up to terms of the third order of smallness, inclusive. For running waves, the solution is constructed using a method analogous to the Stewart method. The same problem is solved using direct numerical integration of the starting equations of motion. The values of the critical amplitudes which bring about the breakdown of a vortical annulus are obtained. It is shown that as a result of the nonlinearity of the equations in solution of the problem with the starting data higher harmonics can separate out intensively. For a spiral vortical filament, a study was made of the nonlinear interaction of the perturbations; it is shown, in particular, that a perturbation which is stable according to the linear theory may become unstable as a result of nonlinear interaction with neighboring perturbations along the length of the wave and of perturbations of the frequency.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–49, May–June, 1973.The authors thank G. I. Petrov for his direction of the work, as well as V. Ya. Shkadov for his interest in the work.     

On the nonlinear development of two-dimensional and three-dimensional perturbations in the presence of Rayleigh-Taylor instability

The nonlinear evolution of two-dimensional and three-dimensional perturbations of finite amplitude in the presence of Rayleigh-Taylor instability is investigated. It is assumed that the problem is one of potential flow. The solution is constructed by the Fourier method [1]. In the two-dimensional case the conformal mapping method [2, 3] is employed, which makes it possible to consider the strongly nonlinear stages of development of the perturbations, including the formation of surfaces with multiply valued dependence of the variables in Cartesian coordinates. The construction of the mappings reduces to the solution of the Hilbert problem, which is given in the form of Schwartz integrals [4]. Explicit expressions for these integrals [5], obtained with the aid of Fourier series, are employed. Effective computational algorithms are developed and a series of numerical investigations is carried out. Inter alia, a destabilizing effect of the short-wave components is detected, the regularizing action of the surface tension is demonstrated, and the characteristic times of nonlinear development of the perturbations and the characteristic spectral distributions are found. The role of three-dimensional effects, characterized by a decrease in the rate of development of perturbations, is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 38–46, March–April, 1985.In conclusion, the authors wish to express their deep gratitude to Yu. B. Rabinovich and S. L. Petrov whose research was used in compiling the calculation program and, moreover, Yu. M. Shtempler for discussing methodilogy and the numerical results.     

Stability of quasilongitudinally propagating solitons in a plasma with hall dispersion

By analogy with the generalization obtained in [16] for the Korteweg-de Vries equation, the derivative nonlinear Schrödinger equation is extended to the weakly non-one-dimensional case. On the basis of the equation obtained the stability of solitons propagating at small angles to the undisturbed magnetic field relative to non-one-dimensional perturbations is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 159–165, March–April, 1987.     

Asymptotic analysis of a pulsating flow of viscous fluid in a symmetric deformed channel

The time-periodic flow of a viscous incompressible fluid in a two-dimensional symmetric channel with slightly deformed walls is considered. The solution of the Navier-Stokes equations is constructed by means of the method of matched asymptotic expansions [1] at large characteristic Reynolds numbers. It is shown that in an unsteady flow a region of nonlinear perturbations surrounds the line of zero velocity inside the fluid. The formation and development of such nonlinear zones with respect to time is considered. An alternation of the topological features of the streamline pattern in the nonlinear perturbation zone is discovered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 17–23, July–August, 1987.The author is deeply grateful to V. V. Sychev for his formulation of the problem and his attentive attitude to my work.     

Finite-amplitude instability of axisymmetric submerged jets

The effects associated with the finite-amplitude instability of submerged jets are numerically modeled. The investigation is carried out within the framework of the Euler inviscid model using the Bubnov-Galerkin method. The stability of an axisymmetric air jet submerged in an infinite fluid volume is considered with allowance for surface tension forces. The nonlinear evolution of a large-scale sinusoidal disturbance, stable according to the linear theory, is traced. The effect of an intense small-scale surface ripple on the development of the large-scale disturbance is studied, together with the reaction of the latter on the amplitude of the small-scale ripple.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 100–107, January–February, 1990.