Stability of convective motion in a vertical layer in the presence of longitudinal vibrations

The influence of vibrations of a cavity containing a fluid on the convective stability of the equilibrium has been investigated on a number of occasions [1]. The stability of convective flows in a modulated gravity field has not hitherto been studied systematically. There is only the paper of Baxi, Arpaci, and Vest [2], which contains fragmentary data corresponding to various values of the determining parameters of the problem. The present paper investigates the linear stability of convective flow in a vertical plane layer with walls at different temperatures in the presence of longitudinal harmonic vibrations of the cavity containing the fluid. It is assumed that the frequency of the vibrations is fairly high; the motion is described by the equations of the averaged convective motion. The stability boundaries of the flow with respect to monotonic perturbations in the region of Prandtl numbers 0 ? P ? 10 are determined. It is found that high-frequency vibrations have a destabilizing influence on the convective motion. At sufficiently large values of the vibration parameter, the flow becomes unstable at arbitrarily small values of the Grashof number, this being due to the mechanism of vibrational convection, which leads to instability even under conditions of weightlessness, when the main flow is absent [3, 4].     

Development of a steady relief at the interface of fluids in a vibrational field

The vibrations of a vessel strongly influence the behavior of the interface of the fluids in it. Thus, vertical vibrations can lead both to the parametric excitation of waves (Faraday ripples) and to the suppression of the Rayleigh-Taylor instability [1–2]. At the present time, the influence of vertical vibrations on the behavior of a fluid surface have been studied in sufficient detail (see, for example, review [3]). The behavior of an interface of fluids in the case of horizontal vibrations has been studied less. An interesting phenomenon has been revealed in the experimental papers [4, 5]: in the case of fairly strong horizontal vibrations of a vessel containing a fluid with a free surface, the fluid collects near one of the vertical vessel walls, the free surface being practically plane and stationary with respect to the vessel, while its angle of inclination to the horizon depends on the vibration rate. But if there is a system of immiscible fluids with comparable but different densities in the vessel, horizontal vibrations lead to the formation of a steady wave relief at the interface. An explanation of the behavior of a fluid with a free boundary was given in [6] on the basis of averaged equations of fluid motion in a vibrational field. The present paper is devoted to an analysis of the behavior of the interface of fluids with comparable densities in a high-frequency vibrational field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 8–13, November–December, 1986.     

喷嘴结构对液氧煤油火箭发动机高频燃烧不稳定性的影响

为了筛选高压补燃循环液氧煤油火箭发动机的喷嘴,在喷注单元低压高频燃烧不稳定性模拟实验系统上开展实验,研究了喷嘴结构对燃烧稳定性边界的影响。实验使用气态空气与氧气的混合物作为氧化剂,加热的煤油蒸汽作为燃料;喷嘴为全尺寸气液同轴直流离心式喷嘴,模拟燃烧室与真实燃烧室的固有声学频率相等。根据测量模拟燃烧室内的脉动压力区分大幅振荡、小幅振荡和稳定工作。研究结果表明,喷嘴长度、缩进室长度和入口节流嘴直径对高频燃烧不稳定性裕量有很大影响,并存在相对最佳值。     

Effect of the Boundary Conditions and Influence of the Rotational Inertia on the Vibrational Modes of an Elastic Ring

We present the small-amplitude vibrations of a circular elastic ring with periodic and clamped boundary conditions. We model the rod as an inextensible, isotropic, naturally straight Kirchhoff elastic rod and obtain the vibrational modes of the ring analytically for periodic boundary conditions and numerically for clamped boundary conditions. Of particular interest are the dependence of the vibrational modes on the torsional stress in the ring and the influence of the rotational inertia of the rod on the mode frequencies and amplitudes. In rescaling the Kirchhoff equations, we introduce a parameter inversely proportional to the aspect ratio of the rod. This parameter makes it possible to capture the influence of the rotational inertia of the rod. We find that the rotational inertia has a minor influence on the vibrational modes with the exception of a specific category of modes corresponding to high-frequency twisting deformations in the ring. Moreover, some of the vibrational modes over or undertwist the elastic rod depending on the imposed torsional stress in the ring.     

Two-dimensional stability of the combustion of condensed systems

The question of the combustion stability of condensed systems relative to curvature of the front is investigated in a linear approximation. Two of the simplest combustion models are examined, a gasless system and a model of flameless combustion of a solid fuel. In the first case, the combustion products are condensed, just as are the initial materials, and in the second the solid fuel is converted into a gas in which no chemical reactions occur. Boundaries of the stability of the stationary combustion mode are found. It is shown that gasless systems are less stable with respect to two-dimensional perturbations than to one-dimensional perturbations. For the flameless combustion model the result depends on the relationship between the thermophysical constants of the initial material and the products. The question of the influence of heat emission on the one-dimensional stability of the gasless composites is considered. An increase in the heat emission diminishes the stable combustion region, where a one-dimensional instability originates earlier than collapse of combustion occurs because of strong heat emission to the wall.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 51–59, September–October, 1971.     

Convective flows in a cylindrical fluid zone in a high-frequency vibration field

Convective flows of a nonuniformly heated fluid in a cylindrical fluid zone in a high-frequency longitudinal vibration field are studied. Vibration frequencies which are high as compared with dissipative decrements and capillary frequencies, but small as compared with acoustic frequencies are considered. The general method formulated earlier for describing the behavior of inhomogeneous fluids under the influence of high-frequency vibrations is used. The interaction between the vibrational flow mechanisms and thermocapillary effects on a free surface is analyzed.Perm'. Marseilles (France). Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 53–61, September–October, 1994.     

Three-dimensional stability theory of deformed bodies. Internal instability

Conclusions In studying internal instability effects for elastic (which is fully obvious) and elastoplastic models of deformable bodies the approximate approach [12, 15] in the three-dimensional stability theory leads to results which disagree quantitatively and qualitatively with the corresponding results of the three-dimensional linearzed stability theory of deformable bodies (the second variant of the theory of small subcritical deformations). In this connection, in studying internal instability effects for various models of deformable bodies, in which elastic or elastoplastic deformations are substantial, the use of this approximate approach is recommended.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 21, No. 11, pp. 3–17, November, 1985.     

Convective combustion of aerosuspensions in fuel that contains the oxidant

The convective combustion of porous gunpowder and high explosives is an intermediate stage in the transition from layered combustion to detonation [1, 2]. The theory of convective combustion of such systems is developed in [3–6]. It has now become necessary to analyze the possibility of convective combustion of aerosuspensions. The present paper develops the theory of the combustion of such systems on the basis of an analysis of the equations of gas dynamics with distributed supply of mass and heat; the problem of nonstationary motion of a convective combustion front is formulated. In the homobaric approximation [7], when the pressure is assumed to be spatially homogeneous, an analytic solution to the problem is found; this determines the law of motion of the front and the distribution of the parameters that characterize the gas and the particles in the combustion zone. Necessary conditions for the transition from convective combustion to explosion are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 49–56, September–October, 1980.I thank R. I. Nigmatulin for helpful comments and advice, and also V. A. Pyzh and V. K. Khudyakov for discussing the work.     

Certain types of vibrationally convective instability of a two-dimensional fluid layer in zero gravity

The stability of a new equilibrium configuration possible in a two-dimensional layer of nonisothermal fluid executing high-frequency vibrations in zero gravity is investigated in the framework of the linear theory. A study is made on the basis of the equations of vibrational convection. Instability with respect to one-dimensional and two-dimensional perturbations is studied. An elementary exact solution is obtained for the one-dimensional perturbations. Vibrationally connective instability of a fluid in zero gravity has been studied in a number of papers [1-3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 4–7, September–October, 1987.The author expresses his gratitude to G. Z. Gershuni for his constant interest in my work.     

Finite deformation of elastic membranes with application to the stability of an inflated and extended tube

A theory is formulated for the finite deformation of a thin membrane composed of homogeneous elastic material which is isotropic in its undeformed state. The theory is then extended to the case of a small deformation superposed on a known finite deformation of the membrane. As an example, small deformations of a circular cylindrical tube which has been subjected to a finite homogeneous extension and inflation are considered and the equations governing these small deformations are obtained for an incompressible material. By means of a static analysis the stability of cylindrically symmetric modes for the inflated and extended cylinder with fixed ends is determined and the results are verified by a dynamic analysis. The stability is considered in detail for a Mooney material. Methods are developed to obtain the natural frequencies for axially symmetric free vibrations of the extended and inflated cylindrical membrane. Some of the lower natural frequencies are calculated for a Mooney material and the methods are compared.     

Unsteady flow of a viscous gas between two solid walls,one of which is free and vibrating at high frequency

The pressure field is investigated in a thin layer of a viscous compressible gas between two walls, one of which is free and executing high-frequency harmonic vibrations. Asymptotic methods are applicable to the case in which one wall vibrates at high frequencies [1]. The motion of the gas, as shown in [2], can be assumed to be nearly isothermal, and the influence of the inertial terms in the equation of motion for the gas can be neglected.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 111–116, March–April, 1971.     

On the motions of a double pendulum with vibrating suspension point

We consider the motions of a system consisting of two pivotally connected physical pendulums rotating about horizontal axes. We assume that the system suspension point, which coincides with the suspension point of one of the pendulums, performs harmonic vibrations of high frequency and small amplitude along the vertical. We also assume that the system has four relative equilibrium positions in which the suspension points and the pendulum centers of mass lie on one vertical line. We study the stability of these relative equilibria. For arbitrary physical pendulums, we obtain stability conditions in the linear approximation. For a system consisting of two identical rods, we solve the stability problem the in nonlinear setting. For the same system, we study the existence, bifurcations, and stability of high-frequency periodic motions of small amplitude other than the relative equilibria on the vertical line. The studies of dynamic stability augmentation in mechanical systems under the action of high-frequency perturbations was initiated in the paper [1], where it was shown that the unstable inverted equilibrium of a pendulum may become stable if the suspension point vibrates rapidly. This idea was developed in [2–10] and other papers, where several aspects of motion of a mathematical pendulum in the case of rapid small-amplitude vibrations of the suspension point were studied in the linear setting and also (without full mathematical rigor) in the nonlinear setting. The motions of the suspension point along an arbitrary oblique straight line [2, 4, 7, 8], along the vertical [3, 5, 6], along the horizontal [9], and in the case of damping [8] were considered. The monograph [10] deals with the stabilization of a pendulum or a system of pendulums under periodic and conditionally periodic vibrations of the suspension point along the vertical, along an oblique straight line, and along an ellipse. A rigorous nonlinear analysis of the existence and stability of periodic motions of the mathematical pendulum under horizontal and oblique vibrations of the suspension point at arbitrary frequencies and amplitudes can be found in [11, 12]. For the case of vertical vibrations of the suspension point at an arbitrary frequency and amplitude, a rigorous stability analysis of the relative equilibria of the pendulum on the vertical was carried out in [13].     

Stability of combustion of powder in a phenomenological model with nonadiabatic flame

A stability criterion for combustion of powder is obtained, taking into account the effect of the processes in the gas phase. It is shown that consideration of the effects of a nonadiabatic flame leads to the stability reserve of combustion being reduced and the natural frequency of vibrations being lowered. The effects thus found are physically explained by the radiation of a part of energy from the combustion zone with thermal and acoustic waves.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 82–87, November–December, 1973.Deceased.     

Development and instability of steady convective flows in a square cavity heated from below and a field of vertically directed vibrational forces

The present paper is devoted to numerical investigation of the spatial structure and stability of secondary vibrational convective flows resulting from instability of the equilibrium of a fluid heated from below. Vibrations parallel to the vector of the gravitational force (vertical vibrations) are considered. As in earlier work [7–9], a region of finite size is used — a square cavity heated from below. It is shown that enhancement of the vibrational disturbance of the natural convective flow may either stabilize or destabilize flows with different spatial structures; it may also stabilize certain solutions of the system of convection equations that are unstable in the absence of vibrational forces. In addition, increase of the vibrational Rayleigh number can lead to a change of the mechanisms responsible for equilibrium instability and oscillatory instability of the secondary steady flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–18, March–April, 1991.I thank G. Z. Gershuni for assistance and extremely fruitful discussions of the results of the paper.     

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.     

Oscillation,Instability and Control of Stepper Motors

A novel approach to analyzing instability in permanent-magnet stepper motors is presented. It is shown that there are two kinds of unstable phenomena in this kind of motor: mid-frequency oscillation and high-frequency instability. Nonlinear bifurcation theory is used to illustrate the relationship between local instability and mid-frequency oscillatory motion. A novel analysis is presented to analyze the loss of synchronism phenomenon, which is identified as high-frequency instability. The concepts of separatrices and attractors in phase-space are used to derive a quantity to evaluate the high-frequency instability. By using this quantity one can easily estimate the stability for high supply frequencies. Furthermore, a stabilization method is presented. A generalized approach to analyze the stabilization problem based on feedback theory is given. It is shown that the mid-frequency stability and the high-frequency stability can be improved by state feedback.     

Secondary vibrational convective motions in a flat vertical layer of liquid

Investigations of the stability of steady-state plane-parallel convective motion between vertical planes heated to different temperatures [1–5] have shown that this motion, depending on the value of the Prandtl number P, exhibits instability of two types. With small and moderate Prandtl numbers, the instability is of a hydrodynamic nature. It is brought about by monotonic perturbations which, in the supercritical region, develop into a periodic, with respect to the vertical, system of steady-state vortices at the interface between the opposing convective flows. Articles [6, 7] are devoted to the numerical investigation of nonlinear secondary steady-state flows. If P>11.4, there appears a new mode of instability, i.e., running thermal waves increasing in the flow; with P>12, this mode becomes more dangerous [4]. This instability is connected with the development of vibrational perturbations, and it can be considered that in the supercritical region the perturbations lead to the establishment of steady-state vibrations. Linear theory has made it possible to determine the boundaries of the regions of stability. In the present article a numerical investigation is made of nonlinear supercritical conditions developing as a result of a loss of stability of the steady-state flow with respect to vibrational perturbations.