Secondary convective motions in a plane inclined layer

A linear theory of stability of a plane-parallel convective flow between infinite isothermal planes heated to different temperature was developed in [1–6]. At moderate Pr values the instability is monotonic and leads to the development of steady secondary motions. These motions for the case of a vertical layer have been investigated by the net [7, 8] and small-parameter [9] methods. In this paper steady secondary motions in an inclined layer are investigated. The small-parameter and net methods are used. The hard nature of excitation of secondary motions in a defined range of tilt angles is established. There are two types of secondary motions, whose regions of existence overlap — vortices at the boundary of countercurrent streams and convection rolls; the hard instability is due to the development of convection rolls. The analog of the Squire transformation obtained in [4] for infinitely small disturbances of a plane-parallel convective flow is extended to secondary motions of finite amplitude.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–9, May–June, 1977.I thank G. Z. Gershumi, E. M. Zhukhovitskii, and E. L. Tarunin for interest in the work and valuable discussion.     

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.     

Stability of wavy conditions in a film flowing down an inclined plane

The nonlinear theory of motion in a film of liquid flowing down an inclined plane predicts the existence of an interval k₀m, inside of which the wave number of periodic wave motion may lie [1]. The condition of the stability of experimentally attained motions imposes a limitation on their wave numbers. In [2] a numerical investigation of the stability of wavy motions was made; in the investigated range of change in the Galileo number and the wave number all the motions were found to be unstable; however, the fastest growing were perturbations imposed on a motion with a determined wave number (“optimal” conditions). In [3] the instability of motions with a wavelength exceeding some limiting value was established in a long-wave approximation. In the present work, within the framework of the two-dimensional problem, an investigation was made of the stability of periodic wavy motions, based on expansion in terms of the small parameter k_m. It is established that, within the interval k₀m, there lies a finite subinterval of wave numbers for which wavy motions are stable. The narrowness of this interval (δk≈0.07 k_m) may be the reason why, in the experiment, with not too great Galileo numbers for fully established periodic wavy motions, no substantial differences in the wave-length are observed [4].     

Shock formation during a sudden “inclusion∝ of external effects

The problem of shock origination in a one-dimensional gas flow is considered in the case when the gas is subjected to external force and thermal effects on a certain section of the flow starting with the time t=0. Prior to the time t=0 the flow is stationary, and there are no external forces and energy sources. Situations are investigated for which the shock formation is associated with intersection of the characteristics from one family with the boundary characteristic separating the domain of unperturbed (stationary) flow from the nonstationary flow domain originating because of the “inclusion∝ of the external effects. This problem is particularly interesting in connection with a study of the initial stage of flow evolution in different magnetohydrodynamic apparatuses. Its solution can be used to explain and comprehend more deeply the results of numerical integration (some examples of such computations are presented in [1]) and of a “stationary∝ analysis which can be used to describe the flow for “large∝ t in a number of cases (see [2]). The analysis executed has much in common with the analysis of the question of shock origination during the advance of a piston in a gas [3] although the perturbing effect in this latter case is a surface and not a volume effect. The solution is carried out by using semicharacteristic variables, which were used in [4] in an investigation of shock formation in the piston problem in chemically reacting mixtures.     

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.     

Convection currents in a vertical layer with a space-modulated temperature distribution on the boundaries

Steady convective motions in a plane vertical fluid layer are investigated. The temperature along the boundaries of the layer varies harmonically and has different average values on each of the boundaries. Thus space-period modulation of the temperature of the walls is assigned along with average lateral heating of the layer. The form of the plane steady motions and regions of existence of through currents and currents of cellular structure are found for various values of the parameters of the problem by the finite difference grid-point method. The dependence of the main characteristics of fluid motion on the Grashof number is determined. The results presented in the article pertain to the case when the period of modulation of the temperature of the boundaries coincides with the wavelength of the critical mode of a plane-parallel current. A numerical investigation of supercritical motions in a vertical layer with plane isothermal boundaries heated to a different temperature was carried out in [1–3]. The effect of a space-periodic inhomogeneity due to curvature of walls on the form and stability of convective motions in a vertical layer with lateral heating was examined in [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 20–25, September–October, 1978.The author thanks E. M. Zhukhovitskii for formulating the problem and supervising the work and G. Z. Gershuni for discussions and useful comments.     

Instability and flow of a weakly conducting fluid in the presence of oxidation-reduction reactions at electrodes and recombination

Experiments show that a weakly conducting fluid in a plane-parallel system of electrodes is set into motion if the field intensity is sufficiently great [1–5]. The loss of stability is due to the formation of charges near the electrodes and the influence of the Coulomb forces on these charges. The formation of the space charges is usually attributed to oxidation-reduction electrode reactions and bulk recombination of the ions formed at the electrodes [1–4]. In the present paper, the stability of a weakly conducting fluid in a plane-parallel system of electrodes with symmetric distribution of the space charge is studied. The methods of the theory of solution bifurcation are used to construct the stationary flow which arises after the loss of stability and to investigate the stability of this flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 20–26, July–August, 1981.     

Oscillations of liquid in a vessel in the form of a rectangular parallelepiped

We consider the forced and the free oscillations of a liquid partially filling a cavity in the form of a rectangular parallelepiped. The characteristics of these oscillations are studied for small deformations of the free surface. It is shown that for definite frequencies and amplitudes of two-dimensional translational motions of the parallelepiped the fundamental of the liquid oscillations is excited in the plane perpendicular to the plane of motion of the vessel. The effect of small linear damping of the liquid oscillations on the shape of the boundaries of the principal region of instability of the liquid oscillations is evaluated. Fairly large oscillations of a liquid in a cylinder were considered in [1]. The same problem for a cavity of arbitrary configuration was studied in [2]. We note also that the conclusions of the study presented here are in qualitative agreement with the basic results obtained by a somewhat different method in [3] for a cavity in the form of a right circular cylinder.     

Modeling of properties and motions of an inhomogeneous one-dimensional continuum of a complicated Cosserat-type microstructure

We consider an approach to modeling the properties of the one-dimensional Cosserat continuum [1] by using the mechanical modeling method proposed by Il’yushin in [2] and applied in [3]. In this method, elements (blocks, cells) of special form are used to develop a discrete model of the structure so that the average properties of the model reproduced the properties of the continuum under study. The rigged rod model, which is an elastic structure in the form of a thin rod with massive inclusions (pulleys) fixed by elastic hinges on its elastic line and connected by elastic belt transmissions, is taken to be the original discrete model of the Cosserat continuum. The complete system of equations describing the mechanical properties and the dynamical equilibrium of the rigged rod in arbitrary plane motions is derived. These equations are averaged in the case of a sufficiently smooth variation in the parameters of motion along the rod (the long-wave approximation). It was found that the average equations exactly coincide with the equations for the one-dimensional Cosserat medium [1] and, in some specific cases, with the classical equations of motion of an elastic rod [4–6]. We study the plane motions of the one-dimensional continuum model thus constructed. The equations characterizing the continuum properties and motions are linearized by using several assumptions that the kinematic parameters are small. We solve the problem of natural vibrations with homogeneous boundary conditions and establish that each value of the parameter distinguishing the natural vibration modes is associated with exactly two distinct vibration mode shapes (in the same mode), each of which has its own frequency value.     

Stability of a plane-parallel convective flow of a liquid in a horizontal layer

We consider the stationary plane-parallel convective flow, studied in [1], which appears in a two-dimensional horizontal layer of a liquid in the presence of a longitudinal temperature gradient. In the present paper we examine the stability of this flow relative to small perturbations. To solve the spectral amplitude problem and to determine the stability boundaries we apply a version of the Galerkin method, which was used earlier for studying the stability of convective flows in vertical and inclined layers in the presence of a transverse temperature difference or of internal heat sources (see [2]). A horizontal plane-parallel flow is found to be unstable relative to two critical modes of perturbations. For small Prandtl numbers the instability has a hydrodynamic character and is associated with the development of vortices on the boundary of counterflows. For moderate and for large Prandtl numbers the instability has a Rayleigh character and is due to a thermal stratification arising in the stationary flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–100, January–February, 1974.     

Investigation of convective motions of a gas due to propagation of a temperature discontinuity along the boundary of a closed region

The Navier-Stokes equations are used in a numerical study of the two-dimensional motions of a compressible gas in a closed rectangular region in a gravity field. The motion of the gas is due to the propagation of a temperature discontinuity along the lower boundary of the region. The mechanism of formation of eddy structures is followed in detail for different velocities of the discontinuity and different ratios of the sides of the region. The method of stabilization is used to obtain different stationary solutions to the problem of convection in a rectangular region heated below. The realization of a particular stationary solution depends on the history of the process. Problems of the convective motion of liquids and gases in closed regions heated below, including questions relating to the nonuniqueness of stationary solutions, are considered in the monograph [1] and the review [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 87–92, September–October, 1980.We thank V. B. Librovich, L. A. Chudov, and G. M. Makhviladze for guidance and helpful discussions.     

ON THE GLOEAL STABILITY OF THE ROTATIONAL MOTION AND THE QUALITATIVE ANALYSIS OF THE BEHAVIOUR OF DISTURBED MOTION OF A RIGID BODY HAVING A LIQUID-FILLED-CAVITY

This paper is a continuation of[1].In this paper we inves-tigate the distribution of steady motions of the liquid-filled-cavity body,decide the stability of each steady motion and findout the corresponding regions of stability and instability.Be-sides.the behaviour of disturbed motion is analysed qualitatively.     

Self-excited yawing vibrations of railway cars

The site [1] presents a video recording of the motion of empty freight railway cars. The video demonstrates that some railway cars experience angular self-excited vibrations in the horizontal plane (i.e., yawing). In the same reference, the main causes of such a “negative dynamics” of railway cars are listed. All causes are related to some defects in railway car manufacturing. We show that such self-excited vibrations are possible even in the case of absolutely ideal railway car without any defects at all. It is shown that, for a certain combination of parameters, the rectilinear (unperturbed) motion of a railway car may experience flutter instability. This type of self-excited vibrations of railway cars has not been studied earlier [2, 3].     

Conditions for nonlinear stability of flows of an ideal incompressible liquid

Stability of steady state flows of an ideal incompressible liquid with homogeneous density with some type of symmetry (translational, axial, rotational, or helical) is considered. Two types of sufficient conditions for nonlinear stability are obtained, which can be proven by constructing two types of functionals which have absolute minima at the given steady state solutions. Each of the functionals used is the sum of the kinetic energy and some other integral, specific to the given class of motion. The first type of stability conditions are a generalization to the case of finite perturbations and a new class of flows of the well known Rayleigh criterion [1] for centrifugal stability of rotating flows relative to perturbations with rotational symmetry. In the same sense the second type of stability conditions generalize another result, also originally proposed by Rayleigh, according to which plane-parallel flow of a liquid is stable in the absence of an inflection point in the velocity profile [1]. A nonlinear variant of the latter condition for the class of planar motions was first obtained in [2]. To systematize the results extensive use is made of the analogy between the effects of density stratification and rotation in the form of [3], The results to be presented relate to stability of a wide class of hydrodynamic flows having the required symmetry. For example, they relate to flows in tubes and channels which rotate or are at rest, and flows with concentrated annular or spiral vortices.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 70–78, May–June, 1986.     

Stability of heat transfer regimes at the front stagnation point of a body in a stream of dissociated gas

The formulation and solution of the stationary problem of heat transfer in the neighborhood of the front point of a body at constant temperature in a stream of dissociated air are given in [1]. In [2], the results are given of numerical solution of this problem in the nonstationary formulation; the establishment of a stationary heat transfer regime was established for all the calculated variants. In the present paper, we investigate the stability of stationary heat transfer regimes at the front stagnation point of a body in a stream of dissociated air using the Lyapunov functional method [3, 4] and the method of [2, 5], which is based on the use of Meksyn's method in boundary-layer theory [6, 7]. It is established that an arbitrarily strong growth of the Damköhler number does not lead to instability and multiplicity of the stationary regimes, in contrast to the case when a hot mixture of gases flows over the front point of a thermostat [2, 5, 8]. Numerical solution of the boundary-layer equations for a wide range of Damköhler numbers confirms the results of the approximate qualitative analysis and shows that in a number of cases the time of establishment of the stationary state is a nonmonotonic function of the Damköhler number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 97–106, September–October, 1979.     

Linear stability of plane couette flow of an upper convected maxwell fluid

We investigate the linear stability of plane Couette flow of an upper convected Maxwell fluid using a spectral method to compute the eigenvalues. No instabilities are found. This is in agreement with the results of Ho and Denn [1] and Lee and Finlayson [2], but contradicts “proofs” of instability by Gorodtsov and Leonov [3] and Akbay and Frischmann [4,5]. The errors in those arguments are pointed out. We also find that the numerical discretization can generate artificial instabilities (see also [1,6]). The qualitative behavior of the eigenvalue spectrum as well as the artificial instabilities is discussed.     

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.     

Stability of convective flow in a vibration field with respect to three-dimensional perturbations

A plane-parallel convective flow in a vertical layer between boundaries maintained at different temperatures becomes unstable when the Grashof number reaches a critical value (see [1]). In [2, 3] the effect of high-frequency harmonic vibration in the vertical direction on the stability of this flow was investigated. The presence of vibration in a nonisothermal fluid leads to the appearance of a new instability mechanism which operates even under conditions of total weightlessness [4]. As shown in [2, 3], the interaction of the usual instability mechanisms in a static gravity field and the vibration mechanism has an important influence on the stability of convective flow. In this paper the flow stability is considered for an arbitrary direction of the vibration axis in the plane of the layer and the stability characteristics with respect to three-dimensional normal perturbations are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 116–122, March–April, 1988.