Vibrational thermal convection in a cavity executing high-frequency rocking motions

The vibrational thermal convection in a cavity executing high-frequency rocking motions is investigated. The equations of vibrational convection are obtained by the method of averaging. It is shown that rocking motions lead to some new and distinctive effects. The convective stability in a plane layer in the presence of such vibrations is investigated on the basis of the obtained equations. A comparison with known experimental data is made. The results of experiments confirm the theoretical conclusions drawn on the basis of the averaged equations of vibrational convection.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 138–144, May–June, 1988.I thank E. M. Zhukhovitskii for helpful discussion.     

Effect of a latitudinal temperature gradient on convective motions arising in a rotating spherical layer

The hydrodynamics of planetary atmospheres and Interiors are frequently directly or indirectly connected with convective motions taking place in rotating liquid spherical layers in the field of a central force. Convective stability in a spherical layer at rest, in a central gravity field, was first discussed in [1, 2]. It was shown that the critical Rayleigh number Rao at which convective instability sets in and the wave number of the critical perturbations depend essentially on the thickness of the layer. As in the plane case, the problem of the convective stability of a spherical layer is found to be degenerate, and the form of the critical perturbations cannot be determined from the linear problem. In actuality, minimization of the Rayleigh number permits establishing only the wave numberl for the spherical harmonic Y _l ^m (θ, ?), realized at the limit of stability; the parameter m remains indeterminate and thus 2l+1 independent convective modes correspond to Rao. In [3] a study was made of the convective stability of a liquid in a slowly rotating thin spherical layer. It was shown that the presence of rotation eliminates the degeneracy; at the limit of stability there arise motions corresponding to the Y _l l (θ, ?) -harmonic with a degenerate maximum at the equator, and propagating in a wave manner toward the side opposite to the rotation. In the present work a study is made of the convective stability of a flow of liquid, arising in a rotating spherical layer due to a nonuniform distribution of the temperatures at one of the boundaries of the layer. In such a statement of the problem it is possible to model large-scale motions in the atmospheres of large planets having internal sources of heat and absorbing solar radiation near the cloud cover of the atmosphere. It is established that, depending on the relationships between the parameters imparting the rotation and the inhomogeneous distribution of the temperature, there is either stabilization or destabilization of the layer in comparison with a fixed layer of the same thickness and with the same, but uniformly distributed heat flux supplied to the layer. A study is made of the form of the corresponding critical perturbations.     

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.     

Compressibility effects under conditions of convective stability of miscible fluids in a porous medium

The effect of compressibility on the convective stability of equilibrium of a binary mixture in a homogeneous porous medium is analyzed. It is shown that the contribution of compressibility significantly increases the equilibrium stability with respect to oscillatory perturbations.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–23, January–February, 1995.     

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.     

Effect of high-frequency vibration on the development of secondary convective conditions

An investigation is made of the development of convective flows of a viscous incompressible liquid, subjected to high-frequency vibration. The nonlinear equations of convection are used in the Boussinesq approximation, averaged in time. The amplitude of the perturbations is assumed to be small, but finite. For a horizontal layer with solid walls the existence of both subcritical and supercritical stable secondary conditions is established. In a linear statement, the problem of stability in the presence of a modulation has been discussed in [1–3]. Articles [4–6] were devoted to investigation of the nonlinear problem. In [4], the method of grids was used to study secondary conditions in a cavity of square cross section. In the case of a horizontal layer with free boundaries [5, 6], the character of the branching is established by the method of a small parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 90–96, March–April, 1976.The authors thank I. B. Simonenko for his useful evaluation of the work.     

Coupled stability problem for plane-parallel convective flow

The linear stability of convective flow in a layer between large masses with arbitrary thermal properties is investigated. When the thermal conductivities of the liquid and the masses are commensurable, the stability problem must be solved in the coupled formulation with allowance for the penetration of the temperature perturbations into the masses.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 18–22, July–August, 1989.The author is grateful to G. Z. Gershuni for his constant interest.     

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.     

Non-Darcian Effects on the Onset of Convection in a Porous Layer with Throughflow

The Brinkman extended Darcy model including Lapwood and Forchheimer inertia terms with fluid viscosity being different from effective viscosity is employed to investigate the effect of vertical throughflow on thermal convective instabilities in a porous layer. Three different types of boundary conditions (free–free, rigid–rigid and rigid–free) are considered which are either conducting or insulating to temperature perturbations. The Galerkin method is used to calculate the critical Rayleigh numbers for conducting boundaries, while closed form solutions are achieved for insulating boundaries. The relative importance of inertial resistance on convective instabilities is investigated in detail. In the case of rigid–free boundaries, it is found that throughflow is destabilizing depending on the choice of physical parameters and the model used. Further, it is noted that an increase in viscosity ratio delays the onset of convection. Standard results are also obtained as particular cases from the general model presented here.     

Convective stability of a medium containing a heavy solid additive

Two problems of convective stability in a medium containing settling heavy solid particles are discussed. A study is made of the stability of steady convective flow of a medium containing an additive between vertical plates heated to different temperatures and also of the stability of a flat layer of a medium containing an additive which is heated from below. It is shown that the presence of settling solid particles has a significant stabilizing effect on convective stability.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 105–115, May–June, 1976.The author thanks E. M. Zhukhovitskii for directing the work, V. E. Nakoryakov and participants in the seminars directed by him, and also A. G. Kirdyashkin for providing valuable discussions of the results.     

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.     

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.     

Convective stability of a vertical layer of a non-newtonian liquid

We consider the convective stability of a non-Newtonian (nonlinearly viscous) liquid in a two-dimensional vertical channel. We solve a nonlinear boundary value problem concerning plane-parallel stationary convection for the case of piecewise-linear and power-law type rheological characteristics. We discuss the problem concerning the stability of equilibrium and of stationary motions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 88–95, September–October, 1973.The authors thank D. V. Lyubimov for his help in carrying out the calculations.     

Thermocapillary and thermogravitational convection in a two-layer system with curved interface

In an earlier study [1], the present authors used the complete nonlinear hydrodynamic equations to investigate thermocapillary convection in a two-layer system. Oscillatory instability of the equilibrium was established for some ratios of the parameters. In the present paper, a study is made of the influence on the thermocapillary convective motions of two different factors — curvature of the interface and gravity. It is established that curvature of the interface can lead to significant changes in the flow structure and hysteresis transitions between convection regimes. In the case of the joint influence of the thermogravitational and thermocapillary instability mechanisms, many different flow regimes are found: steady motions with different directions of rotation of the vortices and periodic and nonperiodic oscillatory motions with different spatial structures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–179, May–June, 1984.We thank E. M. Zhukovitskii for discussing the results.     

Convective auto-oscillations in the components of binary stars

A study is made of the convection in a shell of matter in a star in a binary Keplerian system. Because the axial rotation and orbital revolution are not synchronous, there is a periodic modulation of the gravity field of the star by tidal forces. It is shown that under certain conditions one can have an auto-oscillatory regime in which the mean intensity of the oscillatory convective motions also varies periodically with the time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 149–155, March–April, 1981.     

Rayleigh stability of shear flow in relaxing media

The stability of steady-state flow is considered in a medium with a nonlocal coupling between pressure and density. The equations for perturbations in such a medium are derived in the linear approximation. The results of numerical integration are given for shear motion. The stability of parallel layered flow in an inviscid homogeneous fluid has been studied for a hundred years. The mathematics for investigating an inviscid instability has been developed, and it has been given a physical interpretation. The first important results in flow stability of an incompressible fluid were obtained in the papers of Helmholtz, Rayleigh, and Kelvin [1] in the last century. Heisenberg [2] worked on this problem in the 1920's, and a series of interesting papers by Tollmien [3] appeared subsequently. Apparently one of the first problems in the stability of a compressible fluid was solved by Landau [4]. The first investigations on the boundary-layer stability of an ideal gas were carried out by Lees and Lin [5], and Dunn and Lin [6]. Mention should be made of a series of papers which have appeared quite recently [7–9]. In all the papers mentioned flow stability is investigated in the framework of classical single-phase hydrodynamics. Meanwhile, in recent years, the processes by which perturbations propagate in media with relaxation have been intensively studied [10–12].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 87–93, May–June, 1976.     

Numerical study of convection of a liquid heated from below

The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances. If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value R_c steady-state motions of the liquid are not possible. With R>R_c a steady convection arises, whose amplitude near the threshold is small and proportional to (R–R_c)^1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to R_c, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value R_c. The numerical solution permits following the development of the steady motion which arises with R>R_c in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method.     

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.     

Convective instability of a homogeneous fluidized bed

Convective motions of the phases in a homogeneous fluidized bed are considered. The motions of the phases are described by a simple model of two ideal interpenetrating interacting fluids. The model admits an increasing concentration of the solid particles with the height, which leads to circulatory flows. Approximate equations of motion of the liquid and the solid particles are obtained, and these are analogous to the Boussinesq approximation in the case of natural convection in a pure liquid. The equations contain a parameter which is analogous to the Rayleigh number and characterizes the stability of the layer. An approximate analytic solution to the corresponding eigenvalue problem is found. The spectrum of Rayleigh numbers and the sizes of the convective cells are determined. The obtained results provide an explanation for the existence of the multicenter circulatory motions of the phases frequently observed in a fluidized beds and make it possible to determine the sizes of the circulatory regions. Information about the Rayleigh number spectrum and the sizes of the regions is needed to solve the problem of the scale transition and also to enable the choice of measures to suppress or intensify the displacement processes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 39–46, July–August, 1982.     

Stability of quasistatic equilibrium state of an inhomogeneous viscous layer on an inclined plane

A study is made of the stability against small perturbations [1] of a slow flow of an incompressible inhomogeneous linearly viscous liquid under the influence of a force of gravity on an unbounded inclined plane. Problems of such kind arise in glaciology when one estimates the stability of snow on mountain slopes or determines the catastrophic movement of a glacier; the results can also be applied to solifluction phenomena [2, 3]. Equations for perturbations of parallel flows of linearly viscous fluids in the case of a continuous variation of the viscosity and density across the flow were derived in [4]. An attempt to solve the hydrodynamic problem with allowance for a perturbation of the viscosity was made in [5]; however, in the equations for the perturbations, simplifications resulted in the neglect of terms that take into account perturbations of the viscosity. In the quasistatic formulation considered here in the case when allowance is made for perturbation of the density and viscosity, the equation for the perturbation amplitudes is an ordinary differential equation with variable coefficients; analytic solution of the equation is very difficult, even for long-wave perturbations. In this connection a study is made of the stability of a laminar model; the viscosity and density are constant within each layer. A similar hydrodynamic problem in the long-wave approximation was considered in [6]. In the present paper an exact solution is constructed in the quasistatic formulation for a two-layer model; the solution shows that the viscosity of the lower layer has an important influence on the stability. Essentially, instability is observed when the lower layer acts as a lubricant.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 20–24, November–December, 1973.