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-amplitude convection in mixtures with heat sources proportional to the concentration of one of the components

A numerical investigation has been made into the equilibrium stability with respect to finite perturbations of a mixture with heat sources proportional to the concentration of an active component. The convective motions that develop after the loss of stability were also studied. The equations of thermoconcentration convection were solved by the grid method for a planar region of rectangular shape simulating a convective cell in the horizontal layer. Neutral curves for finite-amplitude perturbations are constructed, the regions of existence of subcritical motions are found, and a comparison with the results of linear theory is made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 10–16, November–December, 1982.We thank E. M. Zhukovitskii for discussing the results of the paper.     

Effect of a permeable partition on convective instability in a rectangular cavity

The equilibrium stability of a fluid, heated from below, in a rectangular cavity with a vertical permeable partition is investigated. The small perturbation problem is solved by the Galerkin-Kantorovich method. The relations obtained for the dependence of the critical Rayleigh numbers on the partition parameters and the cavity dimensions make it possible to identify regions in which either even or odd perturbations, sensitive to only the normal or only the tangential resistance of the partition, respectively, are responsible for equilibrium crisis. The effect of a permeable partition on the convective instability of a horizontal layer of fluid under various heating conditions was considered in [1–3], where a significant dependence of the critical Rayleigh numbers on the properties of the partition was established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 6–10, May–June, 1989.     

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.     

The influence of gas solubility on the stability of a layer of liquid with bubbles and an immobile filling

The instability of a bubbling layer due to the presence of a vertical gradient in the ascent velocity of the bubbles, causing stratification of the layer with respect to density, is considered in [1]. A similar instability mechanism of a fluidized bed is studied in [2]. The stabilizing influence of electrical and magnetic fields on a bubbling layer is shown in [3]. Consideration is given in [4] to the influence of the conditions of supply of the gas on the stability of a bubbling layer with an immobile filling. The present work deals with the stability of the mechanical equilibrium of a horizontal layer of liquid with an immobile filling through which a gas soluble in the liquid is bubbled. It is shown that there exists a critical solubility of the gas at which the mechanical equilibrium is unstable with respect to monotonie perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 68–74, September–October, 1984.The author would like to thank V. P. Myasnikov and V. V. Dil ' man for their interest in this work, and M. H. Rozenberg for assistance with the programming.     

On the convective instability of a liquid in an inclined layer of a porous medium

In this paper we study the stability of the equilibrium of a liquid heated from below, wherein the liquid saturates a planar layer of a porous medium arbitrarily inclined to the direction of gravity. We consider the cases for which the boundaries of the layer are heat-conducting and also thermally insulated. In a horizontal layer with heat-conducting boundaries equilibrium is destroyed by perturbations of cellular structure [1], In a vertical layer the minimum critical temperature gradient corresponds to perturbations of plane-parallel structure. The transition to cellular perturbations in the case of heat-conducting boundaries takes place at an arbitrarily small angle of inclination of the layer to the vertical. For the thermally insulated layer the crisis of equilibrium is connected with plane-parallel perturbations at all angles of inclination.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 127–131, May–June, 1973.The author thanks G. Z. Gershuni for stating the problem and his interest in the work.     

Vibrational Convection in an Inclined Fluid Layer Heated from Below

The stability of mechanical equilibrium of an inclined fluid layer with respect to three-dimensional perturbations under the action of high-frequency vibration is studied. It is shown that under heating from below the spiral perturbations are always the most dangerous for vibration transverse to the layer. For vertical vibration the stability limit is determined by three-dimensional perturbations whose shape depends in a complicated way on the angle of inclination of the layer and the vibrational Rayleigh number. In the limiting case of a thin vertical layer supercritical vibrational-convective motions are calculated numerically and analytically and scenarios of transition from quasi-equilibrium to irregular motions are studied.     

Parametric excitation of Rayleigh convection in the presence of a variable heat flux on the free surface

The onset of Rayleigh convection in a semi-infinite fluid layer is investigated for a heat flux harmonically modulated along the normal to the surface of the fluid. The problem of the evolution of the velocity and temperature perturbations is solved numerically by means of a finite-difference method. The stability limits and the characteristics of the critical perturbations are determined as functions of the Prandtl numbers. The behavior of the critical Rayleigh number is studied for finite layer depths.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 40–44, November–December, 1996.     

Thermocapillary instability of a liquid layer in the presence of a soluble surfactant

The effect of capillarity on the stability of a plane layer of viscous heat-conducting liquid in the presence of a soluble surfactant is investigated. It is found that an increase in surfactant solubility has a stabilizing effect on equilibrium. Monotonic instability is the most dangerous mode in the case of long-wave perturbations, whereas in the short-wave region loss of stability is induced by oscillatory perturbations.Krasnoyarsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–8, January–February, 1996.     

The effect of induced magnetic fields on convective stability of a conductive liquid carrying a current

The stability of equilibrium in a plane layer of an electrically conductive liquid located in electric and magnetic fields is considered. The effect of an unperturbed induced magnetic field and its perturbations on stability of this type is considered. The stability is analyzed nonlinearly.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 102–110, January–February, 1974.     

Parametric Excitation of a Secondary Flow in a Vertical Layer of a Fluid in the Presence of Small Solid Particles

Thermal convection is studied in an inhomogeneous medium consisting of a fluid and a solid admixture under conditions of finite–frequency vibrations. Convection equations are derived within the framework of the generalized Boussinesq approximation, and the problem of flow stability in a vertical layer of a viscous fluid with horizontal oscillations along the layer to infinitely small perturbations is considered. A comparison with experimental data is made.     

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.     

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.     

Convection in a horizontal fluid layer with specific-volume inversion

The effect of the position of the inversion point within the layer on the critical values of the Rayleigh number and the amplitudes of the rectangular-cell convective flows is numerically investigated. The monotonic instability of the mechanical equilibrium of the fluid with respect to small perturbations periodic along the layer is studied by the linearization method. The Lyapunov-Schmidt method is used to construct the secondary steady convective flows. The applicability of these methods in incompressible fluid stability problems was demonstrated in [8–10]. The calculations show that, starting from a certain value of the parameter , the branching is subcritical for any cell side ratio and a fixed wave vector modulus. For smaller values of the nature of the branching depends on the cell side ratio. This points to subcritical branching and hysteresis effects in those cases in which the periodicity of the perturbations is determined by external factors (corrugation of the boundary, spatially periodic temperature modulation, etc.). It is noted that the rectangular convection amplitude tends to zero when the cell side ratio tends to 3, the value at which hexagonal cellular convection is possible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 43–49, January–February, 1989.The author wishes to thank V. I. Yudovich for his interest and useful advice and the participants in the Rostov State University Computational Mathematics Department's Scientific Seminar for discussing the results.     

The stability of the flow of a layer of a viscous liquid with moment stresses along an inclined plane

The stability of the flow of a layer of liquid over an inclined plane, taking account of the spin of the molecules and the internal moment stresses, was discussed in [1], However, in [1], a number of errors were allowed to creep in, which led the authors to untrue qualitative and quantitative results. In the present work, the stability of the flow of a layer with respect to long-wave perturbations is investigated by the method of successive approximations [2, 3] under the assumption that the coefficient of rotational viscosity nr is considerably less than the coefficient of Newtonian viscosity . It is shown that, in a first approximation, internal moment stresses do not affect the stability of the motion, and that the rotation of the particles exerts a destabilizing effect on the flow of the layer with respect to three-dimensional periodic perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, pp. 149–151, September–October, 1977.     

Stability of the boundary layer of a non-Newtonian liquid obeying a power-type rheological law

The stability of the stationary (steady-state) laminar boundary layer of a non-Newtonian liquid obeying a power-type rheological law at a semiinfinite plate situated in a longitudinal flow is analyzed. An approximate formula is derived for estimating the minimum Reynolds number at which the flow loses stability with respect to slight two-dimensional perturbations. Calculations of the point of stability loss for aqueous solutions of carboxyl methyl cellulose are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 121–124, March–April, 1971.     

Convective Instability in Superposed Fluid and Porous Layers with Vertical Throughflow

A closed form solution to the convective instability in a composite system of fluid and porous layers with vertical throughflow is presented. The boundaries are considered to be rigid-permeable and insulating to temperature perturbations. Flow in the porous layer is governed by Darcy–Forchheimer equation and the Beavers–Joseph condition is applied at the interface between the fluid and the porous layer. In contrast to the single-layer system, it is found that destabilization due to throughflow arises, and the ratio of fluid layer thickness to porous layer thickness, , too, plays a crucial role in deciding the stability of the system depending on the Prandtl number.     

Stability of stationary convective flow of a mixture in a vertical porous layer

In contrast to the corresponding viscous flow, the convective flow of a homogeneous liquid in a planar vertical layer whose boundaries are maintained at different temperatures is stable [1]. When a porous layer is saturated with a binary mixture, in the presence of potentially stable stratification one must expect an instability of thermal-concentration nature to be manifested. This instability mechanism is associated with the difference between the temperature and concentration relaxation times, which leads to a buoyancy force when an element of the fluid is displaced horizontally. In viscous binary mixtures, the thermal-concentration instability is the origin of the formation of layered flows, which have been studied in detail in recent years [2–4]. The convective instability of the equilibrium of a binary mixture in a porous medium was considered earlier by the present authors in [5]. In the present paper, the stability of stationary convective flow of a binary mixture in a planar vertical porous layer is studied. It is shown that in the presence of sufficient longitudinal stratification the flow becomes unstable against thermal-concentration perturbations; the stability boundary is determined as a function of the parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–157, January–February, 1980.     

Effect of Rotation on the Stability of Advective Flow in a Horizontal Fluid Layer at a Small Prandtl Number

The stability of advective flow in a rotating infinite horizontal fluid layer with rigid bound-aries is investigated for a small Prandtl number Pr = 0.1 and various Taylor numbers for perturbations of the hydrodynamic type. Within the framework of the linear theory of stability, neutral curves describing the dependence of the critical Grashof number on the wave number are obtained. The behavior of finite-amplitude perturbations beyond the stability threshold is studied numerically.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 29–38.Original Russian Text Copyright © 2005 by Schwarz.     

Stability of unsteady viscous flows

A study is made of the linear stability of plane-parallel unsteady flows of a viscous incompressible fluid: in the mixing layer of two flows, in a jet with constant flow rate, and near a wall suddenly set in motion [1]. The slow variation of these flows in time compared with the rate of change of the perturbations makes it possible to use the method of two-scale expansions [2]. The stability of nonparallel flows with allowance for their slow variation with respect to the longitudinal coordinate was investigated, for example, in [3–6]. The unsteady flows considered in the present paper have a number of characteristic properties of non-parallel flows [1], but in contrast to them are described by exact solutions of the Navier-Stokes equations. In addition, for unsteady planeparallel flows a criterion of neutral stability can be uniquely established by means of the energy balance equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, 138–142, July–August, 1981.I thank G. I. Petrov for suggesting the problem, and also S. Ya. Gertsenshtein and A. V. Latyshev for assisting in the work.