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.     

Convective stability of a horizontal rotating fluid layer with distributed heat sources

The results of investigating the convective instability of a horizontal layer of rotating fluid, created by a temperature difference applied at the boundaries of the layer and by heat sources distributed according to various laws, are presented. It is shown that, when the other parameters of the problem are fixed, an increase in the internal heat release lowers the limits of both monotonic and oscillatory stability of the layer, increases the wave number and reduces the neutral oscillation frequency. An increase in source concentration towards the center of the layer intensifies the effect. As the strength of the internal heat sources and their concentration towards the center of the layer increase, the oscillating convection that develops at the stability limit when the Prandtl number is low and the rotation fairly fast is first replaced by monotonic convection and then ceases altogether.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–28, January–February, 1989.     

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.     

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.     

Convective stability of a horizontal rotating fluid layer with helical turbulence

The problem of the corrective stability of a horizontal layer of turbulent fluid rotating about a vertical axis with a fixed heat flow at the boundaries is investigated in the case in which the intensity of the helical background does not depend on the rate of rotation and the degree of heating.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 33–39, January–February, 1992.The author is grateful to S. S. Moiseev for proposing the subject and to G. Z. Gershuni and D. V. Lyubimov for useful discussions.     

Steady convective motions in a plane horizontal fluid layer with permeable boundaries

In a plane horizontal fluid layer bounded by permeable plane surfaces which are heated to different temperatures and between which transverse flow takes place with uniform velocity, convection occurs at a definite critical Rayieigh number. The study of the disturbance spectrum and the convective stability, made within the framework of linear theory in [1], showed that convective instability in the layer with permeable boundaries, just as in the case of the Rayieigh problem, is associated with the development of monotonie disturbances. It turns out that the transverse motion in the layer leads to a considerable increase of the Rayieigh number. Linear theory does not permit analysis of the development of the disturbances in the supercritical region. Analysis of the developed nonlinear motion can be made only on the basis of the complete nonlinear convection equations.In this investigation we made a numerical study of nonlinear motions in the supercritical region. Calculations were made on a computer via the grid method. Solutions are obtained for the nonlinear equations of motion over a wide range of Rayieigh numbers for different values of the Peclet number, whichdefines the intensity of the transverse motion in the layer.The author wishes to thank E. M. Zhukovitskii for his guidance, and G. Z. Gershuni and E. L. Tarunin for their interest and assistance in the study.     

Stability of steady plane-parallel convective motion of a chemically active medium

A study is made of a vertical plane layer of reacting fluid whose boundaries are maintained at constant equal temperatures. As a result of heating due to a chemical reaction of zeroth order taking place in the fluid a steady plane-parallel convective flow develops in the layer, and if the internal heat release is sufficiently intense this can become unstable. The linear stability of this motion has hitherto been considered only in the hydro-dynamic formulation [1], in which one can ignore the thermal perturbations and their influence on the development of the hydrodynamic perturbations (the region of small Prandtl numbers). In the present paper, the stability boundary is determined for arbitrary values of the Prandtl number and the Frank-Kamenetskii parameter FK characterizing the steady plane-parallel regime. An important difference between this flow and the types of convective motion hitherto studied [2] is that the basic planeparallel flow of the reacting medium is possible only in a definite range of the parameter FK: At values of the parameter exceeding a critical value, there is a thermal explosion — abrupt strong heating of the fluid. This is due to the essentially nonlinear dependence of the heat release of a chemical reaction on the temperature.     

Stability of the convective flow in an inclined layer with heat release at its center

The flow of a viscous incompressible fluid in a plane infinite inclined layer in the presence of internal heat sources concentrated on its axis is considered. The stability of the plane-parallel motion is investigated, the neutral curves are plotted, and the stability regions are determined. The results are compared with the case of uniform distribution of the heat sources. Supercritical fluid flows are calculated numerically.     

Convective heat transfer in a rotating horizontal cylindrical fluid layer

This paper describes the thermal convection and heat transfer in a cylindrical fluid layer rotating around a horizontal axis, with various constant temperatures set at the layer boundaries. The influence of the rotational speed of the cylindrical fluid layer on the convective heat transfer in this layer is studied. The study results are presented as functions of dimensionless parameters that characterize the action of two convective mechanisms: centrifugal and thermal-oscillatory. It is shown that, with low rotational speed, the heat transfer is determined by quasistationary gravitational convection.     

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.     

Convection of a heat-generating fluid in a rotating horizontal cylinder

Thermal convection of a fluid in a horizontal cylinder rotating about its own axis with uniformly volume-distributed internal heat sources is experimentally investigated. The enclosure boundary temperature was kept constant. The threshold of the excitation of convective flows and their structure are studied as functions of the heat-release intensity and the rotation velocity. The experiments are performed with water and water-glycerin solutions. It is shown that rapidly rotating fluid is in a stable quasiequilibrium state, namely, the temperature distribution is axisymmetric and has a maximum at the center of the enclosure. It is found that with decrease in the rotation velocity a convective flow arises thresholdwise, in the form of vortex cells periodically arranged along the axis. The thermal convection in the rotating enclosure is shown to be determined by the effects of two different mechanisms. One of these is due to the centrifugal force of inertia and plays the stabilizing role, while the other, thermovibrational mechanism is connected with nonisothermal fluid oscillations under the action of gravity in the enclosure-fitted reference frame and is responsible for the occurrence of mean thermal convection. The boundaries of the convection generation are plotted in the plane of the governing dimensionless parameters and the heat transfer in the supercritical region is studied.     

Investigation of convection in a horizontal cylindrical layer of a saturated porous medium

The structures of the convective motions and the nature of the heat transfer in a horizontal cylindrical layer are studied numerically for the Forchheimer model of a porous medium in the Boussinesq approximation. New asymmetric solutions of the equations of convection flow through a porous medium are found. Their development, domains of existence, and stability are investigated. One consists of a multivortex structure with asymmetric vortices in the near-polar region. Another asymmetric solution is realized at large Grashof numbers in the form of a convective plume deflected from the vertical. The threshold Grashof number of formation of the asymmetric motions depends on the Prandtl number and the cylindrical layer thickness.     

Stability of advective flow in an inclined plane fluid layer bounded by solid planes with a longitudinal temperature gradient. 1. Unstable stratification

The stability of steady convective flow in an inclined plane fluid layer bounded by ideally heat conducting solid planes is studied in the presence of a homogeneous longitudinal temperature gradient under unstable stratification conditions where the layer is inclined so that the temperature is higher in the lower part than in the upper part. It is shown that the inclination leads to the transition from critical perturbations to long-wavelength helical perturbations. Flow stability maps are given for the entire range of Prandtl numbers and inclination angles corresponding to unstable stratification.     

Stability of convective motion in a two-dimensional vertical fluid layer with permeable boundaries

The stability of the convective motion of a viscous incompressible fluid in a channel between permeable vertical planes heated to different temperatures is considered under the assumption of homogeneous transverse air blasting. Stability boundaries for different values of the Prandtl number Pr and Peclet number Pe that characterize the intensity of transverse motion are numerically determined. The results demonstrate that transverse blasting substantially influences both the hydrodynamic instability mechanism and instability due to the growth of thermal waves in the flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 94–101, January–February, 1976.In conclusion, I wish to express my appreciation to E. M. Zhukhovitskii for supervising the study. and G. Z. Gershuni for useful discussion of the results.     

Stability of convective motion caused by inhomogeneously distributed internal heat sources

The stability of steady convective plane-parallel flow in a vertical layer of viscous incompressible liquid of thickness h is investigated. The motion is caused by heat sources distributed in the liquid with volume density Q = Q₀exp (^–x) (the x axis is taken perpendicular to the boundary layer). The region of instability is determined for various values of the Prandtl number and the parameter N = h characterizing the inhomogeneity of the internal sources. It is shown that with increase in N there is qualitative rearrangement of the stability limit for perturbations of hydrodynamic type and incremental thermal waves.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 140–144, May–June, 1977.     

Effect of Longitudinal Forced Motion on the Stability of Convection in a Plane Vertical Layer with Internal Heat Sources

The effect of longitudinal forced fluid motion on the mechanisms of instability of a convection flow developing in a plane vertical layer in the presence of internal heat sources is considered. It is found that forced motion which intensifies the central stream of the convection flow can lead to moderate stabilization of the hydrodynamic and thermal crisis mechanisms. In the presence of counterstream forced motion the flow stability increases sharply.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 14–17.Original Russian Text Copyright © 2005 by Lobov.     

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.     

Nonlinear investigation of anti-convection and Rayleigh–Benard convection in systems with heat release at the interface

Anti-convection and Rayleigh–Benard convection generated by the joint action of external heating and heat sources (sinks) on the interface in layers with finite thicknesses are studied. Numerical simulations of the finite-amplitude convective regimes have been mage for the real two-liquid system (silicone oil 10 cs – ethylenglycol), convenient for the performance of experiments. The nonlinear boundary value problem was solved by means of the finite-difference method. Anti-convective structures in fluid systems subject to anti-convective instability only in the presence of heat sources (sinks) on the interface, have been obtained. This new type of the anti-convective motion appears in the case where one layer is strongly heated from above, while the temperature gradient in another layer is very weak.     

Stationary convection of a viscoplastic liquid in a vertical layer

A study is made of plane-parallel convective motion of a viscoplastic liquid between parallel vertical planes on which different temperatures are maintained. In contrast to [1], the yield shear stress ₀ is not a constant but is assumed to be a function of the temperature; moreover, above a certain critical temperature T_* the yield shear stress vanishes, so that for T > T_* the liquid is purely Newtonian. The structure of the regions of quasirigid and viscoplastic flow is studied in its dependence on the Theological parameters. The velocity profiles corresponding to the different flow regimes are found, and the boundaries between the regimes and the longitudinal heat flux are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 118–123, March–April, 1980.We thank G. Z. Gershuni and A. A. Nepomnyashchii for a helpful discussion of the work.     

Stability of a vertical liquid film with consideration of the marangoni effect and heat exchange with the environment

The stability of a free vertical liquid film under the combined action of gravity and thermocapillary forces has been studied. An exact solution of the Navier-Stokes and thermal conductivity equations is obtained for the case of plane steady flow with constant film thickness. It is shown that if the free surfaces of the film are perfectly heat insulated, the liquid flow rate through the cross section of the layer is zero. It is found that to close the model with consideration of the heat exchange with the environment, it is necessary to specify the liquid flow rate and the derivative of the temperature with respect to the longitudinal coordinate or the flow rate and the film thickness. The stability of the solution with constant film thickness at small wave numbers is studied. A solution of the spectral problem for perturbations in the form of damped oscillations is obtained.