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.     

Asymptotic analysis of convective flows in a rotating spheroid for vanishing viscosity and thermal conductivity

The six-dimensional model of viscous fluid thermal convection in a uniformly rotating ellipsoid [1] is studied. The limiting case in which the viscosity and thermal conductivity approach zero while the Prandtl number Pr remains finite is considered. Using both asymptotic and numerical methods, it is shown that for Pr > 2 the attractor is a two-dimensional invariant torus or a limit cycle; the corresponding convective flows are either quasiperiodic, with two basic frequencies, or periodic. It is also shown that resonances on the torus play a dominant role in the breakdown of two-dimensional toruses with increasing viscosity and thermal conductivity.Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 35–38, September–October, 1995.     

The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model

A linear stability analysis is carried out to study viscoelastic fluid convection in a horizontal porous layer heated from below and cooled from above when the solid and fluid phases are not in a local thermal equilibrium. The modified Darcy–Brinkman–Maxwell model is used for the momentum equation and two-field model is used for the energy equation each representing the solid and fluid phases separately. The conditions for the onset of stationary and oscillatory convection are obtained analytically. Linear stability analysis suggests that, there is a competition between the processes of viscoelasticity and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Elasticity is found to destabilize the system. Besides, the effects of Darcy number, thermal non-equilibrium and the Darcy–Prandtl number on the stability of the system are analyzed in detail.     

The Onset and Nonlinear Regimes of Convection in a Two-Layer System of Fluid and Porous Medium Saturated by the Fluid

The onset of convection and its nonlinear regimes in a heated from below two-layer system consisting of a horizontal pure fluid layer and porous medium saturated by the same fluid is studied under the conditions of static gravitational field. The problem is solved numerically by the finite-difference method. The competition between the long-wave and short-wave convective modes at various ratios of the porous layer to the fluid layer thicknesses is analyzed. The data on the nature of convective motion excitation and flow structure transformation are obtained for the range of the Rayleigh numbers up to quintuple supercriticality. It has been found that in the case of a thick porous layer the steady-state convective regime occurring after the establishment of the mechanical equilibrium becomes unstable and gives way to the oscillatory regime at some value of the Rayleigh number. As the Rayleigh number grows further the oscillatory regime of convection is again replaced by the steady-state convective regime.     

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.     

Thermal structure of convective vortex lattices

The thermal structure of the convective motions of a rotating plane layer of fluid is experimentally investigated in the regular vortex structure regime. It is found that in such a system the intense vortex motion leads to a temperature distribution such that the mean fluid temperature falls linearly from the bottom of the layer to the surface, the temperature gradient being determined by the rate of rotation and depth of the fluid. By dimensional analysis it is shown that this gradient corresponds to heat transfer in which the Nusselt number isolines are parallel to the convection curve. The horizontal structure of the temperature field is investigated; it corresponds to motion in which the fluid descends within a narrow vortex-sink and rises along the edges of a cylinder which determines the characteristic dimension of the structure in rotating fluid convection.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 160–166, November–December, 1987.The author wishes to thank G. S. Golitsyn for his constant interest in the work.     

Steady thermocapillary motion in a horizontal layer of liquid metal locally heated from above

The article considers stationary thermocapillary convection in a thin horizontal layer of fluid with Prandtl number Pr < 1 when it is being locally heated from above in conditions in which the curvature of the free surface is small. It is shown that the motion has a cellular structure. The size of the convective cell is determined from the solution to the spectral problem to which the integration of the free convection system of equations reduces. If the Maragoni (Péclet) number is sufficiently high, the length of the convective cell turns out to be large in comparison with the thickness of the layer. The convection picture is considered and an expression obtained for the velocity of the developing flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 146–152, November–December, 1984.     

Penetrative convection in the isothermally incompressible fluid approximation

The onset of penetrative convection in an infinite horizontal fluid layer bounded by isothermal rigid or free nondeformable surfaces is numerically examined. It is assumed that the specific volume of the fluid depends quadratically on temperature and reaches a minimum inside the layer. The isothermally incompressible fluid convection model in which, as distinct from the Oberbeck-Boussinesq approximation, the thermal expansion is not assumed to be small is considered. Both the neutral stability curves of the conductive regime and the amplitudes of two-dimensional periodic and three-dimensional doubly-periodic convective flow are calculated. The results are compared with those previously obtained for the equations of penetrative convection in the Boussinesq approximation.Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 40–52, March–April, 1996.     

Convective heat transfer from a rotating or nonrotating axisymmetric body

An analysis of steady laminar mixed-convection heat transfer from a rotating or nonrotating axisymmetric body is presented. A mixed-convection parameter is proposed to serve as a controlling parameter that determines the relative importance of the forced and the free convection. In addition, a rotation parameter is introduced to indicate the relative contributions of the flow forced convection and the rotational forced convection. The values of both these two parameters lie between 0 and 1. Furthermore, the coordinates and dependent variables are transformed to yield computationally efficient numerical solutions that are valid over the entire range of mixed convection from the forced-convection limit (rotating or nonrotating bodies) to the pure free-convection limit (non-rotating bodies) and the entire regime of forced convection from the pure flow forced-convection limit (nonrotating bodies) to pure rotational forced-convection limit (rotating bodies). The effects of mixed-convection intensity, body rotation, fluid suction or injection, and fluid Prandtl number on the velocity profiles, the temperature profiles, the skin-friction parameter, and heat transfer parameter are clearly illustrated for both cases of buoyancy assisting and opposing flow conditions.     

Convection in rotating spherical layers with allowance for centrifugal forces

The investigation of convection in rotating spherical layers with a central gravitational field g(r) is very important for the study of the global motions in the atmospheres of large planets and the convective zones of stars. In recent years, many studies of these questions have been made (they have been reviewed, for example, by Yavorskaya and Belyaev [1]), but the centrifugal convective force has been ignored in all the numerical and analytic investigations. In some cases, for example, for large planets, the centrifugal force may reach an appreciable value, O.1g, and have a strong influence on the convective motion. The present paper studies the occurrence of convection in slowly rotating spherical layers with allowance for centrifugal forces. It is shown that the centrifugal force leads to the appearance in a layer of an axisymmetric flow, at the stability limit of which convective cells of banana or toroidal shape can develop. The latter are possible only in layers with undeformable boundaries at sufficiently large values of the Froude number. Irrespective of the form of the layer and the magnitude of the centrifugal force, the banana-shaped cells propagate in a wavelike manner in the direction opposite to the rotation. In the case of undeformable boundaries, the centrifugal force stabilizes the motion of the fluid as compared with the case of a layer at rest. Deformation of one or both of the boundaries under the influence of the centrifugal force leads to destabilization of the basic flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 14–21, March–April, 1984.     

Coriolis effect on convection for a low Prandtl number fluid

The problem of finite-amplitude thermal convection in a horizontal layer of a low Prandtl number heated from below and rotating about a vertical axis is studied. Linear stability and weak non-linear theories are used to investigate analytically the Coriolis effect on gravity-driven convection. The non-linear steady problem is solved by perturbation techniques, and the preferred mode of convection is determined by a stability analysis. Finite-amplitude results, obtained by using a weak amplitude, correspond to both stationary and oscillatory convections. These amplitude equations permit to identify from the post-transient conditions that the fluid is subject to Pitchfork bifurcation in the stationary convection and Hopf bifurcation in the oscillatory convection. Thereafter, in the small perturbations hypothesis, an amplitude solution is evaluated and drawn in time and space scales.     

Boussinesq approximation in the Rayleigh-Benard problem

Instability of mechanical equilibrium and initiation of plane steady-state convective flows in an infinite horizontal fluid layer heated from below (Rayleigh-Benard problem) are investigated. The convection model for an isothermal incompressible fluid is not assumed to have small thermal expansion (contrary to the Oberbeck-Boussinesq approximation). The influence of a supplementary thermal expansion parameter on the convection process is numerically investigated. The results are compared with the known results for the Oberbeck-Boussinesq approximation. It is shown that subcritical instability is possible if the thermal expansion parameter increases. The linearization and Lyapunov-Schmidt methods are applied.Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–10, September–October, 1995.     

Convection in a two-layer system due to the combined action of the Rayleigh and thermocapillary instability mechanisms

In a two-layer system loss of stability may be monotonic or oscillatory in character. Increasing oscillatory perturbations have been detected in the case of both Rayleigh [1, 2] and thermocapillary convection [3–5]; however, for many systems the minimum of the neutral curve corresponds to monotonic perturbations. In [5] an example was given of a system for which oscillatory instability is most dangerous when the thermogravitational and thermocapillary instability mechanisms are simultaneously operative. In this paper the occurrence of convection in a two-layer system due to the combined action of the Rayleigh (volume) and thermocapillary (surface) instability mechanisms is systematically investigated. It is shown that when the Rayleigh mechanism operates primarily in the upper layer of fluid, in the presence of a thermocapillary effect oscillatory instability may be the more dangerous. If thermogravitational convection is excited in the lower layer of fluid, the instability will be monotonic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–170, January–February, 1987.     

An analytical study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple stress fluid

The onset of double diffusive convection in a two component couple stress fluid layer with Soret and Dufour effects has been studied using both linear and non-linear stability analysis. The linear theory depends on normal mode technique and non-linear analysis depends on a minimal representation of double Fourier series. The effect of couple stress parameter, the Soret and Dufour parameters, and the Prandtl number on the stationary and oscillatory convection are presented graphically. The Dufour parameter enhances the stability of the couple stress fluid system in case of both stationary and oscillatory mode. The effect of positive Soret parameter is to destabilize the system in case of stationary mode while it stabilizes the system in case of oscillatory mode. The negative Soret parameter enhances the stability in both stationary and oscillatory mode. The couple stress parameter enhances the stability of the system in both stationary and oscillatory modes. The Dufour parameter increases the heat transfer while the couple stress parameter has reverse effect. The Soret parameter has negligible influence on heat transfer. Both Dufour and Soret parameters increases the mass transfer while the couple stress parameter has dual effect depending on the value of the Rayleigh number.     

Effect of Coriolis force on the thermosolutal convection threshold in a rotating annular Hele-Shaw cell

In this study, the linear stability analysis is used to determine the onset of thermosolutal convection in fluids confined in rotating annular Hele-Shaw cell. The fluid layer is submitted to radial gradients of temperature and concentration. The effects of both Coriolis force and curvature parameter on the stationary and oscillatory convection are investigated when the Prandtl number is of the order of unity or larger than unity.     

Effect of the adsorption-desorption process intensity on solutal convection near a drop in a horizontal channel

The interaction between gravity convection and Marangoni convection in a horizontal rectangular channel filled with a liquid containing a surfactant and a drop of another liquid is numerically investigated. For large Schmidt numbers the occurring oscillatory regime of solutal convection is analyzed. In the model with a surface phase the effect of the adsorption and desorption processes on the convective flow structure is determined. The corresponding initial and boundary value problem is solved using a difference method.     

Oscillatory rotationally symmetric loss of stability of nonisothermal couette flow

A study is made of the stability of nonisothermal Couette flow — steady flow of a viscous heat conducting fluid between two rotating concentric cylinders heated to different temperatures. The methods of perturbation theory are used to establish conditions sufficient for bifurcation of a neutral curve of oscillatory instability from the neutral curve of monotonic instability. Computer calculations show that for certain values of the parameters of the problem these conditions are realized and there is an oscillatory loss of stability of the nonisothermal Couette flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–80, January–February, 1984.I thank V. I. Yudovich for constant interest in the work.     

On the Linear Stability of Large Stefan Number Convection in Rotating Mushy Layers for a New Darcy Equation Formulation

The Coriolis effect on a solidifying mushy layer is considered. A near-eutectic approximation and large far-field temperature is employed in the current study for large Stefan numbers. The linear stability theory is used to investigate analytically the Coriolis effect on convection in a rotating mushy layer for a new formulation of the Darcy equation. It was found that a large Stefan number scaling allows for the presence of both the stationary and oscillatory modes of convection. In contrast to the problem of a stationary mushy layer, rotating the mushy layer has a stabilising effect on convection. It was observed that increasing the Taylor number or the Stefan number encouraged the oscillatory mode of convection.     

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.     

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.