Finite-amplitude regimes of mixed convection in a vertical layer with undulating boundaries

The method of finite differences is used to construct convective motions in a vertical layer with sinusoidally curved boundaries, fluid being pumped through longitudinally. Apart from steady and oscillation regimes, found earlier by analytical means for small amplitudes of undulation and slow pumping through [1, 2], new, essentially nonlinear, types of motion are discovered in the form of two-stroke cycles, and also of complex multi-revolution cycles which are two-dimensional resonance tori. The regions are determined in which regimes of various types exist.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–20, January–February, 1987.The author is grateful to E. M. Zhukhovitskii for constant interest in the study, and also to V. S. Anishchenko and A. A. Nepomnyashchii for useful discussions.     

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.     

Convection in a low Prandtl number fluid layer rotating about a vertical axis

Two- and three-dimensional convection flows in a horizontal layer of a low Prandtl number fluid heated from below and rotating about a vertical axis are studied numerically with a Galerkin method. Solutions for subcritical steady finite amplitude convection and convection in the form of standing oscillations are obtained. Parameter regimes that appear to be attainable in laboratory experiments have been emphasized. The stability of subcritical two-dimensional steady convection has been investigated and three-dimensional chaotic states of convection have been found.     

Stability of steady nonisothermal flow in a vertical layer with permeable boundaries in the microconvection model

The problem of the stability of steady convective viscous incompressible fluid flow in a vertical layer with boundaries at different temperatures is considered in the presence of transverse blowing through the layer. The complete spectral problem is solved for a silicon melt. The neutral curve is constructed and the critical Grashof number is found. The numerical calculations show that the presence of transverse blowing significantly affects the flow stability. As compared with the Oberbeck-Boussinesq model, in the microconvection model the instability develops at lower wavenumbers.     

Convection and heat transfer in a vertical layer with allowance for radiation from nonisothermic walls

Many studies (for example, [1–5]) consider motion and heat transfer in closed vertical cavities with given different temperatures of the lateral boundaries. The majority of studies cover the case of convection, but of late studies have appeared (for example, [4]) in which joint radiative—convective heat transfer is taken into account. In the present study we consider motion and heat transfer in a rectangular cavity separating two media with given different temperatures. In contrast to [4], the temperature of the lateral boundaries is determined from the condition for interaction with the surrounding medium, and the air in the cavity is assumed to be transparent for the heat radiation of the walls. The problem considered is a mathematical model of the heat transfer through windows, and is necessary for the analysis of methods of improving the heat proofing of buildings.Translated-from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 25–30, 1987.     

Fluid convection in a rotating porous layer under modulated temperature on the boundaries

The linear stability of thermal convection in a rotating horizontal layer of fluid-saturated porous medium, confined between two rigid boundaries, is studied for temperature modulation, using Brinkman’s model. In addition to a steady temperature difference between the walls of the porous layer, a time-dependent periodic perturbation is applied to the wall temperatures. Only infinitesimal disturbances are considered. The combined effect of rotation, permeability and modulation of walls’ temperature on the stability of flow through porous medium has been investigated using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as function of amplitude and frequency of modulation, Taylor number, porous parameter and Prandtl number. It is found that both, rotation and permeability are having stabilizing influence on the onset of thermal instability. Further it is also found that it is possible to advance or delay the onset of convection by proper tuning of the frequency of modulation of the walls’ temperature.     

Thermocapillary instability of a plane layer with a vertical temperature gradient

The oscillating disturbances in a plane layer with a temperature gradient are analyzed. It is shown that for heating from below taking the deformability of the free surface into account leads to the appearance of short-wave oscillatory instability, which becomes the most dangerous mode. Moreover, the interaction of the capillary and thermocapillary instability mechanisms results in the appearance of oscillating disturbances of a new type, which lead to equilibrium crisis at high Marangoni numbers. It is established that when the free boundary is heated, the onset of convection is possible only with respect to oscillatory disturbances.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 19–23, May–June, 1992.     

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.     

Convection development in a liquid layer with a free boundary

In certain calculations of the critical Rayleigh number for a liquid layer with free boundary which is heated from below, the linearization method has been used and it has been assumed that the temperature perturbations disappear at the undisturbed free boundary.Proper linearization shows that the temperature perturbation is proportional to the free surface perturbation, and the latter is proportional to the normal stress perturbation with the proportionality factor F=²/gh³ (g is the free-fall acceleration, is the kinematic viscosity, h is the liquid layer thickness). In §1 we present a formulation of the problem with account for the parameter F; in §2 we consider the linearized equations and the existence of a stability threshold is proved-a positive eigenvalue-and it is established that with an increase in the parameter F/P (P is the Prandtl number) the value of the critical Rayleigh number Ra_* decreases; §3 presents the results of a numerical calculation of Ra as a function of the parameter F/P.Convection development in a liquid layer with a free surface on which a given temperature is maintained was studied in [1, 2]. The value R_*=1100 found for the critical Rayleigh number agrees well with the experimental value. In the calculations made in [1, 2] the linearization method is used, and it is assumed that the temperature perturbations disappear at the undisturbed free boundary. Strictly speaking, this assumption is not correct.Correct linearization shows that the temperature perturbation is proportional to the perturbation of the free boundary, and the latter is proportional to the normal stress perturbation (see below (2.3)).The problem formulation is presented in §1; §2 deals with the linearized equations and the existence (Theorem 2.1) is demonstrated of a stability threshold—which is a simple positive eigenvalue; §3 presents the results of a numerical calculation of R_* as a function of the parameter =F/P.     

Characteristics of the temperature distribution in a supersonic laminar boundary layer

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 63–65, September–October, 1991.     

Microconvection in a vertical layer

In the case of very weak gravity the classical Oberbeck-Boussinesq approximation is not valid for describing thermal gravitational convection. This was pointed out in [1] where a new model was proposed under the assumption that the fluid is isothermal and incompressible. In this model the velocity vector is no longer solenoidal. Below, on the basis of this model we analyze the convective motion in a vertical layer, on the rigid boundaries of which a heat flux that depends on time only is prescribed. It is found that the nonsolenoidal character of the velocity does not lead to considerable restructuring of the steady-state convection. At the same time, the patterns of the unsteady, in particular, periodic convective flow calculated within the framework of the classical and the new models differ significantly.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 76–84, September–October, 1994.     

Convection and overstability in a viscoelastic fluid layer

In this paper, a study is made of the stability of a plane layer of a simple fluid heated from below. A differential (second-order fluid) and a single integral model are applied under various boundary conditions. The conditions for which the principle of the exchange of stabilities is valid are presented for both models and selected boundary conditions. It is shown that the second-order-fluid model proposed by Dunn and Fosdick [10], without surface tension, will not exhibit overstability.     

Double diffusive convection in a porous medium with modulated temperature on the boundaries

The effect of temperature modulation on the onset of double diffusive convection in a sparsely packed porous medium is studied by making linear stability analysis, and using Brinkman-Forchheimer extended Darcy model. The temperature field between the walls of the porous layer consists of a steady part and a time dependent periodic part that oscillates with time. Only infinitesimal disturbances are considered. The effect of permeability and thermal modulation on the onset of double diffusive convection has been studied using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as a function of frequency and amplitude of modulation, Vadasz number, Darcy number, diffusivity ratio, and solute Rayleigh number. Stabilizing and destabilizing effects of modulation on the onset of double diffusive convection have been obtained. The effects of other parameters are also discussed on the stability of the system. Some results as the particular cases of the present study have also been obtained. Also the results corresponding to the Brinkman model and Darcy model have been compared.     

Convective flow with uniform vorticity in a vertical layer

The free convective circulation of liquid in plane vertical slits of circular and square cross section with a longitudinal horizontal temperature gradient at the boundaries was investigated experimentally. It was found that under such heating conditions there is a uniform-vorticity flow with a region of quasirigid rotation, which has the shape of a disk in a circular slit and the shape of a cross in a square slit; in each longitudinal section of this zone the liquid moves along concentric trajectories with constant angular velocity. Dimensionless numbers for the problem were established by tests with various liquids and cavities of different dimensions. In dimensionless form, the angular velocity of the vortex and the temperature gradient in it depend linearly on the temperature difference at the boundaries of the layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 160–165, May–June, 1984.