Modeling of microconvection in a fluid between heat conducting solids

Unsteady convection in a fluid under weak gravity is modeled. Convection in a rectangular domain elongated in the direction of gravity and enclosed between two heat-conducting solids is investigated in the case of heat insulation of the ends of the rectangle and the periodic heat flow through the outer boundaries of the solids. In this case, the condition of zero total heat flux is satisfied. Convective fluid motions are described using two mathematical models: the classical Oberbeck-Boussinesq model and the microconvection model for an isothermally incompressible fluid. Results of the numerical studies confirm the quantitative and qualitative differences between the flow characteristics calculated using the two convection models. Fluid particle trajectories are presented. Effects due to various physical characteristics of the problem are studied.     

Stability of the equilibrium of a flat layer in a microconvection model

The stability of the equilibrium state of a flat layer bounded by rigid walls is studied using a microconvection model. The behavior of the complex decrement for longwave perturbations has an asymptotic character. Calculations of the full spectral problem were performed for melted silicon. Unlike in the classical Oberbeck–Boussinesq model, the perturbations in the microconvection model are not monotonic. It is shown that for small Boussinesq parameters, the spectrum of this problem approximates the spectra of the corresponding problems for a heatconducting viscous fluid or thermal gravitational convection when the Rayleigh number is finite.     

Origination of Microconvection in a Flat Layer with a Free Boundary

Stability of a flat layer with a free boundary in the model of microconvection is studied in the linear approximation of equilibrium. The most important physical case is considered, where the Boussinesq parameter and the Rayleigh number depend linearly on the Marangoni number. It is shown that longwave disturbances always decay. Neutral curves for a wide range of dimensionless parameters are constructed numerically; new (as compared to the Oberbeck–Boussinesq model) growing disturbances are found, which are caused by fluid compressibility. Based on numerical results, the areas of applicability of the microconvection, Oberbeck–Boussinesq, and viscous heatconducting fluid models are established.     

Stationary solution of the equations of microconvection in a vertical layer

The stationary problem of convection in liquids is considered using the model of microconvection developed by V. V. Pukhnachev. Velocity profiles for boundary conditions of different classes are constructed. The solutions of the problem under study and the classical problem based on the Oberbeck—Boussinesq model are compared.     

Hydrodynamic features of the exploitation of highly nonuniform source-type oil formations

The flow of a weakly compressible fluid in a highly nonuniform formation with block structure, classified as source-type, is considered. An analytic solution of the problem of fluid flow into a well in a bounded circular reservoir is obtained. On the basis of this solution the effect of the fluid offtake rate on the depletion of the reservoir is investigated. It is shown that in highly nonuniform media a number of unsteady effects, which cannot be described by the classical model with steady mass transfer, occur.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 113–120, September–October, 1993.     

Microconvection in a Binary System

On the basis of an essentially subsonic-flow approximation, the effect of small density variations of the medium on the development of natural convection in a slightly nonisothermal binary mixture at very small Rayleigh numbers and a constant mean thermodynamic pressure (microconvection) is studied analytically and numerically. Microconvective flows are characterized by a nonsolenoidal velocity field. As compared with microconvection in pure fluid, the presence of an admixture and the thermodiffusion effect result in a number of new interesting opportunities. Attention is mainly focused on essentially unsteady processes (transition processes after two different media are brought into contact and convection in rapidly varying temperature fields), in which the medium volume expansion is very significant. The conditions of onset of monotonic modes of the Rayleigh-Bénard and Bénard-Marangoni long-wave instabilities are also considered.     

Turbulent convection in a plane horizontal layer

The problem of convection in an incompressible fluid between two horizontal planes maintained at a constant temperature without friction on the boundaries is considered. The medium is assumed to be turbulent. A theoretical model is constructed using mathematical modeling of the coherent structure in the turbulent flow. This turbulent convection-model has one empirical constant in the relations closing the generalized Reynolds equations. The problem formulated is solved analytically by means of the Stuart-Landau method. The main characteristics of the finite-amplitude ordered convection are obtained and their dependence on the empirical constant is studied.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 49–56, November–December, 1993.     

A physical mechanism of large-scale structure generation in turbulent convection

The generation of large-scale structures during turbulent convection in a rotating layer of incompressible fluid heated by internal heat sources is considered. The results of a theoretical and experimental investigation of a physical mechanism of large-scale structure formation which operates under conditions of high-intensity small-scale turbulent convection and low boundary heat transfer are discussed. The theoretical investigation is based on a system of evolutionary equations obtained for the transverse space moments of the physical fields, which describes the motion in thin layers of rotating fluid. The stability of the solution of the mathematical model is studied using the small perturbation method. As a result, a condition of existence of longwave instability of the system and a criterion determining the threshold of its onset are obtained. The theoretical conclusions are confirmed by a series of experiments carried out on a laboratory model. The design of the laboratory apparatus and the experimental technique are described.Moscow, Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 20–29, September–October, 1996.     

Instability of the equilibrium state of a liquid with a free surface in the presence of volume heat sources

The problem of the convection of a weakly compressible fluid is considered. In the free convection equations a heat source function is taken into account. The stability of the equilibrium state of a horizontal layer relative to small perturbations is studied using the linearization method. On the basis of numerical calculations it is shown that the mechanical equilibrium state of the fluid is unstable. The neutral curves are plotted and the critical Rayleigh numbers are found. In the calculations values of the physical parameters typical of Lake Baikal were used.     

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.     

Stability of Steady Flow in a Vertical Layer in the Microconvection Model

The stability of steady flow in a vertical gap is analyzed using the perturbation method within the framework of the microconvection model. The resulting spectral problem is not self-conjugate. The stability of the flow to long-wave perturbations is established. It is shown that if the Boussinesq parameter is small, the spectrum of this problem approximates the spectra of the corresponding problems for a viscous heat-conducting fluid and thermogravitational convection with a finite Rayleigh number. The numerical calculations indicate that in the microconvection model the instability develops at smaller wave numbers.     

Linear and non-linear analyses of convection in a micropolar fluid occupying a porous medium

Linear and weakly non-linear analyses of convection in a micropolar fluid occupying a high-porosity medium are performed. The Brinkman–Eringen momentum equation is considered. The linear and non-linear analyses are, respectively, based on the normal mode technique and truncated representation of Fourier series. The linear theory for a two-phase system reiterates that the preferred mode of convection is stationary as in the case of a single-phase system. An autonomous system of differential equations representing cellular convection arising in the study is considered to analyse the critical points. The Nusselt number is obtained as a function of micropolar and porous medium parameters.     

Impact on a plate on the surface of a fluid

Questions of the interaction between solid and elastic structures with an ideal fluid which are associated with the initial stage of the impact and penetration of bodies in the fluid were considered in [1–4]. Results are presented below of an analysis of a central impact on a solid weightless plate which is on the surface of a compressible fluid. The impact velocity is much less than the speed of sound in the medium. Computations are performed by a finite-difference Lagrange method according to a program for plane motions of a continuous medium [5] by using a volume artificial viscosity of Neumann-Richtmayer type [6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–145, May–June, 1978.     

The effect of thermal boundary conditions and density stratification on the appearance of convection in the atmospheres of planets and stellar envelopes

In the present article a study is made of the effect of density stratification and thermal boundary conditions which are a consequence of the parametrization of the small-scale turbulent convection in the spherical layers of a compressible gas on the emergence of large-scale convection.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 149–158, March–April, 1985.     

Numerical simulation of the dynamics of free-burning flames,fire whirls,and firestorms.

The paper presents the results of testing a promising package of mathematical models for predicting the dynamics of certain formations in free burning fires, based on a generalized system for time-dependent two-dimensional compressible viscous turbulent gas flows taking combustion into account. The dynamics of flames and fire whirls is investigated using the control volume method. For the first time, firestorms occurring during large fires are numerically simulated; this phenomenon is initiated by the convection of the background vertical component of the atmospheric circulation in the fire influence zone.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 52–61, November–December, 1994.     

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.     

Hydrothermal convection in a thin porous ring

Natural convection of the fluid in a thin porous ring on whose boundaries steady temperature distributions are maintained is considered. For this problem on the basis of the two-dimensional equations an integrodifferential equation is obtained in the zeroth approximation in terms of a small parameter, namely the relative thickness of convection. A parametric numerical investigation of the flow and temperature fields is carried out.Makhachkala, Kaspiisk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 4–8, November–December, 1994.     

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 effect of vertical vibrations on the onset of convection in a planar rotating layer

The effect of vertical vibrations on the convection in a rotating planar fluid layer heated from below was studied. In this case a modulation parameter, the acceleration due to gravity, appears in the problem. The modulation of the parameter may have a significant effect on the onset of convective instability. Parameter modulation in nonrotating layers has been investigated in earlier work [1–3]. The presence of rotation significantly increases the complexity of the mathematical problem, introducing an additional dependence of the solution on the Taylor number Ta and the Prandtl number Pr. Furthermore, an oscillatory convection regime can occur at the stability limit in rotating fluids with Pr < 1. Parameter modulation in the rotating fluid may not only lead to a change in the stability limit and critical wavelength but also to a change in the eigenfrequency of the oscillatory convection. Rauscher and Kelly [4] examined the effect of parameter modulation on the convective stability of a rotating fluid only for the particular case of a sinusoidal variation in the temperature gradient with a small amplitude for Pr = 1, i.e., the effect of modulation was studied on only a steady convection regime.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 12–22, July–August, 1984.     

On the propagation of pressure waves in saturated porous media

The problem of the propagation of pressure waves through compressible porous media saturated with a slightly compressible fluid is considered. By using Darcy's law the problem is reduced to a mixed initial-boundary value problem for an equation of the heat conduction type with a nonlinear term. The method of quasi-characteristics is used to solve this equation numerically. Solutions of the wave propagation problem for media with different permeability coefficients are presented. A solution of the inverse problem of determining the permeability coefficient using wave-pulse test data is constructed on the basis of a set of solutions of the direct problem.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 81–84, November–December, 1996.