Soret-induced Convection Research Articles

Soret-induced convection of ternary fluid in a horizontal porous layer heated from below

The present paper deals with the investigation of the Soret-induced convection of a three-component mixture of hydrocarbons in a horizontal porous layer. This problem is important for geological applications. The first part of the paper is devoted to the linear stability of the conductive state. The longwave instability is studied analytically by the expansion into the power series with respect to the wave number. A new long-wave oscillatory instability mode existing at negative separation ratios is found out. It is shown that this mode is more dangerous than the long-wave monotonic instability mode in the entire range of its existence. The instability to the perturbations with nonzero wave numbers is studied numerically by the shooting method. Stability map is obtained. The results confirm the predictions of the longwave analysis. In the second part of the paper, the nonlinear convection regimes are studied by the finite difference method. The calculations give the results consistent with the linear stability analysis. It is found that the primary bifurcation as a result of which the conductive state losses its stability is supercritical. The transformations of the convective flow structure with the change in the Rayleigh–Darcy number are accompanied by the hysteresis phenomena.

Onset and Nonlinear Regimes of Convection of Ternary Mixture in a Rectangular Porous Cavity Taking into Account the Soret Effect

The onset and nonlinear regimes of the Soret-induced convection in a mixture of dodecane, isobutylbenzene, and tetralin, taken in equal mass fractions and placed in a rectangular porous cavity with rigid impermeable boundaries elongated in the horizontal direction and heated from below, have been numerically investigated. The vertical boundaries of the cavity are thermally insulated. The components of the mixture under consideration are representatives of the main groups of chemical compounds that are oil components. The values of the porosity and permeability of the medium were chosen to be close to the values of real media, such as sand, sandstone, or limestone. The area of such a configuration simulates a hydrocarbon field. Due to the Soret effect (thermal diffusion), the light components of the mixture with positive separation ratios (dodecane and isobutylbenzene) are accumulated in the hot region, whereas the heavy component (tetralin) is accumulated in the cold region. In the case of heating from below (provided by the presence of geothermal gradient under natural conditions), this process may induce convection. It was found that a steady flow occurs in the cavity at a certain value of Rayleigh number; this flow becomes oscillatory with an increase in the Rayleigh number. With a further increase in the Rayleigh number, irregular oscillations arise. Several monotonic and oscillatory regimes, characterized by different spatial scales (from one to ten vortices), were found in the range of Rayleigh numbers under consideration. The oscillations of instantaneous flow characteristics in oscillatory regimes have a complex shape. A steady-state flow regime with an asymmetric structure was found. The ranges of Rayleigh numbers in which the found regimes may exist and the time dependences of the integral characteristics during the formation of steady flows were determined. The effect of supercriticality on the flow structure and concentration distributions of mixture components were analyzed.

Nonlinear regimes of Soret-driven convection of ternary fluid with fixed vertical heat flux at the boundaries.

Nonlinear regimes of the Soret-induced convection of a ternary mixture in a rectangular cavity with the vertical heat flux at the boundaries are studied numerically. Linear stability analysis performed earlier for a plane horizontal layer has shown that in a wide range of parameters the longwave instability is dominant. Nonlinear calculations performed in the present work for the parameter ranges where the monotonic longwave instability is dominant according to the linear stability analysis, show that at large supercriticalities the multi-vortex stationary flows could be realized. With the decrease of absolute value of the Rayleigh number Ra the series of flow structure transformations takes place. This transformation is accompanied by the decrease of a number of the vortices such that near the threshold the flow becomes two-vortex, i.e. the flow with maximum possible wavelength (minimum possible wave number) is realized. Thus, the nonlinear calculations confirm the conclusions of the linear stability analysis that the longwave instability is dominant. For the parameter ranges where according to the linear stability analysis the oscillatory longwave instability is dominant, the nonlinear calculations show that at small values of the net separation ratio the oscillatory regimes could be observed just in a narrow range of Ra close to the convection threshold and with the growth of net separation ratio the range of Ra where they can be observed is extended.

Onset and nonlinear regimes of convection of ternary mixture in a rectangular porous cavity taking into account Soret effect

A numerical study has been conducted to investigate the onset and nonlinear regimes of Soret-induced convection of a mixture of dodecane, isobutylbenzene and tetralin at equal mass fraction proportions in a rectangular porous cavity with solid boundaries elongated in a horizontal direction and heated from below. The vertical boundaries of the cavity are insulated. The components of the mixture are representatives of the main groups of chemical compounds that make up the oil. The values of porosity and permeability of the medium were chosen close to the values of real media, such as sands, sandstones or limestone. The area of such configuration simulates a real hydrocarbon field. Due to the Soret effect, the light components of the mixture, dodecane and isobutylbenzene, which have positive separation ratios, accumulate in the hot region, and the heavy component of the mixture, tetralin, in the cold region. This leads to the onset of convection in the case of heating from below, which in natural conditions is ensured by the presence of a geothermal gradient. It has been found that at a certain value of the Rayleigh number a steady flow occurs in the cavity, which, with an increase in the Rayleigh number, is replaced by an oscillatory one. With a further increase in the Rayleigh number, irregular oscillations occur. In the considered interval of the Rayleigh numbers, several monotonic and oscillatory regimes characterized by different spatial scales (from one to ten vortexes) were found. The form of oscillations of instantaneous flow characteristics in oscillatory regimes has a complex shape. The monotonic flow regime with an asymmetric structure was revealed. The intervals of the Rayleigh numbers, in which the obtained regimes may exist, and the time dependences of the integral characteristics occurred in the process of formation of steady flows were determined. The effect of supercriticality on the flow structure and the concentration distribution of mixture components were studied.

Nonlinear Regimes of the Soret-Induced Convection of Ternary Fluid in a Square Porous Cavity

We present the results of the numerical investigation of the Soret-induced convection in a ternary liquid mixture consisting of dodecane, isobutylbenzene and tetralin, taken in equal portions. The mixture is placed into a square porous cavity with rigid impermeable boundaries heated from below. The lateral boundaries are adiabatic. The problem under consideration is a model of natural hydrocarbon reservoir with porous medium, and the components of mixture are representatives of the main groups of chemical compounds comprising oil. Due to the thermodiffusion effect, dodecane and isobutylbenzene as the lighter components of this mixture with positive separation ratios are accumulated in the warmer domain of the cavity, and the heavy component, tetralin, is accumulated in the colder domain, which may lead to the development of convection. The calculations are performed for the parameters of porous medium close to the real parameters of oil fields and temperature gradient that correspond to geothermal gradient. They provide data on the temporal evolution of the characteristics of the flow and component separation. We also analyze the onset and development of single-vortex and two-vortex instability modes with the growth of the Rayleigh number $$Ra_{\mathrm{{por}}}$$ . It is found that at a certain value of the supercriticality, the stationary flow regime is replaced by the oscillatory regime. At even higher values of the Rayleigh number, the chaotic oscillations take place. The transitions between single-vortex and two-vortex flows are also observed. For porosity equal to 0.1, the two oscillatory regimes at different oscillation amplitudes are excited. With the porosity growth, the region of the existence of the oscillatory regimes becomes narrower.

Vibration effect on the Soret-induced convection of ternary mixture in a rectangular cavity heated from below

This paper presents the results of numerical simulation of the Soret-induced convection of ternary mixture in the rectangular cavity elongated in horizontal direction in gravity field. The cavity has rigid impermeable boundaries. It is heated from the bellow and undergoes translational linearly polarized vibrations of finite amplitude and frequency in the horizontal direction. The problem is solved by finite difference method in the framework of full unsteady non-linear approach. The procedure of diagonalization of the molecular diffusion coefficient matrix is applied, allowing to eliminate cross-diffusion components in the equations and to reduce the number of the governing parameters. The calculations are performed for model ternary mixture with positive separation ratios of the components. The data on the vibration effect on temporal evolution of instantaneous and average fields and integral characteristics of the flow and heat and mass transfer at different levels of gravity are obtained.

Onset and non-linear regimes of Soret-induced convection in binary mixtures heated from above.

The paper deals with the investigation of the onset and non-linear regimes of convection of liquid binary mixtures with negative Soret effect heated from above. The linear stability of a convectionless state in a horizontal layer is studied by the numerical solution of the linearized problem on the temporal evolution of small perturbations of the unsteady base state. Non-linear regimes of convection are investigated by the numerical solution of the non-linear unsteady equations for a horizontally elongated rectangular cavity. The calculations are performed for water-ethanol and water-isopropanol liquid mixtures and for colloidal suspensions. The dependences of the instability onset time and wave number of the most dangerous perturbations on the solutal Rayleigh number (gravity level) obtained by a linear stability analysis and non-linear calculations are found to be in a very good agreement. A favorable comparison with the existing experimental and numerical data is presented.

Onset of Soret-induced convection in a horizontal layer of ternary fluid with fixed vertical heat flux at the boundaries.

The paper deals with the investigation of the onset and weakly nonlinear regimes of the Soret-driven convection of ternary liquid mixture in a horizontal layer with rigid impermeable boundaries subjected to the prescribed constant vertical heat flux. It is found that there are monotonous and oscillatory longwave instability modes. The boundary of the monotonous longwave instability in the parameter plane Rayleigh number Ra - net separation ratio [Formula: see text] at fixed separation ratio of one of solutes consists of two branches of hyperbolic type. One of the branches is located at [Formula: see text], the other one at [Formula: see text]. The oscillatory longwave instability exists at [Formula: see text] only for the heating from below and at [Formula: see text] there exist two oscillatory longwave instability modes: one at [Formula: see text] and the other at [Formula: see text]. Corrections to the Rayleigh number obtained in the higher order of the expansion show that the longwave perturbations can be most dangerous at any values of [Formula: see text]. The numerical solution of the linear stability problem for small perturbations with finite wave numbers confirms this conclusion. The weakly nonlinear analysis shows that all steady solutions are unstable to the modes of larger wavelength and stable to the modes of smaller wavelength, i.e. the solution with maximal possible wavelength is realized.

ТЕРМОКОНЦЕНТРАЦИОННАЯ КОНВЕКЦИЯ БИНАРНЫХ И ТРЕХКОМПОНЕНТНЫХ СМЕСЕЙ В КВАДРАТНОЙ ПОЛОСТИ ПРИ НАГРЕВЕ СВЕРХУ

In the present work the Soret-induced convection of binary and ternary mixtures with negative separation ratios in a square cavity heating from above in the gravity field is investigated. The parameters of the binary mixtures coincide with the parameters of binary mixtures constituting the ternary mixture, which allows one to observe the effect of adding one more solute to the binary mixture. Analysis of the temporal dependences of the integral characteristics of the convection of binary and ternary mixtures shows that an increase in the separation ratio of the binary mixture value or the net separation ratio of the ternary mixture value leads to an acceleration of the instability onset. The magnitude of the jump in the intensity of the flow that accompanies the instability onset also increases with the separation ratio modulus of the mixture. Dependences of the time of the instability onset, the maximum value of the modulus of the stream function attained at the moment of instability onset, and the corresponding to this time moment components concentration difference between the centers of horizontal boundaries on the separation ratio of the binary mixture and on the separation ratio of the second component at a fixed separation ratio of the first solute for the ternary mixture are presented. The temporal evolution of the structures of the emerging flow and the fields of solutes concentration are observed. It is shown that the adding of one more solute to the binary mixture leads to an acceleration of the instability onset, an increase in the intensity of the flow and a decrease in the concentration difference between the horizontal boundaries.

Convection of a binary fluid in a shallow porous cavity heated and salted from the sides

This paper investigates natural convection in a shallow porous cavity filled with a binary fluid. Constant fluxes of heat and concentration are imposed on the vertical walls of the enclosure. Both double-diffusive convection and Soret-induced convection are considered. An analytical solution, valid for shallow enclosures, is derived on the basis of the parallel flow approximation. The work focuses, among other things, on the existence of multiple solutions when the buoyancy ratio is in the vicinity of φ=−1, for which a trivial steady state solution corresponding to the rest state exists. For this particular value of φ, it is well known that the onset of motion occurs above a subcritical Rayleigh number. The present analytical model reveals that, under certain conditions, such a subcritical Rayleigh number also exists when φ<−1. In the range of the governing parameters considered in this study, a good agreement is found between the analytical predictions and the numerical results obtained by solving the full governing equations. Heat, solute and flow characteristics predicted by the analytical model are found to agree well with a numerical study of the full governing equations.

Double-diffusive and Soret-induced convection in a micropolar fluid layer

This paper reports an analytical and numerical study of natural convection in a shallow rectangular cavity filled with a micropolar fluid. Neumann boundary conditions for temperature and concentration are applied to the horizontal walls of the enclosure, while the two vertical ones are assumed insulated. The governing parameters for the problem are the thermal Rayleigh number, Ra, Prandtl number, Pr, Lewis number Le, buoyancy ratio, φ, aspect ratio of the cavity, A, and various material parameters of the micropolar fluid, K, B, λ and n. For convection in an infinite layer (A≫1), analytical solutions for the stream function, temperature, concentration and microrotation are obtained using a parallel flow approximation in the core region of the cavity and an integral form of the energy and constituent equations. The critical Rayleigh numbers for the onset of supercritical and subcritical convection are predicted explicitly by the present model. Also, results are obtained from the analytical model for finite amplitude convection for which the flow and heat transfer are presented in terms of the governing parameters of the problem. Numerical solutions of the full governing equations are reported for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations. The influence of the material parameters on the flow and heat and solute transfers is demonstrated to be significant.

Double-diffusive and Soret-induced convection of a micropolar fluid in a vertical channel

This paper reports an investigation of the fully developed natural convection heat and mass transfer of a micropolar fluid in a vertical channel. Asymmetric temperature and concentration boundary conditions are applied to the walls of the channel. The cases of double diffusion and Soret-induced convection are both considered. The governing parameters for the problem are the buoyancy ratio and the various material parameters of the micropolar fluid. The resulting non-dimensional boundary value problem is solved analytically in closed form using MAPLE software. A numerical solution of the time dependent governing equations is demonstrated to be in good agreement with the analytical model. The influence of the governing parameters on the fluid flow as well as heat and solute transfers is demonstrated to be significant.

Density maximum effect on Soret-induced natural convection in a square porous cavity

This paper reports a numerical study on the effect of density maximum on Soret-induced convection in a square porous cavity. Dirichlet boundary conditions for temperature are applied to the vertical walls of the enclosure, while the two horizontal ones are assumed impermeable and insulated. The non-dimensional equations for momentum, energy and concentration are solved by a finite volume method with power-law scheme for convection and diffusion. A parametric study is undertaken as a function of the main dimensionless group characterizing the problem, namely the thermal Rayleigh number, RT, the solutal Rayleigh number, RS, the Lewis number Le and an extremum parameter γ, which quantifies the effect of the nonlinear equation of state. The results are presented in the form of streamlines, isotherms and isoconcentration lines for various values of the governing parameters. Comprehensive Nusselt number data are presented as functions of the governing parameters mentioned above.

Natural convection in a horizontal binary fluid layer bounded by thin porous layers

This paper reports an analytical and numerical study of natural convection in a horizontal binary fluid layer confined between two horizontal porous walls. The cavity is heated from the bottom by a constant heat flux while the long side walls are impermeable and adiabatic. The Beavers–Joseph slip condition on velocity is applied at the interface between the fluid and porous layers. Both double-diffusive convection and Soret-induced convection are considered. An analytical model, based on the parallel flow approximation, is proposed for the case of a shallow layer. The flow and heat and mass transfer variables are obtained in terms of the governing parameters of the problem. The critical Rayleigh numbers for the onset of supercritical and subcritical convection are predicted for various hydrodynamic boundary conditions. The results for a fluid layer bounded by solid walls and free surfaces emerge from the present analysis as limiting cases. The study is completed by a numerical solution of the full governing equations.

Onset of convection in Soret-driven instability

It is common knowledge that light fluids rise while heavy fluids sink in the gravity field. The most obvious case is the isothermal Rayleigh-Taylor instability when a heavy fluid is placed on top of a light one. In the nonisothermal case, while heating from above, the density stratification is stable in a pure liquid. However, unstable density stratification might be established in a binary mixture with a negative Soret effect in the case of heating from above: the heavier liquid is accumulated on the top of the lighter one. Due to the large differences between viscous, thermal, and diffusion times the system has a tendency to fingering buoyant instabilities. At some moment the flow may be initiated. Near the onset of convection the flow pattern has a columnar convective structure: for a relatively low applied temperature difference Delta T the lighter and colder liquid is drawn up in the central part of the cell and the heavier liquid flows down along the walls. For finite size systems the situation is reversed at higher Delta T. Here we present results of three-dimensional direct numerical simulations of heat and mass transfer in a system with a negative Soret effect. While the development of Soret-induced convection is similar for a wide class of liquids: water based mixtures, colloidal, and polymer solutions, the parameters of the chosen system correspond to a realistic binary mixture of water (90%) and isopropanol (10%) enabling comparison of theoretical predictions with planned experimental studies.

Natural convection of a binary mixture confined in a slightly inclined tall enclosure

This paper presents an analytical and numerical study of natural convection of a binary mixture confined in a tall enclosure, slightly inclined about the gravity field. The cavity is heated from the bottom by a constant heat flux while the long side walls are impermeable and adiabatic. Both double-diffusive convection and Soret-induced convection are considered. The basis of the analytical approximation is an assumption of parallel flow over a large portion of the layer. The existence of multiple steady states, for small enough inclinations around the vertical plane, is demonstrated. Numerical confirmation of the stable analytical results is also presented.

Double-diffusive and Soret-induced convection in a shallow horizontal porous layer

This paper reports an analytical and numerical study of the natural convection in a horizontal porous layer filled with a binary fluid. A uniform heat flux is applied to the horizontal walls while the vertical walls are impermeable and adiabatic. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the horizontal walls (double-diffusive convection, $a\,{=}\,0$ ) or by temperature gradients (Soret effects, $a\,{=}\,1$ ). The governing parameters for the problem are the thermal Rayleigh number, $R_T$ , the Lewis number, $Le$ , the solutal Rayleigh number, $R_S$ , the aspect ratio of the cavity, $A$ , the normalized porosity of the porous medium, $\varepsilon$ , and the constant $a$ . The onset of convection in the layer is studied using a linear stability analysis. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities are determined in terms of the governing parameters. For convection in an infinite layer, an analytical solution of the steady form of the governing equations is obtained by assuming parallel flow in the core of the cavity. The critical Rayleigh numbers for the onset of supercritical, $R_{\hbox{\scriptsize\it TC}}^{\hbox{\scriptsize\it sup}}$ , or subcritical, $R_{\hbox{\scriptsize\it TC}}^{\hbox{\scriptsize\it sub}}$ , convection are predicted by the present theory. A linear stability analysis of the parallel flow pattern is conducted in order to predict the thresholds for Hopf bifurcation. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical prediction and the numerical simulations.