Solutal Buoyancy Forces Research Articles

Effects of variable density for film evaporation on laminar mixed convection in a vertical channel

A numerical investigation was conducted to study mixed convection in a vertical parallel-plate channel with evaporation of thin liquid films on wetted walls. Air–water vapor and air–hexane vapor mixtures, assumed as ideal gases, are considered under various boundary conditions. Steady laminar, two-dimensional flows are examined in detail for large mixture density changes between the inlet and outlet sections of the channel. Comparisons with the usual problem formulations based on the Boussinesq approximation are discussed. The elliptic flow model used allow to predict flow reversal as well as recirculation cells in the entrance region. The evaporation of water and hexane into a downward laminar stream of dry air leads to various flow structures according to the interfacial mass fraction, W v,w , and differences in the molecular weights of the species. For water evaporation, the thermal and solutal forces are opposing. In the entrance region, evaporation produces a significant increase in axial velocity at the core region in comparison with pure forced flow. For W v,w larger than ≈0.2, upward velocities may be observed in the wall regions due to solutal buoyancy forces near the wetted surfaces. For hexane evaporation, the solutal force acts downward. Mass diffusion produces both a strong flow acceleration in the boundary layers and flow recirculations at the channel center for large mass evaporation rates.

Natural convection in a shallow cavity containing two superposed layers of immiscible binary liquids

This paper reports an analytical study of the natural convection in a shallow rectangular cavity containing two superposed layers of binary immiscible fluids. A uniform heat flux is applied to the horizontal walls of the enclosure 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) or by a temperature gradient (Soret effects). An analytical solution of the steady form of the governing equations is derived using a parallel flow approximation. The critical Rayleigh numbers for the onset of critical, $${R_{\rm TC}^{\rm sup}}$$ , or subcritical, $${R_{\rm TC}^{\rm sub}}$$ , convection are predicted by the present theory. For finite amplitude convection the present model yields explicitly the flow patterns, Nusselt and Sherwood numbers in terms of the governing parameters of the problem. In the limit of a single fluid layer the present theory is found to be in agreement with the results reported in the literature.

Multiple steady states in a porous enclosure partially heated and fully salted from below

The Darcy model with the Boussinesq approximation is used to study natural convection in a porous medium saturated by a binary fluid. The geometry considered is a square cavity whose portion of the bottom surface is isothermally heated, the upper surface is cooled at a constant temperature and all other surface are adiabatic. The solutal buoyancy forces are assumed to be induced by the imposition of uniform concentration on the upper and lower boundaries. The governing parameters for the problem are the thermal Rayleigh number, R T , the Lewis number, Le, the buoyancy ratio, φ, the dimensionless length of the bottom plate, B T , the aspect ratio of the cavity A, the normalized porosity of the porous medium, ε, and the relative position of the heating element with respect to the vertical centerline of the cavity, δ T . Two main convective modes are studied, namely single and double-cell convection, and their features are described. Also the possible existence of tricellular flows is demonstrated. Maximum streamfunction and global Nusselt and Sherwood number are presented as functions of the external parameters. The existence of up to three steady-state solutions for a given set of the governing parameters is demonstrated.

Natural convection in an enclosure with localized heating and salting from below

Two-dimensional, double diffusion, natural convection in a rectangular enclosure filled with binary fluid saturating porous media is investigated numerically. Multiple motions are driven by the external temperature and concentration differences imposed across horizontal walls with the simultaneous presence of discrete heat and contaminant sources. The general Brinkman-extended Darcy model is adopted to formulate the fluid flow in the cavity. The fluid, heat and moisture transport through the isotropic porous layer are analyzed using the streamlines, heatlines and masslines, and the heat and mass transfer potentials are also explained by the variations of overall Nusselt and Sherwood numbers. The numerical simulations presented here span a wide range of the main parameters (thermal Rayleigh numbers, strip pitches and Darcy number) in the domain of destabilizing solutal buoyancy forces. It is shown that the heat and mass transfer potential can be promoted or inhibited, depending strongly on the permeability of porous medium, the strip pitch, the thermal and solutal Rayleigh numbers.

Multiple steady flows in confined gaseous double diffusion with discrete thermosolutal sources

A detailed numerical study has been conducted to investigate the steady natural convection in a binary mixture in rectangular enclosures with the simultaneous presence of discrete heat and moisture sources. The Prandtl number and Lewis number used are 0.7 and 0.8, respectively. The heat and mass transfer characteristics of the horizontal enclosure are analyzed using the heat lines and mass lines, and the heat and moisture transfer potentials are also explained by the variations of overall Nusselt and Sherwood numbers. Numerical results are particularly presented to illustrate the effects of the buoyancy ratio, thermal Rayleigh number, and strip pitch on the multiple steady flow patterns and the associated heat and mass transfer, for both destabilizing and stabilizing solutal buoyancy forces. The effects of the buoyancy ratio are found to be rather significant on the flow pattern and heat and mass transfer especially for the destabilizing solutal forces. Additionally, in the regime that both thermal and concentration buoyancy forces are destabilizing, the heat and mass transfer potential can be promoted or inhibited, depending strongly on the thermal Rayleigh number and strip pitch.

Boundary domain integral method for the study of double diffusive natural convection in porous media

The main purpose of this paper is to present a boundary domain integral method (BDIM) for the solution of natural convection in porous media driven by combining thermal and solutal buoyancy forces. The Brinkman extension of the classical Darcy equation is used for the momentum conservation equation. The numerical scheme was tested on a natural convection problem within a square porous cavity, where different temperature and concentration values are applied on the vertical walls, while the horizontal walls are adiabatic and impermeable. The results for different governing parameters (Rayleigh number, Darcy number, buoyancy ratio and Lewis number) are presented and compared with published work. There is a good agreement between those results obtained using the presented numerical scheme and reported studies using other numerical methods.

Influence of a magnetic field on natural convection in a shallow porous enclosure saturated with a binary fluid

An analytical and numerical investigation is conducted to study the effect of an electromagnetic field on natural convection in a horizontal shallow porous cavity filled with an electrically conducting binary mixture. A uniform heat flux is applied to the horizontal walls of the layer while the vertical walls are adiabatic. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the horizontal boundaries (double diffusive convection) or by temperature gradients (Soret induced convection). Governing parameters of the problem under study are the thermal Rayleigh number, R T , Hartmann number, Ha, Lewis number, Le, buoyancy ratio, ϕ, aspect ratio, A, and normalized porosity of the porous medium, ɛ. An analytical solution, valid for a shallow layer (A ≫ 1) is derived on the basis of the parallel flow approximation. 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.

NATURAL CONVECTION OF A BINARY MIXTURE IN A VERTICAL CLOSED ANNULUS

This article reports an analytical and numerical study of natural convection of a binary mixture within a vertical closed annulus. Neumann boundary conditions for temperature are applied to the vertical walls of the enclosure, while the short walls are insulated. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the vertical walls (double-diffusive convection, a = 0) or by temperature gradients (Soret effect, a = 1). The governing parameters for the problem are the thermal Rayleigh number RT, Prandtl number Pr, Lewis number Le, buoyancy ratio ϕ, aspect ratio A, constant a, and curvature parameter η. An analytical solution, based on the assumption of parallel flow over a large portion of enclosure, is derived. Numerical confirmation of the analytical results is also presented.

Influence of Dufour effect on entropy generation in double diffusive convection

The influence of Dufour effect on entropy generation in double diffusive convection is numerically investigated in this paper. The dimensionless total entropy generation is evaluated as a function of: the buoyancy ratio N, a parameter characterising the solutal buoyancy forces by report to the thermal ones; the Dufour parameter KF, a parameter characterising the influence of the diffusion-thermo effect; the thermal Grashof number. More details of the investigation is focused for the special case when N = −1.

Numerical simulation of double diffusive laminar mixed convection in a horizontal annulus with hot, solutal and rotating inner cylinder

A numerical investigation of double-diffusive laminar mixed convection within a two-dimensional, horizontal annulus has been carried out. The inner cylinder was considered to rotate in an anti-clockwise direction to introduce the forced convection effect. In addition, the solutal and thermal buoyancy forces are sustained by maintaining the inner and outer cylinders at uniform temperatures and concentrations, but their values for the inner are higher than the outer. The laminar flow regime is considered under steady state conditions. Moreover, the transport equations for continuity, momentum, energy and mass transfer are solved using the Patankar–Spalding technique. The streamlines, isotherms and isoconcentrations as well as both local and average Nusselt and Sherwood numbers were studied. The study covers a wide range for 10 2 ⩽ Ra T ⩽ 10 6 , 0.1 ⩽ Le ⩽ 10 and − 20 ⩽ N ⩽ 20 . Through this investigation, the following parameters are kept constant: Prandtl number at 0.7, the rotational Reynolds number at 100 and the radius ratio at 0.5. The predicted results for both average Nusselt and Sherwood numbers were correlated in terms of Lewis number, thermal Rayleigh number and buoyancy ratio. A comparison was made with the published results and a good agreement was found.

Multiplicity of Solutions Induced by Thermosolutal Convection in a Square Porous Cavity Heated from Below and Submitted to Horizontal Concentration Gradient in the Presence of Soret Effect

ABSTRACT Soret effect on fluid flow and heat and mass transfer induced by double-diffusive natural convection in a square porous enclosure, submitted to cross gradients of temperature and concentration, is studied numerically. The cavity is heated from below and cooled from the top, while its vertical walls are adiabatic and maintained at constant but different concentrations. Numerical results are presented for governing parameters varying in the following ranges: 40 ≤ R T ≤ 1,000, Le = 10, − 0.2 ≤ N ≤ 0.2 and − 30 ≤ M ≤ 30, where R T , Le, N, and M are the thermal Rayleigh number, the Lewis number, the buoyancy ratio, and the Soret parameter, respectively. The effect of the buoyancy ratio and the Soret parameter on the maintenance and disappearance of the multiple steady-state solutions obtained in the case of purely thermal convection is analyzed. It is found that only one monocellular flow mode persists at large values of N (or − N), both in the presence and in the absence of the Soret effect. Some flow modes are destroyed by the solutal buoyancy forces in the absence of the Soret effect (M = 0) but reappear in some range of the parameter M. The Soret effect may affect considerably the heat and mass transfer in the medium; it leads to an enhancement or to a reduction of the mass transfer, depending on the flow structure and the sign of M. There are situations where a solution transfers the solute toward the wall with the highest concentration. Such behavior is observed when the temperature gradient and the magnitude of the Soret parameter M are such that the thermodiffusion flux opposes and overcomes the convection flux, resulting in a net mass flux directed from the least concentrated wall toward the most concentrated one.

Numerical simulation of double-diffusive mixed convection within a rotating horizontal annulus

A numerical investigation of double-diffusive mixed convection within a two-dimensional, horizontal annulus has been carried out. The outer cylinder was made to rotate in an anti-clockwise direction to introduce the forced convection effect. In addition, the solutal and thermal buoyancy forces are sustained by maintaining the inner cylinder at a uniform but higher concentration and temperature values, respectively. The flow is considered to operate in the laminar regime under steady state conditions. Moreover, the transport equations are solved using the Galerkin weighted residual method by incorporating a non-uniform mesh size. The heat and mass transfer rates were closely examined using several dimensionless groups in a wide domain of operating conditions. The considered domains in this investigation are as follows: 5 ⩽ Re ⩽ 150 , 0.01 ⩽ Le ⩽ 10 , 10 3 ⩽ Ra ⩽ 10 5 , − 15 ⩽ N ⩽ 15 , 0.7 ⩽ Pr ⩽ 10 , 0.5 ⩽ σ ⩽ 5 and − 0.75 ⩽ ε ⩽ 0.75 . The whole flow regimes were defined according to the relative values of Reynolds number and solutal and thermal Grashof number. Furthermore, the predictions of the average Nusselt and Sherwood numbers were obtained for the operating range of the Lewis and buoyancy ratio numbers.

Natural Convection of a Binary Fluid in a Slightly Inclined Shallow Cavity

ABSTRACT This article reports an analytical and numerical study of the natural convection in an inclined shallow cavity filled with a binary fluid. Newmann boundary conditions for temperature are applied to the long side walls of the enclosure, while the two short ones are assumed to be impermeable and insulated. The solutal buoyancy force are induced either by the imposition of constant fluxes of solute on the 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,RaT, the Lewis number Le, the buoyancy ratio ϕ, the inclination of the cavity Θ, the Prandtl numberPr, the aspect ratio of the cavity A, and the constant a. For convection in an infinite layer (A > > 1), an analytical solution of the steady form of the governing equations is obtained on the basis of the parallel flow approximation. The critical Rayleigh numbers for the onset of supercritical and subcritical convection are predicted by the present model. Also, it is demonstrated that, for small enough inclinations around the horizontal plane, multiple steady states exist, some of which are unstable. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. Good agreement is observed between the analytical prediction and the numerical simulations.

Convective heat and solute transfer in partially porous cavities

This paper deals with natural convection driven by combined thermal and solutal buoyancy forces in a binary fluid. The configuration under study is a confined enclosure partially filled with a vertical porous layer. The mathematical description of the problem is based on a one-domain formulation of the conservation equations. The set of numerical results presented here quantitatively shows the influence of the porous layer on the flow structure and on heat and species transfer in the enclosure. The paper is focused on the analysis of the influence of the characteristic parameters governing double diffusive convection, namely the ratios of solutal and thermal parameters: the diffusivities and the buoyancy forces. Heat and mass transfer is analyzed as a function of the permeability of the porous layer. It is shown that the coupling of the flow penetration in the porous layer with the combined buoyancy forces induces a specific behavior of the flow structure and average heat transfer in the enclosure.

Double-diffusive natural convection in a porous enclosure partially heated from below and differentially salted

This paper reports numerical results of two-dimensional double-diffusive natural convection in a square porous cavity partially heated from below while its upper surface is cooled at a constant temperature. The vertical walls of the porous matrix are subjected to a horizontal concentration gradient. The parameters governing the problem are the thermal Rayleigh number ( Ra=100 and 200), the Lewis number ( Le=0.1, 1 and 10), the buoyancy ratio (−10⩽ N⩽10) and the relative position of the heating element with respect to the vertical centerline of the cavity ( δ=0 and 0.5). The effect of the governing parameters on fluid characteristics is analyzed. The multiplicity of solutions is explored and the existence of asymmetric bicellular flow is proved when the heated element is shifted towards a vertical boundary ( δ=0.5). The solutal buoyancy forces induced by horizontal concentration gradient lead to the elimination of the multiplicity of solutions obtained in pure thermal convection when N reaches some threshold value which depends on Le and Ra.

Modeling of thermosolutal convection during Bridgman solidification of semiconductor alloys in relation with experiments

Thermosolutal convection during vertical Bridgman directional solidification of Ga 1− x In x Sb alloys has been studied by numerical simulation. The transient analysis of heat, momentum and species transport has been performed by using the finite element code FIDAP ®. In the case of vertical Bridgman configuration, the thermal convection is driven by the radial temperature gradients. The solute (InSb) rejected at the solid–liquid interface, which is heavier than the GaSb component, damps the thermally driven convection. The solutal effect on the melt convection has been analyzed for low ( x=0.01) and high ( x=0.1) doped Ga 1− x In x Sb alloys. It is found that the damping effect is negligible for Ga 0.99In 0.01Sb alloy grown at low pulling rates ( V= 1 μm/s ), but cannot be neglected if the pulling rate is increased. In the case of concentrated alloys, the low level of convection intensity leads to an increase of radial segregation and interface curvature during the whole growth process as also shown by experiments. The effect of solutal buoyancy force on the melt convection is analyzed for the horizontal Bridgman configuration under microgravity conditions. An inverse but lower solutal effect on the melt convection, as compared with vertical Bridgman arrangement, is observed. The results are in good agreement with the experimental data, and show that convective transport can be observed even for low (2×10 −6 g 0) residual gravity levels.

Double diffusive convection in a porous enclosure submitted to cross gradients of temperature and concentration

Numerical results of two dimensional double diffusive natural convection in a square porous cavity submitted to cross gradients of heat and solute concentration are reported in this study. The parameters governing the problem are the thermal Rayleigh number (100⩽ Ra⩽200), the Lewis number (0.1⩽ Le⩽10) and the buoyancy ratio (−10⩽ N⩽10). The effects of the governing parameters on the flow structure and heat and mass transfer are analyzed. It is demonstrated that the solutal buoyancy force induced by horizontal concentration gradients eliminates the multiplicity of solutions obtained in pure thermal convection when N exceeds some critical value, which depends on Le and Ra. For N>0/( N<0), the monocellular trigonometric/(clockwise) solution is maintained for all the explored ranges of the governing parameters considered in this study.

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.

Thermal convection in binary fluid mixtures with a weak concentration diffusivity, but strong solutal buoyancy forces.

Thermal convection in binary liquid mixtures is investigated in the limit where the solutal diffusivity is weak, but the separation ratio is large. Representative examples are colloidal suspensions such as ferrofluids. With a grain size being large on molecular length scales, the particle mobility is extremely small, allowing to disregard the concentration dynamics in most cases. However, this simplification does not hold for thermal convection: Due to the pronounced Soret effect of these materials in combination with a considerable solutal expansion, the resulting solutal buoyancy forces are dominant. Indeed, convective motion is found to set in at Rayleigh numbers well below the critical threshold for single-component liquids. A nonlinear analysis demonstrates that the amplitude quickly saturates in a state of stationary convective motion.

COMPLEX DYNAMICS OF RHYTHMIC PATTERNS AND SEDIMENTATION OF ORGANIC CRYSTALS: A NEW NUMERICAL APPROACH

This article deals with the mathematical modeling and numerical simulation of the complex phenomena related to the crystallization of an organic macromolecular substance due to solubility modulation (lysozyme). Novel mathematical models and numerical strategies are introduced to simulate the protein nucleation and further precipitation (or resolution) according to the concentration distribution and in particular to take into account the motion of the crystals due to the combined effect of gravitational force and viscous drag if the sedimenting process is allowed (protein reactor free of gel). The numerical simulations show that gellified lysozyme reacts to produce "spaced deposits." The phenomenon is characterized by a certain degree of periodicity in time and in space (Liesegang patterns). If the gel matrix is not present, crystals settle in the growth chamber. In this case, due to solutal buoyancy forces, a very complex convection pattern arises whose dynamics depend on the nucleation phenomena occurring in the bulk of the protein chamber.