双自由面溶质?热毛细液层的不稳定性

溶质?热毛细对流是流体界面的浓度和温度分布不均导致的表面张力梯度驱动的流动, 它主要存在于空间微重力环境、小尺度流动等表面张力占主导的情况中, 例如晶体生长、微流控、合金浇筑凝固、有机薄液膜生长等. 对其流动进行稳定性分析具有重要意义. 本文采用线性稳定性理论研究了双自由面溶质?热毛细液层对流的不稳定性, 得到了两种负毛细力比(η)下的临界Marangoni数与Prandtl数(Pr)的函数关系, 并分析了临界模态的流场和能量机制. 研究发现: 溶质?热毛细对流和纯热毛细对流的临界模态有较大的差别, 前者是同向流向波、逆向流向波、展向稳态模态和逆向斜波, 后者是逆向斜波和逆向流向波. 在Pr较大时, Pr增加会降低流动稳定性; 在其他参数下, Pr增加会增强流动稳定性. 在中低Pr, 溶质毛细力使流动更加不稳定; 在大Pr时, 溶质毛细力的出现可能使流动更加稳定; 在其他参数下, 溶质毛细力会减弱流动稳定性. 流动稳定性不随η单调变化. 在多数情况下, 扰动浓度场与扰动温度场都是相似的. 能量分析表明: 扰动动能的主要能量来源是表面张力做功, 但其中溶质毛细力和热毛细力做功的正负性与参数有关.      

Control of Thermo- and Solutocapillary Flows in FZ Crystal Growth by High-Frequency Vibrations

The paper deals with the numerical investigation of the possibilities to control convective flows in the liquid bridge in zero gravity conditions applying axial vibrations. The surface tension is assumed to be dependent both on the temperature and on the solute concentration. The free surface deformations and the curvature of the phase change surfaces are neglected but pulsational deformations of the free surface are accounted for. The first part of the paper concerns axisymmetric steady flows. The calculations show that the evolution of convective flow with the variation of thermal Marangoni number at a fixed value of the solutal Marangoni number is accompanied by the hysteresis phenomenon, which is related to the existence of two stable steady regimes in a certain parameter range. One of these regimes is thermocapillary dominated, it corresponds to the two-vortex flow, and the other is solutocapillary dominated, it corresponds to the single-vortex flow. Under vibrations, the range of the Marangoni numbers where hysteresis is observed becomes narrower and is shifted to the area of larger values. The second part of the paper concerns the stability of axisymmetric thermo-and solutocapillary flows and the transition to three-dimensional regimes. Significant mutual influence of flows generated by each process on the stability of the other is discovered. Stability maps in the parametric plane for the thermal Marangoni number, the solutal Marangoni number, are obtained for different values of vibration parameters. It is shown, that vibrations exert a stabilizing effect, increasing critical Marangoni numbers for all modes of instability. However, this effect is different for different modes and at high vibration intensity destabilization is possible. Consequently, vibrations can modify the scenario of the transition to the three-dimensional mode.     

The effect of an insoluble surfactant on the skin friction of a bubble

In this paper, we develop a novel moving mesh method suitable for solving axisymmetric free-boundary problems, including the Marangoni effect induced by surfactant or temperature variation. This method employs a body-fitted grid system where the gas–liquid interface is one line of the grid system. We model the surfactant equation of state with a non-linear Langmuir law, and, for simplicity, we limit ourselves to the situation of an insoluble surfactant. We solve complicated dynamic boundary conditions accurately on the gas–liquid interface in the framework of finite-volume methods. Our method is used to study the effect of a surfactant on the skin friction of a bubble in a uniaxial flow. For the limiting case where the surface diffusivity is zero, the effect of a tangential stress generated by the surface tension gradient, allows us to explain a new phenomenon in high concentration regimes: larger surface tension, but also larger deformation. Furthermore, this condition leads to the formation of boundary layers and flow separation at high Reynolds numbers. The influence of these complex flow patterns is examined.     

Direct numerical simulation of thermocapillary flow based on the Volume of Fluid method

A numerical method for direct simulation of thermal Marangoni effects at dynamically deformable interface of two-phase incompressible fluids is developed. The approach is based on the Volume of Fluid (VOF) method with special focus on the numerical treatment of the temperature surface gradient because of its decisive role as the driving force of the flow. The surface gradient calculation is split into computing its length and direction in order to satisfy the correct thermal boundary condition at the interface without losing mobility of the interface. The method is applied to three different types of thermocapillary flow, namely thermocapillary migration of a droplet in an ambient fluid with linear temperature gradient, thermocapillary convection in a liquid layer under linear temperature gradient along the interface, and Marangoni convection due to Bénard–Marangoni instability. In the first case, different aspects of the dynamics of the migration are considered for validation such as the terminal migration velocity, the initial acceleration and quantification of the wall effects. The simulation also considers high convective heat transfer and covers a wide range of Marangoni numbers up to 5000, where good agreement with both theoretical and experimental results is achieved. In the second case, the convection velocity in the liquid layer is compared with an analytical result. In the final application, pattern formation due to the Bénard–Marangoni instability in a liquid layer in square geometry of small aspect ratio is investigated for realistic Biot number and dynamically deformable fluid interface. The results show good agreement with experiments from literature, where our numerical simulation also predicts cell pattern for a particular aspect ratio which is outside the limitation of the above cited experimental work.     

Effect of an insoluble surfactant film on the stability of the concentration-driven Marangoni flow

The stability and the structure of the concentration and capillary driven Marangoni flow from a localized source is experimentally investigated in the presence of an adsorbed layer of an insoluble surfactant. It is found that the presence of the surfactant on the interface leads to the instability of the main axisymmetric flow with the result that a secondary azimuthally-periodic flow with a multivortex structure is developed. The structure of the convective motion on the interface is studied as a function of the Marangoni flow intensity and the surface density of the surfactant. The azimuthal wavenumber is shown to increase with the Marangoni number and to decrease with increase in the surface density of the surfactant. It is established that there exists a threshold value of the surface density of the surfactant at which the surface flow does not occur.     

Experimental study of Marangoni bubble migration in normal gravity

Marangoni convection, whether thermal or solutal, is known to have a profound impact on many technological processes involving gas inclusions in a liquid phase. Evidently, similar phenomena may arise both in thermocapillary and solutocapillary situations, due to similarity of the motion driving mechanisms. However, the fact that the characteristic times of heat and surfactant diffusion generally differ by several orders of magnitude lends singularity to the behavior of Marangoni convection in inhomogeneous mixtures. Moreover, in the solutocapillary case one can meet the action of some additional effects associated with dissolution of the surfactant in a liquid, its adsorption at the interface and evaporation into a gas phase. This paper presents a comparative analysis of the results of ground experiments studying the behavior of air bubbles in a liquid under the action of thermocapillary and solutocapillary forces. The use of original experimental techniques makes it possible to eliminate the influence of gravity effects. A new Marangoni phenomenon—solutocapillary bubble migration—was detected and investigated. The results of studying thermal and concentration convective flows and bubble motion, in relation to bubble size, time, liquid layer thickness and fluids properties, are presented and discussed.     

Molecular dynamics study of the liquid-vapor interface of lithium bromide aqueous solutions

The molecular dynamics simulations of the liquid–vapor interface of LiBr aqueous solutions were carried out to investigate the structural and thermophysical properties. As concerns the structural properties, the results of molecular dynamics simulation show that the ions exist in the liquid apart from the surface and this tendency becomes strong as the solute concentration is lowered. This phenomenon is due to the desorption of ion. The calculated values such as density or surface tension agree with experimental ones. As concerns thermophysical properties, the number of water molecules in the bulk gas decreases with an increase of the solute concentration. This result represents the depression of vapor pressure. In addition, in order to investigate the dynamic process of water vapor absorption into LiBr aqueous solution, the molecular dynamics simulation under non-equilibrium condition was carried out. The results show that when the solute concentration is low and the temperature is also low, almost all incident water molecules become trapped at the solution surface and then easily diffuse into the bulk liquid, and when the solute concentration is high and temperature is also high, most incident water molecules become trapped at the solution surface, and the sequent processes are very complicated. Received on 28 September 1998     

Experimental evaluation of Marangoni stress and surfactant concentration at interface of contaminated single spherical drop using spatiotemporal filter velocimetry

Spatiotemporal filter velocimetry (SFV) was extended to Lagrangian measurements with boundary-fitted measurement areas, and was applied to flows about single spherical drops of glycerol-water solution falling in stagnant silicon oil under clean and contaminated conditions to examine its applicability to the estimation of the Marangoni stress and surfactant concentration at a moving interface. Effects of bulk concentration of surfactant on the velocity field, the Marangoni stress and the surface concentration of surfactant were discussed from the measured data. As a result, we confirmed that accurate velocity distribution in the vicinity of the interface measured by SFV enables us to evaluate interfacial velocity and interfacial shear stresses and to estimate the Marangoni stress, interfacial tension and surfactant concentration at the interface with the assumption of negligible surface viscosity. The flow inside the drop and the interfacial velocity become weak due to the Marangoni stress caused by the gradient of surfactant concentration at the interface as the bulk concentration of surfactant increases. These results demonstrate that SFV is of great use in experimental analysis of adsorption and desorption kinetics at a moving interface.     

Marangoni instability in a liquid layer bounded by two coaxial cylinder surfaces

A liquid layer, confined between two coaxial cylinder surfaces, has either a gasliquid interface on the inside and is heated from the outer (solid) boundary, or it has a gas-liquid interface on the outside and is heated from the inner (solid) boundary. Neglecting gravity and using a standard normal-mode approach, we analyse surface-tension driven instability (Marangoni instability) of the motionless steady state in which the temperature depends on the radial coordinate only. Numerical results for the critical Marangoni number and corresponding wave-number pair are presented for various values of the curvature of the interface. This curvature turns out to exert a significant influence on the onset of Marangoni convection flows. Further, the stability behaviour of the system is found to be quite variable, depending on whether the interface is on the inside or on the outside of the layer and whether it is well-conducting or nearly-isolated.     

Mass transfer enhancement at deformable droplets due to Marangoni convection

One of the key design parameters in liquid/liquid extraction is the mass transfer coefficient. A complex list of parameters including fluid dynamics, drop size distribution, chemical properties of the involved species, local interfacial instabilities (Marangoni convection) is required in order to determine the transient evolution of the mass transfer coefficient. The influence of Marangoni convection on single drop mass transfer cannot yet be described in an analytical manner, and empirical correlations available in literature fail to predict the mass transfer process. In the present study, experimental investigations on deformable single droplets in the toluene/acetone/water system are presented which shows strong interfacial instabilities. Parameters varied are the drop diameter, the initial solute concentration and the mass transfer direction. Experimental results are compared with the well-known models by Kronig and Brink and Handlos and Baron. The Kronig and Brink model cannot describe Marangoni dominated systems, but comparisons reveal the influence of deformation on the mass transfer enhancement. In contrary, with a slight modification to the Handlos and Baron model, the mean droplet concentration of the transferred component was successfully modelled as a function of Fourier number.     

磁场对双扩散液层热毛细对流的影响

通过数值模拟的方法对磁场作用下的双扩散液层热毛细对流进行了研究, 模型中同时考虑了热毛细效应和溶质毛细效应的存在. 研究结果显示, 外部磁场能够有效削弱液层内热毛细对流的强度, 改变热毛细对流的对流结构; 随着磁场强度的增大, 液层内热毛细对流的对流强度逐渐减小, 热质传递过程中扩散效应逐渐得到增强; 最终, 溶质浓度沿水平方向呈梯度分布. 因此, 当磁场强度足够大时能够实现晶体生长中所需的纯扩散条件.     

Convective and diffusive surfactant transfer in multiphase liquid systems

Laser interferometry was used to investigate diffusive and convective mass transfer in a multicomponent fluid mixture with a liquid–liquid or liquid–gas interface. For this purpose, an immobile gas bubble or insoluble fluid droplet, having the shape of a short cylinder with a free lateral surface, was inserted into a thin liquid layer. In the case of non-uniform distribution of the dissolved surfactant component, the Marangoni convection near the drop/bubble was initiated by the surface tension inhomogeneities, depending on the surfactant concentration. The applied experimental techniques allowed us to study the structure and evolution of the convective flows and concentration fields in a liquid layer, which due to its small thickness were nearly two-dimensional. Making use of both the vertical and horizontal orientation of the liquid layer, we investigated the mass transfer process at different levels of the interaction between gravity and capillary forces. During the experiments, we detected new solutocapillary phenomena, which were found to be caused by oscillatory regimes of solutal convection occurring around air bubbles and chlorobenzene drops in heterogeneous aqueous solutions of alcohol with a vertical surfactant concentration gradient. The role of the oscillatory instability in the processes of drop saturation by the surfactant from its water solution and an inverse process of surfactant extraction from the drop into the surrounding homogeneous fluid (water) was determined. A reasonable explanation for the driving mechanisms of the discovered effects has been proposed.     

Topological structure bifurcation of temperature field of deformable drop in Marangoni migration

The unsteady processes of the Marangoni migration of deformable liquid drops are simulated numerically in a wider range of Marangoni number (up to Ma = 500) in the present work. A steady terminal state can always be reached, and the scaled terminal velocity is a monotonic function decreasing with increasing Marangoni number, which is generally in agreement with corresponding experimental data. The topological structure of flow field in the steady terminal state does not change as the Marangoni number increases, while bifurcation of the topological structure of temperature field occurs twice at two corresponding critical Marangoni numbers. A third critical value of Marangoni number also exists, beyond which the coldest point jumps from the rear stagnation to inside the drop though the topological structure of the temperature field does not change. It is found that the inner and outer thermal boundary layers may exist along the interface both inside and outside the drop if Ma > 70. But the thickness decreases with increasing Marangoni number more slowly than the prediction of potential flow at large Marangoni and Reynolds numbers.     

Phase-field modeling of interfacial dynamics in emulsion flows: Nonequilibrium surface tension

A weakly nonlocal phase-field model is used to define the surface tension in liquid binary mixtures in terms of the composition gradient in the interfacial region so that, at equilibrium, it depends linearly on the characteristic length that defines the interfacial width. Contrary to previous works suggesting that the surface tension in a phase-field model is fixed, we define the surface tension for a curved interface and far-from-equilibrium conditions as the integral of the free energy excess (i.e., above the thermodynamic component of the free energy) across the interface profile in a direction parallel to the composition gradient. Consequently, the nonequilibrium surface tension can be widely different from its equilibrium value under dynamic conditions, while it reduces to its thermodynamic value for a flat interface at local equilibrium. In nonequilibrium conditions, the surface tension changes with time: during mixing, it decreases as the inverse square root of time, while in the linear regime of spinodal decomposition, it increases exponentially to its equilibrium value, as nonlinear effects saturate the exponential growth. In addition, since temperature gradients modify the steepness of the concentration profile in the interfacial region, they induce gradients in the nonequilibrium surface tension, leading to the Marangoni thermocapillary migration of an isolated drop. Similarly, Marangoni stresses are induced in a composition gradient, leading to diffusiophoresis. We also review results on the nonequilibrium surface tension for a wall-bound pendant drop near detachment, which help to explain a discrepancy between our numerically determined static contact angle dependence of the critical Bond number and its sharp-interface counterpart from a static stability analysis of equilibrium shapes after numerical integration of the Young-Laplace equation. Finally, we present new results from phase-field simulations of the motion of an isolated droplet down an incline in gravity, showing that dynamic contact angle hysteresis can be explained in terms of the nonequilibrium surface tension.     

等热流密度加热垂直下降液膜的研究

利用等热流密度加热条件下降膜流动的三维模型方程进行线性稳定性分析和数值模拟。线性稳定性分析表明,模型方程在小到中等Reynolds数下都适用,并且流向不稳定性增长率随着Reynolds数和Marangoni数增加而增加,展向不稳定性增长率则随着Marangoni数增加而增加,随着Reynolds数增加而减小,流向和展向对扰动波数都存在一个不稳定区间。三维数值模拟表明,在等热流密度加热条件下,液膜在随机扰动的情况下最终会形成带孤立波的三维溪流状结构,液膜与气体的换热也因溪流状结构的出现而加强;在随机扰动的基础上引入占优势地位的展向最不稳定扰动会使得换热增强,液膜会提前破裂;在随机扰动的基础上引入占优势地位的流向最不稳定扰动时,液膜的换热会增强,但不会提前破裂;在随机扰动的基础上同时引入占优势地位的流向和展向最不稳定扰动时,换热会加强且液膜会提前破裂。     

Experimental study of the stability of long axisymmetric liquid bridges between solid supports at different temperatures

The stability of axisymmetric, long liquid bridges held captive between two coaxial, circular solid disks kept at different temperatures is considered. Because of the temperature difference between the supporting disks, a thermally-induced surface tension gradient and its associated flow (Marangoni convection) appear in the liquid column, modifying (decreasing) the capillary stability of the bridge. The influence on the stability limits of long, axisymmetric liquid bridges of the combined effect of gravity acceleration and thermally induced surface tension gradients was experimentally analyzed by using very small size liquid bridges (between disks 1 mm in diameter). Experimental results are compared with available analytical results.     

Numerical prediction of turbulent thermocapillary convection in superposed fluid layers with a free interface

The present paper introduces a new numerical method for predicting the characteristics of thermocapillary turbulent convection in a differentially-heated rectangular cavity with two superposed and immiscible fluid layers. The unsteady Reynolds form of the Navier–Stokes equations and energy equation are solved by using the control volume approach on a staggered grid system using SIMPLE algorithm. The turbulence quantities are predicted by applying the standard k–ε turbulence model. The level set formulation is applied for predicting the topological changes of the interface separating the two fluid layers and to provide an accurate and robust modeling of the interfacial normal and tangential stresses. The computational results obtained showed good agreement when compared with the previous experimental, numerical and analytical benchmark data for different validation cases in both laminar and turbulent regimes. The present numerical method is then applied to predict the velocity and temperature distribution in two immiscible liquid layers with undeformable interface for a wide range of Marangoni numbers. The laminar-turbulent transition is demonstrated by obtaining the turbulence features at high interfacial temperature gradient which is characterized by high Marangoni number. The effect of increasing Marangoni number on the interface dynamics in turbulent regime is also investigated.     

Numerical prediction of turbulent thermocapillary convection in superposed fluid layers with a free interface

The present paper introduces a new numerical method for predicting the characteristics of thermocapillary turbulent convection in a differentially-heated rectangular cavity with two superposed and immiscible fluid layers. The unsteady Reynolds form of the Navier–Stokes equations and energy equation are solved by using the control volume approach on a staggered grid system using SIMPLE algorithm. The turbulence quantities are predicted by applying the standard k–ε turbulence model. The level set formulation is applied for predicting the topological changes of the interface separating the two fluid layers and to provide an accurate and robust modeling of the interfacial normal and tangential stresses. The computational results obtained showed good agreement when compared with the previous experimental, numerical and analytical benchmark data for different validation cases in both laminar and turbulent regimes. The present numerical method is then applied to predict the velocity and temperature distribution in two immiscible liquid layers with undeformable interface for a wide range of Marangoni numbers. The laminar-turbulent transition is demonstrated by obtaining the turbulence features at high interfacial temperature gradient which is characterized by high Marangoni number. The effect of increasing Marangoni number on the interface dynamics in turbulent regime is also investigated.     

Stability of thermocapillary advective flow in a slowly rotating liquid layer under microgravity conditions

The stability of thermocapillary flow developed in a slowly rotating fluid layer under microgravity conditions is investigated. Both boundaries of the layer are free and assumed to be plane. The tangential thermocapillary Marangoni force exerts on the boundaries, where heat transfer takes place in accordance with the Newton law, the temperature of the medium in the neighborhood of the boundaries being a linear function of the coordinates. The axis of rotation is perpendicular to the liquid layer, rotation is weak so that the centrifugal force can be neglected. Being the solution of the Navier-Stokes equations, the thermocapillary flow in question can be described analytically. The neutral curves which describe the wavenumber dependence of the critical Marangoni number for various Taylor numbers and various directions of the horizontal temperature gradient on the layer boundaries are obtained within the framework of the linear stability theory. The behavior of finite-amplitude perturbations beyond the stability threshold is studied numerically.     

Absolute and convective instability of a liquid sheet with transverse temperature gradient

The spatial–temporal instability behavior of a viscous liquid sheet with temperature difference between the two surfaces was investigated theoretically. The practical situation motivating this investigation is liquid sheet heated by ambient gas, usually encountered in industrial heat transfer and liquid propellant rocket engines. The existing dispersion relation was used, to explore the spatial–temporal instability of viscous liquid sheets with a nonuniform temperature profile, by setting both the wave number and frequency complex. A parametric study was performed in both sinuous and varicose modes to test the influence of dimensionless numbers on the transition between absolute and convective instability of the flow. For a small value of liquid Weber number, or a great value of gas-to-liquid density ratio, the flow was found to be absolutely unstable. The absolute instability was enhanced by increasing the liquid viscosity. It was found that variation of the Marangoni number hardly influenced the absolute instability of the sinuous mode of oscillations; however it slightly affected the absolute instability in the varicose mode.