Porous Media and the Bénard–Marangoni Problem

The Rayleigh–Bénard instability for a clear fluid has its equivalent for a liquid saturated porous matrix. The phenomenological Darcy momentum law cannot give rise by itself to an instability analogous to that of Bénard–Marangoni, but the Brinkman model at least allows it. A critical Marangoni number exists leading to cellullar patterns and, for realistic values of the permeability, it is proportional to the inverse of this last parameter.     

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.     

Effects of a.c. Electric Field and Rotation on Bénard–Marangoni Convection

The coupled buoyancy and thermocapillary instability, the Bénard–Marangoniproblem, in an electrically conducting fluid layer whose upper surface is deformed and subject to a temperature gradient is studied. Both influences of an a.c. electric field and rotation are investigated. Special attention is directed at the occurrence of convection both in the form of stationary motion and oscillatory convection. The linear stability problem is solved for different values of the relevant dimensionless numbers, namely the a.c. electric Rayleigh number, the Taylor, Rayleigh, Biot, Crispation and Prandtl numbers. For steady convection, it is found that by increasing the angular velocity, one reinforces the stability of the fluid layer whatever the values of the surface deformation and the applied a.c. electric field. We have also determined the regions of oscillatory instability and discussed the competition between both stationary and oscillatory convections. This revised version was published online in July 2006 with corrections to the Cover Date.     

Growth rates of the Marangoni instability in a layer of elastic liquid

The problem of thermocapillary (Marangoni) convection in a layer of viscoelastic liquid is considered. The stability boundary for this problem has been previously calculated in various cases by a number of authors. Here attention is fixed on the magnitude of the growth rate in the parameter regime corresponding to instability. Two noteworthy features are pointed out. First, there are anomalously large values of the growth rate at or near the limiting special case of a Maxwell fluid. Second, the complex values of the growth rate (corresponding to overstability, or the onset of instability via oscillatory motion) coalesce into real (positive) values at moderately supercritical values of the Marangoni number, suggesting that overstability might be elusive to observation.     

液滴热毛细迁移问题的研究进展

液滴或气泡的迁移现象无论是在流体力学的基础研究中,还是在材料加工,化学工程等实际应用中都是一个很重要的课题。在微重力环境中,如果在液滴或气泡所在的母液中外加一个温度场,则液滴或气泡就会由于表面张力分布的不均匀而发生迁移运动。这种运动被称为Marangoni迁移或热毛细迁移运动。本文综述了液滴或气泡的热毛细迁移问题历史研究中理论分析,数值模拟以及实验方面的主要结果,阐述了该问题的研究发展过程。目前液滴迁移问题的研究状况,理论分析解还只限于线性及弱非线性的定常问题,数值模拟工作已经得到了在热对流作用比较小的时候液滴的非定常迁移过程,但是对于热对流影响很大的情况(Marangoni数大于100)则尚未得到过与实验中观测到的相一致的理论结果。本文在总结前人研究的基础上,同时给出了在对于热对流作用较大时液滴热毛细迁移非定常问题的最新的数值模拟的结果,并对该问题在此情况下产生的新的变化也给予了分析。最后,文中分析了当前研究中所存在的问题并进一步展望了液滴热毛细迁移问题在未来的发展方向。      

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.     

Effect of vertical heat transfer on thermocapillary convection in an open shallow rectangular cavity

In order to understand the effect of the vertical heat transfer on thermocapillary convection characteristics in a differentially heated open shallow rectangular cavity, a series of two- and three-dimensional numerical simulations were carried out by means of the finite volume method. The cavity was filled with the 1cSt silicone oil (Prandtl number Pr = 13.9) and the aspect ratio ranged from 12 to 30. Results show that thermocapillary convection is stable at a small Marangoni number. With the increase of the heat flux on the bottom surface, thermocapillary convection transits to the asymmetrical bi-cellular pattern with the opposite rotation direction. The roll near the hot wall shrinks as the Marangoni number increases. At a large Marangoni number, numerical simulations predict two types of the oscillatory thermocapillary flow. One is the hydrothermal wave, which is dominant only in a thin cavity. The other appears in a deeper cavity and is characterized by oscillating multi-cellular flow. The critical Marangoni number for the onset of the oscillatory flow increases first and then decreases with the increase of the vertical heat flux. The three-dimensional numerical simulation can predict the propagating direction of the hydrothermal wave. The velocity and temperature fields obtained by three-dimensional simulation in the meridian plane are very close to those obtained by two-dimensional simulation.     

Bénard-Marangoni instability in a viscoelastic Jeffreys' fluid layer

Coupled buoyancy (Bénard) and thermocapillary (Marangoni) convection in a thin fluid layer of a viscoelastic fluid are studied. The viscoelastic fluid is modeled by Jeffreys' constitutive equation. The lower surface of the layer is in contact with a rigid heat-conducting plate while its upper surface is subject to a temperature-dependent surface tension. The critical temperature difference between both boundaries corresponding to the onset of convection is calculated. The role of the various viscometric coefficients is discussed. In the appendix it is shown that Jeffreys' constitutive relation is easily derived from thermodynamic considerations based on extended irreversible thermodynamics.     

Axisymmetric thermocapillary convection in a slowly rotating annular cylindrical container

In a slowly rotating annular cylindrical container the free liquid surface (liquid-gas interface) is subjected to a temperature gradient in radial direction. The temperature dependent surface tension creates a shear stress on the interface which is transmitting a thermocapillary convection in the bulk of the liquid. For constant temperature T ₁ of the inner and T ₂ of the outer wall a steady Marangoni convection takes place, exhibiting a double vortex ring of equal directional flow. For time-oscillatory temperatures of the walls a time-dependent thermocapillary convection appears, which will create on the free liquid surface various wave patterns. They shall, depending on the forcing frequency of the temperature, exhibit resonance peaks. The velocity distribution and the response magnitude inside the container has been determined. Received on 3 September 1997     

Marangoni convection in an annular cylindrical tank

If the free liquid-gas interface of a liquid in an annular container is subjected to a temperature gradient the temperature dependent shear stress on the free liquid surface creates by viscous traction a thermocapillary convection in the bulk of the liquid. For constant temperature T ₁ of the inner and T ₂ of the outer wall a steady Marangoni convection takes place, while for time-oscillatory temperatures of the walls a time-dependent thermocapillary convection appears, which will create on the free liquid surface wave patterns. They shall, depending on the forcing frequency of the temperature, exhibit resonance peaks. The velocity distribution and the response magnitude inside the container has been determined. Received on 11 July 1997     

液封液桥内热毛细对流的数值模拟

建立了液封液桥(不相溶混的双层同轴液柱)内热毛细对流的物理数学模型，采用涡量-流 函数法、利用有限差分格式对微重力条件下液封液桥内热毛细对流进行了数值模拟，得到了 双层液柱主流区的温度场和流场，证实了液封技术能削弱液桥主流区的热毛细对流，从而提 高浮区晶体生长质量，找出了液封厚度以及内、外层流体物性参数比对液桥内热毛细对流的 影响规律.     

Experimental and numerical study of Marangoni convection in shallow liquid layers

An experimental and numerical study of thermal Marangoni convection in shallow liquid layers was carried out for a range of temperature differences and layer depths. This was done to permit earth based experiments to be undertaken in situations where Marangoni convection dominated the flow. Particle image velocimetry (PIV) was used to obtain the flow patterns and velocity vectors. The experimental results were compared to numerical models created using FLUENT V6. Both results are in good agreement. The liquid free surface profile due to the presence of the menisci is shown to be critical for good quantitative validation. The layer depths are also proven to be shallow enough for Marangoni convection to dominate over buoyancy under normal gravity conditions.     

Marangoni Mixed Convection Boundary Layer Flow

The paper deals with a steady coupled dissipative layer, called Marangoni mixed convection boundary layer, which can be formed along the interface of two immiscible fluids, in surface driven flows. The mixed convection boundary layer is generated when besides the Marangoni effects there are also buoyancy effects due to gravity and external pressure gradient effects. We shall use a model proposed by Golia and Viviani (L’ Aerotecnica missili e Spazio 64 (1985) 29–35, Meccanica 21 (1986) 200–204) wherein the Marangoni coupling condition has been included into the boundary conditions at the interface. The similarity equations are first determined, and the pertinent equations are solved numerically for some values of the governing parameters and the features of the flow and temperature fields as well as the interface velocity and heat transfer at the interface are analysed and discussed.     

Numerical and experimental study of the heat transfer and fluid flow by thermocapillary convection around gas bubbles

 At liquid–gas or liquid–liquid interfaces thermocapillary or Marangoni convection develops in the presence of a temperature or concentration gradient along the interface. This convection was not paid much attention up to now, because under terrestrial conditions it is superimposed by the strong buoyancy convection. In a microgravity environment, however, it is the remaining mode of natural convection. During boiling in microgravity it was observed at subcooled conditions. Therefore the question arises about its contribution to the heat transfer. Thus the thermocapillary convection was intensively studied at single gas bubbles in various liquids both experimentally and numerically. Inside a temperature gradient chamber, the overall heat transfer around single bubbles of different volume was measured with calorimetry and the liquid flow with PIV and LDV. In parallel to the experiment, a 2-dimensional mathematical model was worked out and the coupled heat transfer and fluid flow was simulated with a CV-FEM method both under earth gravity level and under microgravity. The results are described in terms of the dimensionless Nusselt-, Peclet-, Marangoni-, Bond- and Prandtl-number. Received on 23 August 1999     

Thermocapillary instability of the equilibrium of a flat layer containing internal heat sources

In the absence of body forces, a factor which has a strong influence on the equilibrium stability of a nonuniformly heated liquid is the dependence of the coefficient of surface tension on the temperature and the thermocapillary effect generated by it. If the equilibrium temperature gradient is sufficiently great, then the presence of the thermocapillary forces on the free surface can lead to the occurrence of convective motion. The monotonie instability of the equilibrium of a flat layer was investigated in [1–3]. Analysis of nonmonotonic disturbances [4] showed that in the case of an undeformable free surface there is no oscillatory instability. In [5] it was found that oscillatory instability is possible if there is a nonlinear dependence of the coefficient of surface tension on the temperature. The present paper is devoted to numerical investigation of the equilibrium stability of a flat layer with respect to arbitrary disturbances. It is shown that for a deformable free boundary there appears an additional neutral curve, which corresponds to monotonie capillary instability. In addition, when the capillary convection mechanism is taken into account, there appears an oscillatory instability, which becomes the most dangerous in the region of small Prandtl and wave numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–31, March–April, 1991.I thank V. K. Andreev for a helpful discussion of the work.     

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.     

Three‐dimensional numerical simulation of Marangoni instabilities in liquid bridges: influence of geometrical aspect ratio

Oscillatory Marangoni convection in silicone oil–liquid bridges with different geometrical aspect ratios is investigated by three‐dimensional and time‐dependent numerical simulations, based on control volume methods in staggered cylindrical non‐uniform grids. The three‐dimensional oscillatory flow regimes are studied and compared with previous experimental and theoretical results. The results show that the critical wavenumber (m), related to the azimuthal spatio‐temporal flow structure, is a monotonically decreasing function of the geometrical aspect ratio of the liquid bridge (defined as the ratio of length to diameter). For this function, a general correlation formula is found, which is in agreement with the previous experimental findings. The critical Marangoni number and the oscillation frequency are decreasing functions of the aspect ratio; however, the critical Marangoni number, based on the axial length of the bridge, does not change much with the aspect ratio. For each aspect ratio investigated, the onset of the instability from the axisymmetric steady state to the three‐dimensional oscillatory one is characterized by the appearance of a standing wave regime that exhibits, after a certain time, a second transition to a travelling wave regime. The standing wave regime is more stable for lower aspect ratios since it lasts for a long time. This behaviour is explained on the basis of the propagation velocity of the disturbances in the liquid phase. For this velocity, a general correlation law is found as a function of the aspect ratio and of the Marangoni number. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

基于POD-Galerkin降维方法的热毛细对流分岔分析

作为流动与传热相互耦合的非线性过程, 热毛细对流有着复杂的转捩过程, 探究流场和温度场随参数变化而发生的分岔现象, 是热毛细对流研究的一个重要课题. 基于本征正交分解的POD-Galerkin降维方法可以通过提取特征模态, 构建低维模型, 实现流场的快速计算. 数值分岔方法可以通过求解含参数动力系统的分岔方程, 直接计算稳定解和分岔点. 探究了将直接数值模拟方法、POD-Galerkin降维方法、数值分岔方法的优势结合, 以提高热毛细对流转捩过程分析效率的可行性. 利用直接数值模拟得到的流场和温度场数据, 构建了不同体积比下, 二维有限长液层热毛细对流的POD-Galerkin低维模型, 在低维模型上采用数值积分及数值分岔方法计算了分岔点, 得到了低维方程的分岔图. 在一定参数范围内, 在低维模型上模拟热毛细对流, 对雷诺数和体积比进行参数外推, 通过与直接数值模拟的结果对比, 验证了低维模型的准确性与鲁棒性. 说明了低维方程可以定性反映原高维系统的流动特性, 而定量方面, 由低维模型和直接数值模拟计算得到的周期解频率的相对误差大约为5%. 验证了利用POD-Galerkin降维方法研究热毛细对流的可行性.      

Thermocapillar and thermogravitatory waves in a convection problem

We study, from a numerical point of view, a thermal convection problem in a cylindrical annulus where a dynamic flow is imposed through a non-zero temperature gradient at the bottom. Experimentally, many interesting dynamical behaviours have been discovered in this system, which are controlled by heat related parameters and buoyant and thermocapillary instability mechanisms. By setting the Marangoni or the Rayleigh numbers equal to zero we explore the origin of different stationary, oscillatory, and codimension two stationary-oscillatory structures and their connection to either thermocapillary or thermogravitatory effects. We find that waves are possible in both cases if heat related parameters are conveniently tuned. PACS 47.27.Te, 02.60.Cb, 47.20.y