Influence of free surface curvature of a liquid layer on the critical Marangoni convection

The Pearson instability was suggested to discuss the onset of Marangoni convection in a liquid layer of large Prandtl number under an applied temperature difference perpendicular to the free surface in the microgravity environment. In this case, the temperature distribution on the curved free surface is non-uniform, and the thermocapillary convection is induced and coupled with the Marangoni convection. In the present paper the effect of volume ratio of the liquid layer on the critical Marangoni convection and the corresponding spatial variation of the convection structure in zero-gravity condition were numerically investigated by two-dimensional model.     

Linear Stability Analysis of Thermocapillary Flow in Rotating Shallow PoolsHeated from Inner Wall

Thermocapillary flow of silicon melt(Pr=0.011)in shallow annular pool heated from inner wall was simulated at the dimensionless rotation ratewranging from 0 to 7000.The effect of pool rotation on the stability of the thermocapillary flow was investigated.The steady axisymmetric basic state was solved by using the spectral element method;the critical stability parameters were determined by linear stability analysis;the mechanism of the flow instability was explored by the analysis of energy balance.A stability diagram,exhibiting the variation of the critical Marangoni number versus the dimensionless rotation ratewwas presented.The results reveal that only one Hopf bifurcation point appeared in the intervals ofω<3020 andω>3965,and the corresponding instability was caused by the shear energy,which was provided by the thermocapillary force and pool rotation,respectively.In addition,the competition between thermocapillary force and pool rotation leads to three Hopf bifurcation points in the range of 3020<ω<3965 with the increase of Marangoni number.     

蒸发液滴内部非稳态流动的数值计算

在气液两相流VOF(volume of fluid,VOF)模型的基础上耦合CSF(continuum surface force,CSF)表面张力模型,建立了高温平板上的铺展液滴与高温空气中悬浮液滴蒸发过程中内部非稳态流动模型,对液滴蒸发过程中内部非稳态流动进行了研究。基于相变理论,采用用户自定义函数将流体相变模型加入非稳态流动模型中进行耦合计算,获得了高温平板上的铺展液滴与高温空气中悬浮液滴蒸发过程中的内部流动及变化过程。液滴蒸发过程中非稳态内部流动由液滴表面的温度梯度引发,Marangoni流动在液滴内部形成的时间非常短,流体从液滴表面高温区域流向低温区域。计算结果表明:高温平板上随着液滴蒸发的进行,液滴内部一直保持两个对称的涡流,Marangoni流动比较稳定;高温空气环境中随着液滴蒸发的进行,液滴内部四个涡流逐渐转变成两个对称的涡流;液滴内部温度分布因Marangoni流动加强传热而变得均匀,同时由于温度分布变得均匀,Marangoni流动被削弱。     

Effects of controller and cubic temperature profile on onset of Bénard–Marangoni convection in ferrofluid

This study investigated the instability of Bénard–Marangoni convection in a horizontal layer of ferrofluid under the influence of a linear feedback control and cubic temperature profile. A linear stability analysis was performed. A single-term Galerkin technique was used to obtain the critical Marangoni number and critical Rayleigh number. The possibility of delaying the onset of convection by the application of linear feedback control is demonstrated.     

The influence of preferential diffusion and stretch on the burning intensity of a curved flame front with fuel spray

In our most recent paper on Bunsen spray flames, only a completely prevaporized mode of a normal Bunsen flame was considered; inverted Bunsen flame and droplet size effects had not been examined yet. In the present study, we consider two flame structures: normal and inverted Bunsen flames, and two spray modes: completely and partially prevaporized burning, by the method of large activation energy asymptotics. In this way, a complete parametric study of flame tip intensification or extinction (opening) can be conducted. Four parameters are used in the analysis. The first two are the droplet size and amount of liquid-fuel loading, which indicate internal heat loss for a rich spray but heat gain for a lean spray. The other two are the stretch and Lewis number (Le). Stretch is negative for a normal Bunsen flame but positive for an inverted Bunsen flame. Stretch strengthens (or weakens) the burning intensity of the Le>1 (or Le<1) normal Bunsen flame but decreases (or increases) the burning intensity of the Le>1 (or Le<1) inverted Bunsen flame. Burning intensity of the flame tip intensifies (or weakens) when the lean (or rich) spray has a smaller droplet size or a larger amount of liquid loading. For lean and rich ethanol-spray normal Bunsen flames with Le>1 or a rich methanol-spray inverted Bunsen flame with Le<1, closed-tip solutions are obtained. Conversely, stretch weakens the burning intensities of lean and rich ethanol-spray inverted Bunsen flames with Le>1, or rich methanol-spray normal Bunsen flames with Le<1, eventually leading to tip opening. Moreover, the opening becomes wider (or narrower) as the droplet size decreases or liquid loading increases for the rich (or lean) sprays. Note that for lean ethanol-spray normal (or inverted) Bunsen flame with Le>1, if liquid loading is large enough and droplet size is sufficiently small, there exists flame transition from normal (or inverted) Bunsen through planar to inverted cone (or normal Bunsen) flame. Finally, the critical value of droplet size, at which there exists a planar flame rather than a normal (or an inverted) Bunsen flame, increases with increasing liquid loading.     

液层厚度比对环形腔内双层液体的热毛细对流稳定性的影响

为了了解微重力条件下、水平温度梯度作用时,上部为固壁的环形腔内双层流体系统中液层厚度比对流动稳定性的影响,采用隐式重启Arnoldi方法(IRAM)对环形池内5cSt硅油/HT-70双层流体的热对流过程进行了线性稳定性分析,获得了不同液层厚度比下的临界Marangoni数、临界波数、临界相速度,并通过计算特征向量,得到了临界Marangoni数附近液-液界面的热流体波形态。     

Thermal and solutal effects on convection inside a polymer solution droplet on a substrate

The convective flow inside polymer solution droplets drying on a lyophobic substrate is numerically studied. The evaporating droplet is presumed as a hemisphere shrinking with time at the constant contact angle. The thermal and solutal effects are simultaneously considered in the computation. The thermal Marangoni convection is induced due to the quick thermal diffusion, and this convection transports the solute resulting in the solutal Marangoni flow. The solutal dependence corresponds to our previous experimental work, but the flow pattern does not. Consideration of the pseudo evaporation rate distribution depending on the contact angle yields to the flow pattern correspondence.     

环形池内具有自由表面的双层流体热毛细对流

采用数值模拟方法研究微重力条件下环形双层液体内存在水平温度梯度时的热毛细对流及其稳定性。流体为5cSt硅油/HT-70,外壁被加热、内壁被冷却,下固壁和上自由表面均绝热。结果表明:当Ma较小时,流动为稳定的轴对称流动;随着Ma和深宽比的增大,流动加强,等温线发生强烈的非线性变形;当Ma超过临界值后,流动转化为非稳定的多胞流动;随着Ma和深宽比的增大,速度振荡增大并向热壁方向运动,多胞流动结构占据区域拓展;流动转变的临界Ma随着深宽比的增大而减小。     

Stabilization of thermocapillary instability in a fluid layer with internal heat source

The onset of steady thermocapillary (surface-tension-driven Marangoni) instability in a horizontal fluid layer in the presence of a uniform heat generation is considered theoretically using stability theory. The fluid layer, heated from below, is bounded above by a deformable free upper surface and below by a rigid plane boundary. The stability of the fluid layer is investigated and subjected to a feedback control on the temperatures of the boundaries. An exact solution for marginal stability owing to an exchange of stabilities has been obtained. Long wavelength instability may coexist with a finite wavelength instability for certain sets of parameter values, often referred to as frontier points.     

Effect of Marangoni number on thermocapillary convection and free-surface deformation in liquid bridges

Floating zone technique is a crucible-free process for growth of high quality single crystals. Unstable thermocapillary convection is a typical phenomenon during the process under microgravity. Therefore, it is very important to investigate the instability of thermocapillary convection in liquid bridges with deformable free-surface under microgravity. In this works, the Volume of Fluid(VOF) method is employed to track the free-surface movement. The results are presented as the behavior of flow structure and temperature distribution of the molten zone. The impact of Marangoni number(Ma) is also investigated on free-surface deformation as well as the instability of thermocapillary convection. The free-surface exhibits a noticeable axisymmetric(but it is non-centrosymmetric) and elliptical shape along the circumferential direction. This specific surface shape presents a typical narrow ‘neck-shaped' structure with convex at two ends of the zone and concave at the mid-plane along the axial direction. At both θ = 0° and θ = 90°, the deformation ratio ξ increases rapidly with Ma at first, and then increases slowly. Moreover, the hydrothermal wave number m and the instability of thermocapillary convection increase with Ma.     

High pressure droplet vaporization; effects of liquid-phase gas solubility

Numerical results are presented for an n-hexane droplet initially at 300 K evaporating into nitrogen, for ambient pressures, P_x, 1–100 atm and ambient temperatures, T_x, 500–1500 K. At low P_x (<30 atm), droplet lifetimes predicted with or without liquid-phase gas solubility are very close. At high P_x, the model neglecting solubility either underestimates the droplet lifetime (low T_x) or breaks down by failing to predict vapor-liquid equilibria (high T_x). At high enough P_x, heat-up is extremely important throughout the entire droplet lifetime. In a fuel rich environment, relatively low T_x, and high T_x, substantial condensation occurs before the onset of vaporization.     

Topological structure evolvement of flow and temperature fields in deformable drop Marangoni migration in microgravity

Using the level-set method and the continuum interface model, the axisymmetric thermocapillary migration of a deformable liquid drop immerged in an immiscible bulk liquid with a temperature gradient is simulated numerically with constant material properties of the two phases. Steady terminal state of the motion can always be reached. The dimensionless terminal migration velocity decreases monotonously with the increase of the Marangoni number. Good agreements with space experimental data and most of previous numerical studies in the literature are evident. The terminal topological structure of flow field, in which a recirculation identical to Hill’s vortex exists inside the drop, does not change with the Marangoni number. Only slight movement of the location of vortex center can be observed. On the contrary, bifurcations of the terminal topological structure of temperature field occur twice with increasing Marangoni number. At first, the uniform and straight layer-type structure of temperature field at infinitesimal Reynolds and Marangoni numbers wraps inside of the drop due to convective transport of heat as the Marangoni number increases, resulting in the emergence of an onion-type local cooler zone around the center of the drop beyond a lower critical Marangoni number. Expanding of this zone, particularly in the transverse direction, with the increasing of the Marangoni number leads to a cap- or even shell-type structure. The coldest point within the liquid drop locates on the axis. There is a middle critical Marangoni number, beyond which the coldest point will jump from the rear stagnation into the drop, though the topological structure of the temperature field does not change. The second bifurcation occurs at an upper critical Marangoni number, where the shell-type cooler zone inside drops ruptures from the central point and then a torus-type one emerges. The coldest point departs from the axis, and the so-called “cold-eye” appears in the meridian. It is also found that the inner and outer thermal boundary layers along the interface may exist both inside and outside the drop if Ma > 70. But the thickness decreases with the increasing Marangoni number more slowly than the prediction of potential flow at large Marangoni and Reynolds numbers. A velocity shear layer outside the drop is also introduced formally, of which modality may be affected by the convective transports of heat and/or momentum.     

Heat convection within evaporating droplets in strong aerodynamic interactions

The temperature field within evaporating ethanol droplets is investigated, relying on the two-color laser induced fluorescence (LIF) measurement technique and on a Direct Numerical Simulation (DNS). The configuration studied corresponds to a monodisperse droplet stream in a diffusion flame sustained by the droplet vapor. An experimental probe volume, small compared to the droplet size, is used to characterize the temperature field within the droplets, whereas DNS takes into account key aspects of the droplet heating and evaporation such as the non-uniform and transient stress, and the mass and heat transfer coefficients at the droplet surface. These investigations reveal that the frictional stresses are strongly reduced due to the small spacing between the droplets. They also show that the Marangoni effect has a significant influence on the internal motion and hence on the internal temperature field.     

Predicting Marangoni convection caused by transient gas diffusion in liquids

The onset of convection driven by surface tension during gas diffusion in a liquid is investigated. Gas diffusion at the gas-liquid interface results in the variation of concentration of the solute that may cause an increase in surface tension leading to Marangoni convection. The onset of convection for unsteady-state gas desorption can be predicted from the maximum transient Ma_t, which is here derived by analogy with its equivalent in thermal convection. It is a function of the transient Biot number (Bi_D) for interfacial gas diffusion, which depends strongly on the state of vapour-liquid equilibrium at the interface. The transient Marangoni numbers, critical times for stable mass diffusion and the critical sizes of convection cells have been formulated. The desorption of ethyl-ether from chloro-benzene in L.M. Blair’s [The onset of cellular convection in a fluid layer with time-dependent density gradients, PhD thesis, University of Illinois, Urbana, 1968] experiments is liquid phase-controlled, hence, the highly soluble system is characterized by Bi_D = 0. Therefore, his experiments that were initiated with a step-change in pressure cannot be analyzed by a step-function boundary that is characterized by Bi_D = ∞. The surface concentration may change very slowly, it has been approximated to be about 0.1% of the initial pressure change at the point of onset of convection. The average critical Marangoni number for this condition was estimated to be 53.3, which is fairly close to the theoretical value of 67 for an interface with a Biot number of 0. Therefore, the high value of 3100 calculated by I.F. Davenport and C.J. King [The initiation of natural convection caused by time-dependent profiles, Lawrence Berkeley Lab, Report NBR LBL-600, 1972] is wrong, who wrongly assumed a fixed surface-concentration boundary that is applicable only to a sparingly soluble solute. The critical sizes of convection cells predicted by theory are generally less than 1 mm for reported critical times of less than 20 s, they would be difficult to measure.     

The effects of Prandtl number on wavy weld boundary

The effects of Prandtl number on two-dimensional thermocapillary convection and molten pool shape induced by negative surface tension coefficient in welding or melting with a time-dependent and distributed incident flux are numerically predicted in this study. This work is also applicable for predicting the quasi-steady three-dimensional thermocapillary convection. In this model, the time-dependent incident flux is specified as a function of scanning speed and energy distribution parameter. The computed flow patterns and molten pool shapes under the flat free surface are found to have distinct regions for different Marangoni and Prandtl numbers. Prandtl number indicates the thickness ratio between momentum and thermal boundary layers, whereas Marangoni number with negative surface tension coefficient induces an outward surface flow. Rather than the enhanced pool depth resulted from melting from an induced vortex cell near the bottom in the case of Prandtl number much less than unity, the molten pool shape for Prandtl number much greater than unity can produce a thin and narrow edge. The variations of Peclet number and dimensionless beam power with flow and temperature fields and fusion zone shapes are similar. The dimensionless peak surface speed versus product of Marangoni and Prandtl number, which involve predicted peak speed and temperature and molten pool width on the surface, agree with scale analysis and experimental data provided in the literature.     

Mixed Thermocapillary and Forced Convection Heat Transfer Around a Hemispherical Bubble in a Miniature Channel—A 3D Numerical Study

It has been established that for certain conditions, such as microgravity boiling, thermocapillary Marangoni flow has associated with it a significant enhancement of heat transfer. Typically, this phenomenon was investigated for the idealized case of an isolated and stationary bubble resting atop a heated solid that is immersed in a semi-infinite quiescent fluid or within a two-dimensional cavity. This article presents a three-dimensional numerical study that investigates the influence of thermal Marangoni convection on the fluid dynamics and heat transfer around a bubble during laminar flow of water in a minichannel. This mixed thermocapillary and forced convection problem is investigated for channel liquid inlet velocity of 0.01 m/s to 0.03 m/s and Marangoni numbers in the range of 10 to 300 under microgravity conditions. Three-dimensional effects become particularly important on the side and rear regions of the bubble. The thermocapillary forces accelerate the flow along almost the entire bubble interface. The hot core fluid from the heated bottom wall region is forced inward and propelled upward into the thermocapillary jet above the bubble. It can be quantified that the influence of thermocapillary flow on heat transfer enhancement shows an average increase by 40% at the downstream of the bubble and by 60% at the front and rear regions. This heat transfer enhancement depends mainly on the temperature differential as the driving potential for thermocapillary flow and bulk liquid velocity.     

Impact of Marangoni instabilities on the fluid dynamic behaviour of organic droplets

Marangoni instabilities in dispersed liquid/liquid systems occur if the local solute concentration varies over the interface. The additional shear stress at the interface between a droplet and an ambient phase generates complex convection patterns which increase temporarily the global drag coefficient of the drop and thus retard the drop rise velocity. When Marangoni effects get weaker, the shear forces decrease and the drop reaccelerates. In the present experimental study, the transient drop rise velocity has been intensely investigated in the system toluene_(d)/acetone_(s)/water_(c) for different initial solute concentrations and different drop diameters. Both mass transfer directions have been considered. The reacceleration time as an indicator of the end of Marangoni dominance can be expressed as a function of drop diameter and initial solute concentration.     

Effects of the liquid polarity and the wall slip on the heat and mass transport characteristics of the micro-scale evaporating transition film

A mathematical model is developed to describe the micro-/nano-scale fluid flow and heat/mass transfer phenomena in an evaporating extended meniscus, focusing on the transition film region under non-isothermal interfacial conditions. The model incorporates polarity contributions to the working fluid field, a slip boundary condition on the solid wall, and thermocapillary stresses at the liquid-vapor interface. Two different disjoining pressure models, one polar and one non-polar, are considered for water as the working fluid so that the effect of polar interactions between the working fluid and solid surface can be exclusively examined on heat and mass transfer from the thin film. The polar effect is examined for the thin film established in a 20-μm diameter capillary pore. The effect of the slip boundary condition is separately examined for the thin film developed in a two-dimensional 20-μm slotted pore. The analytical results show that for a polar liquid, the transition region of the evaporating meniscus is longer than that of a non-polar liquid. In addition, the strong polar attraction with the solid wall acts to lower the evaporative heat transfer flux. The slip boundary condition, on the other hand, increases evaporative heat and mass flux and lowers the liquid pressure gradients and viscous drag at the wall. The slip effect shows a more pronounced enhancement as superheat increases. Another thing to note is that the slip effect of elongating the transition region can counteract the thermocapillary action of reducing the region and a potential delay of thermocapillary driven instability onset may be anticipated.     

Marangoni-Bénard instabilities under non-steady conditions. Experimental and theoretical results

The onset of Marangoni-Bénard instability under microgravity conditions is studied experimentally using a sounding rocket (TEXUS 21) launched in the microgravity program developed by the European Space Agency. Due to the short available time, the experiment is conducted under unsteady conditions, i.e. the temperature gradient inside the liquid phase is not constant. For such a non-linear temperature profile, we compare the experimental and the theoretical value of the critical non-oscillatory Marangoni number. This simple analysis provides an excellent agreement in contradistinction with earlier results obtained during Apollo 14 and 17 flights.     

Thermogravitational and thermocapillary convection heat transfer in concentric and eccentric horizontal,cylindrical annuli filled with two immiscible fluids

Laminar thermogravitational convection in concentric and eccentric horizontal, cylindrical annuli, filled with two immiscible fluids (water/air, water/silicon oil 10, water/silicon oil 100, Freon 113/ water) is studied numerically. Streamline and temperature distributions, local and average equivalent thermal conductivities are obtained over a wide range of Rayleigh number. The influence of thermocapillary convection (Marangoni convection) is similarly demonstrated for the water/air system in an annular enclosure.