Numerical Investigation of Thermocapillary Convection in a Liquid Layer with Free Surface

Effects of Marangoni number, aspect ratio and gravity level on thermocapillary convection in a liquid layer is investigated numerically, in which the level set method is employed to capture free surface deformation. The computational results show that, with the increase of Marangoni number the free surface deformation is increased and it can lead to free surface rupture if the Marangoni number is large enough. The end walls has a damping effect on the free surface deformation, and as the aspect ratio (A =L/(0.5H)) decreases the deformability of free surface is reduced. The gravity can damp the free surface deformation, particularly as gravity level varies from 0.0001g ₀ to g ₀ the free surface deformability decreases steeply.     

The effect of a uniform magnetic field on the onset of steady Bénard-Marangoni convection in a layer of conducting fluid

In this paper we use a combination of analytical and numerical techniques to analyse the effect of a uniform vertical magnetic field on the onset of steady Bénard-Marangoni convection in a horizontal layer of quiescent, electrically conducting fluid subject to a uniform vertical temperature gradient. The critical values of the Rayleigh and Marangoni numbers for the onset of steady convection are calculated and the latter is found to be critically dependent on the non-dimensional Crispation and Bond numbers. The stability of the layer to long wavelength disturbances is analysed and the two different asymptotic limits of strong surface tension (small Crispation number) and strong magnetic field (large Chandrasekhar number) are investigated. In the latter case analytical results for the critical Rayleigh and Marangoni numbers are obtained and are found to be in excellent agreement with the results of numerical calculations. We conclude that the presence of the magnetic field always has a stabilising effect on the layer. Treating the Marangoni number as the critical parameter we show that if the free surface is non-deformable then any particular disturbance can be stabilised with a sufficiently strong magnetic field, but if the free surface is deformable and gravity waves are excluded then the layer is always unstable to infinitely long wavelength disturbances with or without a magnetic field. Including gravity has a stabilising effect on the long wavelength modes, but not all disturbances can be stabilised no matter now strong the magnetic field is.     

The onset of steady Bénard-Marangoni convection in a two-layer system of conducting fluid in the presence of a uniform magnetic field

The onset of steady Bénard-Marangoni convection in two horizontal liquid layers of electrically conducting immiscible fluids subjected to a uniform vertical magnetic field and temperature gradient is analysed by means of a combination of analytical and numerical techniques. The free surface can be either deformable or nondeformable and the interface between the fluids is always assumed to be flat. The effect of the lower layer on the critical values of Rayleigh, Marangoni and wave numbers for the onset of steady convection is investigated. When the free surface is nondeformable, the critical parameters for the onset of pure Marangoni convection are increased, whereas for the onset of pure Bénard convection they are decreased compared to the single-layer model. The results for a single-layer and for two-layers are qualitatively similar for Bénard-Marangoni convection when the free surface is deformable. All disturbances can be stabilized with sufficiently strong magnetic field when the free surface is nondeformable. If the free surface is allowed to deform and gravity waves are excluded, then the layers are always unstable to disturbances with sufficiently small wave number with magnetic field. Inclusion of gravity waves has a stabilizing effect on certain disturbances of small wave number in the presence of weak or moderate magnetic field.     

Numerical Simulation of Thermocapillary-Buoyancy Convection in Detached Solidification

In order to understand the characteristics of thermocapillary-buoyancy convection in detached solidification, we conducted a series of numerical simulations of thermocapillary flow of CdZnTe melt with different aspect ratio and gap width under gravity using the finite-difference method. The results indicate that, at a small Marangoni number, the buoyancy has a great effect on the temperature distribution; however, its influence on the maximum stream function value is so little that it can be neglected. When Marangoni number exceeds a threshold value, the steady flow converts into the unstable convection. In addition, the buoyancy makes the critical Marangoni numbers decrease by one order of magnitude comparing the results obtained in gravity and those in microgravity condition.     

Numerical Investigation of Bubble Induced Marangoni Convection: Some Aspects of Bubble Geometry

Thermocapillary or Marangoni convection is a surface tension driven flow that occurs when a gas–liquid or vapor–liquid interface is subjected to a temperature gradient. In the past, the contribution to local heat transfer arising from Marangoni convection has been overlooked as insignificant since under earth gravity it is overshadowed by buoyant convection. This study numerically investigates some aspects of bubble size and shape on local wall heat transfer resulting from Marangoni convection about individual bubbles on a heated wall immersed in a liquid silicone oil layer (Pr = 110) of depth 5 mm. It was found that increasing bubble volume causes an increase in the area over which Marangoni convection has affect. Heat transfer therefore increases with bubble size. Over the effective area, the surface averaged hot wall heat transfer is not affected greatly by bubble shape. The surface averaged heat transfer over the effective area on both the hot and cold walls is affected dramatically by bubble size, but the increase is more profound on the cold wall.     

The Influence of Gravity and Confinement on Marangoni Flow and Heat Transfer Around a Bubble in a Cavity: A Numerical Study

Marangoni thermocapillary convection and its contribution to heat transfer during boiling has been the subject of some debate in the open literature. Despite extensive research efforts there still remains insufficient quantitative information regarding the impact of thermocapillary flow on the heat transfer. As a result, this paper aims to present a numerical investigation of the heat transfer enhancement due to Marangoni thermocapillary convection under both earth gravity (1-g) and zero gravity (0-g) conditions. A hemispherical bubble of fixed shape is considered atop a heated top wall of a domain with variable height. The heat transfer enhancement is quantified for Marangoni numbers in the range of 100 ≤ Ma ≤ 3,000 for channel heights of 1.5 ≤ H/R_b?≤ 7.5 which, for the 1-g cases, correspond with a Raleigh number range of 51 ≤ Ra_H?≤ 6.5 × 10⁴. For the most confined cases the flow and heat transfer were found to be very similar for the 0-g and 1-g cases. Also, the 0-g test cases were found to be very sensitive to increasing domain height whereas the 1-g simulations were far less sensitive.     

Evaluation of some self-sustained capillary effects taking place in slag at the interface during desulphurization process

In the paper are presented and analyzed some specific problems of instability and of Marangoni convection in desulphurizing slags at 1873.15 K, due to the presence of sulphur, during liquid steel treatments. Starting from the quantity sulphide capacity, a limit of sulphur solubility in a homogeneous liquid slag is established. The thermodynamic effect of sulphur in the slag is evaluated using an enthalpy of interaction of sulphur containing the balance of the partial molar enthalpy of mixing for CaS and CaO. The sulphur effect in slag, on the enhancement of the mass transfer coefficient through the interface is evaluated based on the expression of the concentration coefficient of the surface tension related to the mole fraction, the solutal Marangoni number and of the sulphur mass transfer enhancement parameters. It is concluded that during desulphurization, self-sustained capillary effects are present in slags.     

Combined influence of throughflow and Soret effect on the onset of Marangoni convection

The onset of Marangoni convection with throughflow and the Soret effect in a top-free and bottom-rigid horizontal fluid layer is studied using the normal mode method for different types of thermal and solutal boundary combinations. The bottom surface is either conducting or insulating to temperature and solute concentration perturbations. The resulting eigenvalue problem is solved exactly by assuming that stationary convection is exhibited at the neutral state. It is found that the destabilizing behavior of a small amount of throughflow described by Nield (J Fluid Mech 185:353–360, 1987) becomes more significant in the presence of Soret effect for some boundary combinations. The results are consistent with the existing results in the literature.     

Effect of Surface Evaporation on Steady Thermocapillary Convection in an Annular Pool

In order to understand the effect of surface evaporation on thermocapillary convection in an annular pool, a series of numerical simulation on thermocapillary convection of the fluids with Prandtl number from 0.01 to 50 in the pure vapor environment were carried out. The results show that thermocapillary convection is always coupled with the evaporation process on the free surface. With the increase of evaporation Biot number, the surface temperature decreases, and the evaporation mass flux near the hot wall increases obviously. However, near the cold wall, the evaporation mass flux increases first, and then decreases. When Marangoni number is small, the total evaporation mass rate at free surface increases with the increase of evaporation Biot number; when Marangoni number is larger, it increases first and then approaches a constant value. The aspect ratio of the annular pool has a positive influence on the thermocapillary convection strength and the total evaporation mass rate. With the increase of Prandtl number, the surface temperature rises gradually and the evaporative mass flux increases, and the thermocapillary convection cell moves gradually toward the outer wall and the free surface. This effect decreases with the increase of evaporation Biot number When evaporation Biot number is smaller, the total evaporation mass rate increases with the Prandtl number; when Biot number is larger, Prandtl number has little impact on the total evaporation mass rate.     

Modelling of transition mechanisms from steady to oscillatory Marangoni flows in a liquid bridge configuration

In this paper, the transition mechanism from steady to oscillatory flow of Marangoni convection is investigated. Two-dimensional simulations for high Pr number fluids with several liquid bridge sizes were conducted in order to clarify the effect of temperature distribution on free surface upon the transition phenomena. The dependency of liquid bridge size on the onset of oscillatory flow was also evaluated. The results show that the formation of velocity distribution on the free surface is related to liquid bridge size and that it has a great influence on the onset of oscillation. From these results, a basic model for understanding the transition mechanism of Marangoni convection is proposed. In this transition model, the temperature distribution on the free surface was evaluated over a wide range of Pr numbers. A useful dimensionless parameter which indicates the onset of oscillation, the effective Marangoni number, is also proposed and evaluated.     

Thermo-capillary and residual natural convection in an orbiting spherical liquid system

The influence of a given temperature distribution applied at the surface of a spherical liquid system in a circular orbit has been determined analytically. Three basic convectional flows inside the liquid have been investigated: Marangoni convection, residual gravity natural convection and the natural convection due to self-attraction. For small drops thermocapillary convection is dominant, while for large liquid spheres the convection due to self-attraction is predominant for orbits around the sun, while for orbits of smaller diameter, as for instance around the earth, the natural convection due to the residual gravity is dominant for large spherical drops.     

Numerical Simulation of Thermocapillary Convection in Detached Solidification under Microgravity

In order to investigate the fundamental characteristics of thermocapillary convection in the detached solidification under microgravity, the finite-difference method was adopted to perform the numerical simulations. The results show that the flow of molten liquid is steady with the low Marangoni number and it only exists in the vicinity of the free surface. Moreover, with the Marangoni number increasing, the flow is expanded toward the inner part of molten liquid gradually, and at the same time the flow velocity on the free surface increases. However, the flow is unstable when the Marangoni number exceeds the critical Marangoni number.     

Report on Microgravity Experiments of Dynamic Surface Deformation Effects on Marangoni Instability in High-Prandtl-Number Liquid Bridges

This paper reports an overview and some important results of microgravity experiments called Dynamic Surf, which have been conducted on board the International Space Station from 2013 to 2016. The present project mainly focuses on the relations between the Marangoni instability in a high-Prandtl-number (Pr=?67 and 112) liquid bridge and the dynamic free surface deformation (DSD) as well as the interfacial heat transfer. The dynamic free surface deformations of large-scale liquid bridges (say, for diameters greater than 10 mm) are measured with good accuracy by an optical imaging technique. It is found that there are two causes of the dynamic free surface deformation in the present study: the first is the time-dependent flow behavior inside the liquid bridge due to the Marangoni instability, and the second is the external disturbance due to the residual acceleration of gravity, i.e., g-jitter. The axial distributions of DSD along the free surface are measured for several conditions. The critical parameters for the onset of oscillatory Marangoni convection are also measured for various aspect ratios (i.e., relative height to the diameter) of the liquid bridge and various thermal boundary conditions. The characteristics of DSD and the onset conditions of instability are discussed in this paper.     

Hydrothermal wave in a shallow liquid layer

The oscillatory thermocapillary convection and hydrothermal wave in a shallow liquid layer, where a temperature difference is applied between two parallel sidewalls, have been numerically investigated in a two-dimensional model. The oscillatory thermocapillary convection and hydrothermal wave appear if the Marangoni number is larger than a critical value. The critical phase speed and critical wave number of the hydrothermal wave agree with the ones given analytically by Smith and Davis in the microgravity environment, and it travels in the direction opposed to the surface flow. Another wave traveled downstream in addition to the hydrothermal wave traveled upstream was observed in the case of earth gravity condition.     

微重力条件下熔体中对流过程的数值模拟

微重力条件下生长优质晶体遇到的最大问题是要控制晶体生长的条件，抑制由于重大的减弱而引起的熔体中的热毛细时流。但是，用实验来解决这些问题费用高，周期也长，而且有时完全用实验来模拟也是很困难的。用数值计算方法来模拟微重力条件下熔体中的对流过程是空间晶体生长研究的一个重要的方向，计算结果对控制空间生长晶体和抑制熔体中的对流有指导意义。对微重力条件下熔体中对流发生、发展的过程进行了数值研究。以有限差分法研究了沿上表面为自由表面的水平区域不同边界条件下的熔体中的对流过程。     

Long-Wavelength Marangoni Convection in a Liquid Layer with Insoluble Surfactant: Linear Theory

The subject of this paper is the long-wave Marangoni convection in a horizontal liquid layer with insoluble surfactant absorbed on the free surface. The surfactant is convected by interfacial velocity field and diffuses over the interface but not into the bulk of the fluid. The layer is subjected to a transverse temperature gradient. The buoyancy effects are negligible as compared to the Marangoni forces. We consider both cases of flat nondeformable and deformable surface. The linear stability analysis of this system is performed. It is shown that in both cases of the upper surface monotonic and oscillatory modes exist. Convection thresholds are determined and the critical Marangoni numbers for monotonic as well as for oscillatory mode are obtained. It is shown that the monotonic long-wave instability is more dangerous than oscillatory one only for small elasticity numbers, if the Lewis number is small.     

Casting process for hypermonotectic alloys under terrestrial conditions

Bearing materials are generally heterogeneous materials, containing hard as well as soft, phases. Hypermonotectic AlPb and AlBi alloys, especially, are considered as exceptionally qualified bearing materials if they also contain additional hard phases to decrease wear. Based on the considerable differences in the density of the decomposed fluid phases at high temperatures and the high velocity of separation, such alloys cannot, to date, be manufactured under terrestrial conditions. The results of microgravity experiments for the manufacturing of suitable sample material with a fine phase dispersion of the monotectic phases were rather disappointing. The cause of the rapid phase separation and local enrichment under microgravity conditions was found to be the Marangoni convection, the effects of which, to date, have been underestimated. The results of these space experiments are now utilized in a terrestrial casting process, whereby a comparatively high Marangoni convection is superposed in the opposite direction to the sedimentation action of gravity, thereby partially compensating the effects of gravity. Thus, cast strips of AlSiPb and AlSiBi alloys could be manufactured, the lead and bismuth phases being present in a characteristic fine dispersion overthe length of the cast strips. The first tribological laboratory tests give an indication of the excellent suitability of such advanced bearing material for the future.     

Effect of rotation on surface tension driven flow during protein crystallization

The effect of rotation on surface tension gradient driven flow, also known as Marangoni convective flow, during protein crystallization is modeled and studied computationally under microgravity conditions, where the surface tension gradient force is the main significant driving force. The main parameters are the solutal Marangoni number M_c, representing the surface tension gradient force and the Taylor number Ta representing the rotational effect. The numerical computations for various values of the parameters and low gravity levels indicated nontrivial competing effects, due to surface tension gradient, centrifugal and Coriolis forces on the flow adjacent to the protein crystal interface and the associated solute flux. In particular, for given values of M_c, certain values of Ta were detected where the Sherwood number (Sh), representing the convective solute flux, and the convective flow effects are noticeably reduced. These results can provide conditions under which convective flow transport during the protein crystallization approaches the diffusion limited transport, which is desirable for the production of higher quality protein crystals.     

Effect of throughflow on Marangoni convection in micropolar fluids

Summary The effect of throughflow on the onset of Marangoni convection in a horizontal layer of micropolar fluid flow bounded below by a rigid isothermal surface and above by a nondeformable free adiabatic surface, for marginal state, is studied. The determination of the critical Marangoni number entails solving the eigenvalue problem numerically for which the Single-term Galerkin method is employed.     

The onset of steady Marangoni convection in a spherical geometry

In this paper we use a combination of analytical and numerical techniques to analyse the onset of steady Marangoni convection in a spherical shell of fluid with an outer free surface surrounding a rigid sphere. In so doing we correct the formulation of the problem and the results presented by Cloot & Lebon (Microgravity sci. technol. 3 (1) 1990: 44–46). We find that if the free surface of the layer is non-deformable then the layer is always stable when heated from the outside and is unstable when heated from the inside if the magnitude of the (positive) non-dimensional Marangoni number is sufficiently large. If the free surface of the layer is deformable then the layer is always unstable when heated from the inside. It is stable when heated from the outside if C _r < r ₂/4, but if C _r > r ₂/4 then it is unstable if the magnitude of the (negative) Marangoni number is sufficiently large, where C _r is the non-dimensional Crispation number and r ₂ the non-dimensionalradius of the undisturbed outer free surface of the fluid.