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.     

Heat Transfer Through the Interface and Flow Regimes in Liquid Bridge Subjected to Co-Axial Gas Flow

We present results of an extensive numerical study on the thermocapillary (Marangoni) convection and a heat transfer through the interface in a liquid bridge of Pr?=?68. The geometry of the physical problem is a cylindrical and non-deformable liquid bridge concentrically surrounded by an annular gas channel under conditions of zero gravity. The gas flow is co- or counter-directed with respect to the Marangoni flow. The forced gas flow along the interface provides two actions: via shear stresses and heat exchange. Usually the cooling of the interface enhances the flow while the heating slows down. This general trend may not hold when shear and thermocapillary stresses are comparable. The results show that when gas enters from the cold side the heat transfer through the interface is considerably larger than that when gas enters from the hot side.     

Thermal Convection in an Annular Two-Layer System Under Combined Action of Buoyancy and Thermocapillary Forces

A set of two-dimensional numerical simulations of thermal convection of two immiscible liquid layers with a non-deformable interface was carried out in an annular cavity with the outer heated cylinder and the inner cooled cylinder using the finite-volume method. Bottom and top surfaces were bounded by two rigid and heat-insulated walls. The results show that the conversion of the thermal convection depends on the Marangoni number, the aspect ratio and the curvature of the annular cavity. For a sufficiently small Marangoni number, the flow is steady and there is one cell in each fluid layer. When the Marangoni number exceeds the critical value, the convective flow becomes instable and generates an unsteady multi-cell structure. In the case of the closed annular cavity, the value of the critical Marangoni number is higher than that in the infinite layer, and decreases with the increase of the aspect ratio.     

Oscillatory marangoni convection around bubbles and drops in heterogenous solutions of surfactants

Experimental investigation was performed to study the concentration convection around stationary gas bubbles and insoluble drops in a thin liquid layer placed in a vertical Hele-Shaw cell. The bubbles or drops, squeezed between the two parallel cell walls, took the shape of short cylinders with free lateral surfaces. The cell was filled in with an aqueous solution of a surface-tension active fluid (surfactant) with vertically stratified concentration. A special wire frame prevented bubbles from rising up under the buoyancy force, thus modelling the microgravity conditions. A convective motion in the mixture develops at the bubble or drop interface, due to the solutocapillary Marangoni forces. Owing to a small thickness of the liquid layer (∼1mm), the arising flows and surfactant concentration distributions are nearly two-dimensional so that it is possible to investigate their structure and evolution by interferometric technique. The experiments revealed the development of oscillatory convection around the drop interface, which was similar to that observed in bubble tests. The period and duration of oscillations were determined in relation to time, surfactant concentration gradient and concentration Marangoni number. The analysis of bubble and drop behavior showed that the existence of self-oscillatory modes is related to the specific interaction between the solutocapillary and soluto-gravitational mechanisms of motion.     

Stability of Liquid Bridges Between Coaxial Equidimensional Disks to Axisymmetric Finite Perturbations: A Review

This paper reviews the dynamics of breaking or oscillating axisymmetric liquid bridges, and estimates of the energy which is needed to break a liquid bridge. We consider a liquid bridge spanning two coaxial equal disks with sharp edges and held by surface-tension forces. The liquid volume is assumed to be conserved under perturbations, and the contact lines are pinned to the disk edges. The perturbations are finite and axisymmetric. An analysis is based on the one-dimensional models previously used in capillary jet theory and last several decades for study a liquid bridge dynamics. According to the scientific project JEREMI (Japanese and European Research Experiment on Marangoni Instabilities), the first stage of the space experiment on ISS will involve an isothermal liquid bridge with a gas blowing parallel to the axial direction of the bridge. The geometry corresponds to a cylindrical volume liquid bridge coaxially placed into an outer cylinder with solid walls. The gas enters the annular duct bounded by the outer cylinder and the internal system consisting of supporting vertical rods and the liquid bridge. Considering that the bridge is small (the rod’s radii are 3 mm) and the gas velocity is typically (0.25 ÷ 0.37) m/s, the perturbations cannot be considered small. Thus, one may assume that the amplitude of the liquid bridge perturbations is sufficiently large that departures from linearity must be considered.     

Linear-stability Analysis of Thermocapillary Flow in an Annular Two-layer System with Upper Rigid Wall

The linear-stability analysis of thermocapillary flow in the annular immiscible two-layer system of 5cSt silicone oil and HT-70 with a radial temperature gradient was carried out. The annular two-layer system is heated at the outer cylindrical wall and cooled at the inner wall, the bottom and top surfaces are bounded by two rigid and heat-insulated walls. The influences of the liquid layer depth and radius ratio between the cold inner wall and the hot outer wall on stability are thoroughly investigated. The critical Marangoni number, critical wave number and critical phase velocity are obtained. In addition, the mode of bifurcation for the hydrothermal wave is predicted at different liquid layer depth, and the temperature disturbance pattern of hydrothermal wave at interface is also exhibited.     

Convective Auto-Oscillations Near a Drop–Liquid Interface in a Horizontal Rectangular Channel

The concentration convection in an isothermal liquid near a drop (or an air bubble) clamped between the vertical walls of a horizontal channel is studied numerically within the framework of two simple mathematical models: with and without the surface phase at the drop–liquid interface formed by adsorption/desorption process. The interaction between the buoyancy and the Marangoni convective flows is responsible for the onset of auto-oscillation regime. Such oscillations have been experimentally investigated in other works. In our numeric experiments, more than 20 outbursts of the Marangoni convection were observed. The surfactant distributions obtained numerically at different oscillation phases agree well the experimen tal data.     

Accumulation of solid particles in convective flows

Accumulation of solid particles suspended in unsteady convective flows is under theoretical investigation. The principal goal is to understand and interpret recent experiments by D. Schwabe [1,2]. Providing that volume particle concentration, nonisothermality, and relative size of particle are small, an effective single-fluid theoretical model is developed. The peculiarity of the obtained model is taking into account the distinction between fluid and particle inertia. This model is further applied to study particle accumulation in different flow setups: in a model oscillatory flow in a canal heated from below and subjected to the modulated gravity and in the Marangoni flow in a half-zone under microgravity conditions. These problems are investigated numerically by means of finite difference technique. We demonstrate, that the developed theoretical model properly describes generic features of particle accumulation in unsteady flows. Particularly, heavy particles tend to leave the centers of vortices, where the flow vorticity is maximal, and accumulate at their periphery. From numerical simulations in a floating zone, we try to clarify particle dynamics in Schwabe’s setup.     

Influence of interfacial heat exchange on the flow organization in liquid bridge

The appearance and development of thermoconvective oscillatory flows in a cylindrical column with a free lateral surface (so-called liquid bridge) filled with 5 cSt silicone oil are investigated experimentally. The experiments were carried out under terrestrial conditions for a wide range of volumes of liquid bridges. Heat transfer from the interface changed the threshold of socillatory instabilities. The sensitivity of thermo-convective flows to the interfacial heat exchange was found to be strongly depending on the liquid bridge volume. Slender liquid bridges (under filled zone with respect to the straight cylinder) are rather stable to external disturbances. On the contrary, fat liquid bridges are extremely sensitive to the thermal environment in the gas phase.     

Finite element simulation of combined buoyancy and thermocapillary driven convection in open cavities

Thermocapillary-induced and buoyancy-driven convective flows that commonly occur in crystal growth are numerically simulated using Galerkin finite element method. The physical domain comprises of a open cavity with aspect ratio one and differentially heated vertical walls. The top gas–melt interface is free to deform subject to 90° contact angle boundary conditions at the two vertical walls. The unsteady two-dimensional Navier–Stokes equations are discretized in time using Chorin-type splitting scheme and pressure is determined from the Poisson's equation. The free surface is taken to be resting on vertical spines and its evolution in time is determined from the kinematic free surface equation. The governing equations for heat and momentum are solved in the Arbitrary Lagrangian Eulerian frame of reference to handle the moving boundary. The influence of Grashof number, Marangoni number, Bond number, Ohnesorge number and Prandtl number on the flow field and heat transfer is investigated.     

Effect of Welding Parameters on GTA Weld Shape for Pure Iron Plate under Ar-O2 Mixed Shielding

Weld shape variation for different welding parameters is investigated on pure iron plate under gas tungsten arc (GTA) welding with argon-oxygen mixed shielding. Results showed that small addition of oxygen to the argon base shielding gas can effectively adjust the oxygen adsorption to the molten pool. An inward Marangoni convection occurs on the pool surface when the oxygen content in the weld pool is over the critical value, 80×10-6, for pure iron plate under Ar-0.3%O2 mixed shielding. Low oxygen content in the weld pool changes the inward Marangoni to an outward direction under the Ar-0.1%O2 shielding. The GTA weld shape depends to a large extent on the pattern and strength of the Marangoni convection on the pool surface, which is determined by the content of surface active element, oxygen, in the weld pool and the welding parameters. The strength of the Marangoni convection on the liquid pool is a product of the temperature coefficient of the surface tension (dσ/dT) and the temperature gradient (dT/dr) on the pool surface. Different welding parameters will change the temperature distribution and gradient on the pool surface, and therefore, affect the strength of Marangoni convection and the weld shape.     

Development of an advanced A-TIG (AA-TIG) welding method by control of Marangoni convection

A new type of tungsten inert gas (TIG) welding has been developed, in which an ultra-deep penetration is obtained. In order to control the Marangoni convection induced by the surface tension gradient on the molten pool, He gas containing a small amount of oxidizing gas was used. The effect of the concentration of O₂ and CO₂ in the shielding gas on the weld shape was studied for the bead-on-plate TIG welding of SUS304 stainless under He–O₂ and He–CO₂ mixed shielding gases. Because oxygen is a surface active element for stainless steel, the addition of oxygen to the molten pool can control the Marangoni convection from the outward to inward direction on the liquid pool surface. When the oxygen content in the liquid pool is over a critical value, around 70 ppm, the weld shape suddenly changes from a wide shallow shape to a deep narrow shape due to the change in the direction of the Marangoni convection. Also, for He-based shielding gas, a high welding current will strengthen both the inward Marangoni convection on the pool surface and the inward electromagnetic convection in the liquid pool. Accordingly, at a welding speed of 0.75 mm/s, the welding current of 160 A and the electrode gap of 1 mm under the He–0.4%O₂ shielding, the depth/width ratio reaches 1.8, which is much larger for Ar–O₂ shielding gas (0.7). The effects of the welding parameters, such as welding speed and welding current were also systematically investigated. In addition, a double shielding gas method has been developed to prevent any consumption of the tungsten electrode.     

Effect of inertia on the Marangoni instability of two-layer channel flow, Part I: numerical simulations

A numerical method is developed for simulating the flow of two superposed liquid layers in a two-dimensional channel confined between two parallel plane walls, in the presence of an insoluble surfactant. The algorithm combines Peskin’s immersed-interface method with the diffuse-interface approximation, wherein the step discontinuity in the fluid properties is replaced by a transition zone defined in terms of a mollifying function. A finite-difference method is implemented for integrating the generalized Navier–Stokes equation incorporating the jump in the interfacial traction, and a finite-volume method is implemented for solving the surfactant transport equation over the evolving interface. The accuracy of the overall scheme is confirmed by successfully comparing the numerical results with the predictions of linear stability analysis and numerical simulations based on a boundary-element method for Stokes flow. Results for selected case studies suggest that inertial effects have a mild effect on the growth rate of the surfactant-induced Marangoni instability.     

Effect of gas stream swirls on the instability of viscous annular liquid jets

Summary. A linear temporal instability analysis has been carried out for a viscous annular liquid jet moving in two swirling gas streams of unequal velocities with the gas stream swirling motion represented by free-vortex rotation. It is found that two modes of unstable surface waves exist, the para-sinuous and para-varicose mode. The results of the two limiting flow situations, which are a cylindrical liquid jet in a swirling gas stream and a swirling gas jet in a liquid stream, indicate that their instabilities are associated with the para-varicose mode on the outer interface and para-sinuous mode on the inner interface of the annular liquid jet, respectively. It is shown that the centripetal force induced by the inner gas stream rotation is destabilizing and enhances the jet instability, while the centripetal force produced by the outer gas stream rotation is stabilizing and reduces the instability of annular liquid jets. It is interesting to find that for a para-varicose mode an increase in the outer gas rotation not only reduces the upper cut-off wave number, but also increases the lower cut-off wave number, leading to the significant reduction in the unstable wave number range. The stabilizing effect of the outer gas rotation is much more significant for para-varicose mode, and the destabilizing effect of the inner gas rotation is much more influential for para-sinuous mode. In general, the para-sinuous mode has a much larger growth rate and is predominant in the annular liquid jet breakup process. Therefore, increasing the inner gas stream rotation can significantly enhance the breakup of annular liquid jets for practical spray applications.     

Conjugated Effect of Joule Heating and Magnetohydrodynamic on Laminar Convective Heat Transfer of Nanofluids Inside a Concentric Annulus in the Presence of Slip Condition

In the current study, the conjugated effect of Joule heating and magnetohydrodynamics (MHD) on the forced convective heat transfer of fully developed laminar nanofluid flows inside annular pipes, under the influence of MHD field, has been investigated. The temperature and nanoparticle distributions at both the inner and outer walls are assumed to vary in the direction of the fluid. Furthermore, owing to the nanoparticle migrations in the fluid, a slip condition becomes far more important than the no-slip condition of the fluid–solid interface, which appropriately represents the non-equilibrium region near the interface. The governing equations—obtained by employing the Buongiorno’s model for nanofluid in cylindrical coordinates—are converted into two-point ordinary boundary value differential equations and solved numerically. The effects of various controlling parameters on the flow characteristics, the average Nusselt number and the average Sherwood number have been assessed in detail. Additionally, the effect of the inner to outer diameter ratio on the heat and mass transfer rate has been studied. The results obtained indicate that, in the presence of a magnetic field when the fluid is electrically conductive, heat transfer will be reduced significantly due to the influences of Joule heating, while the average mass transfer rate experiences an opposite trend. Moreover, the increase in the slip velocity on both the walls causes the average heat transfer to rise and the average mass transfer to decrease.     

Effects of Reduced Gravity Conditions on Bubble Dispersion Characteristics in the Bubble Column

Bubble-liquid turbulent flow has an excellent heat and mass transfer behaviors than single gas or liquid flow. In order to analyze the effects of normal and reduced gravity on cold bubble-liquid two-phase turbulent flow in bubble column a second-order moment cold bubble-liquid two-phase turbulent model was developed to disclose the bubble dispersion characteristics. Under the reduced gravity condition, volume fraction caused by the decrease of buoyance force is larger than normal gravity level due to bigger bubble solid volume. In addition, bubble frequency is also decreased by in decrease of buoyance force. Normal and shear stresses have strongly anisotropic characteristics at every directions and have larger values under normal gravity than reduced gravity. The liquid turbulent kinetic energy has the two-peak bimodal distribution and weaker than bubble turbulent kinetic energy with one peak unimodal, which is caused by vigorous wake fluctuations. The correlation of fluctuation velocities between bubble and liquid has clearly anisotropic behaviors Under reduced gravity, the bubble motion has a little impact on liquid turbulent flow caused by slight buoyancy force, however, it will greatly reduce the liquid turbulent intensity due to energy cascade transport, which was transformed into bubbles or dissipated by interface friction. Bubble formation and detachment mechanisms affected by gravity conditions lead to the different levels of bubble dispersion distributions.     

硅单晶Czochralski法生长全局数值模拟II．质量传递特性

使用有限元方法对炉内的质量传递过程进行了全局数值模拟，研究了硅单晶Czochralski(Cz)法生长时氧传输的基本特性．结果表明：在小型硅Cz炉中，晶体中的氧浓度主要取决于熔体的流型和气相传质速率；安装在热区的气体导板可有效强化气相传质系数并改变熔体的流型，Marangoni效应可将自由界面处低氧浓度的熔体带至结晶界面，使晶体中氧浓度降低．     

Convectional controlled crystal–melt interface using two-phase radio-frequency electromagnetic heating

The radio frequency floating-zone growth of massive intermetallic single crystals is very often unsuccessful due to an unfavourable solid–liquid interface geometry enclosing concave fringes. This interface depends on the flow in the molten zone. A tailored magnetic two-phase stirrer system has been developed which enables the controlled influence on the melt flow ranging from intense inwards to outwards flows. Depending on the phase shift between the two induction coils, a transition from a double vortex structure to a single vortex structure is created at a preferable phase shift of 90°. This change in the flow field has a significant influence on the shape of the solid–liquid interface. Due to their attractive properties for high temperature applications such as high melting temperature, low density, high modulus and good oxidation resistance, the magnetic system was applied to the crystal growth of TiAl alloys.     

Three-dimensional axisymmetric model for convection in laser-melted pools

Abstract

A three-dimensional axisymmetric model of the fluid flow and heat transfer in a laser-melted pool is developed. The model corresponds to the limiting case when the scanning velocity is small compared with the recirculating velocity. This model is also valid for spot welding. Non-dimensional forms of the governing equations are derived, from which four dimensionless parameters are obtained: the Marangoni number, the Prandtl number, the dimensionless melting temperature, and the radiation factor. Their effects and significance are discussed, and numerical solutions are obtained. The position and shape of the solid/liquid interface are obtained by an iterative scheme. The quantitative effects of the dimensionless parameters on pool shape are presented. In the presence of the flow field, the heat transfer becomes convection dominated. The effect of convection on isotherms within the molten pool is discussed, and experimental results are presented.

MST/535     

Marangoni convection in V-shaped containers

This paper presents a numerical study of the time evolution of Marangoni convection in two V-shaped containers involved in the microgravity experiments reported in Hoefsloot et al. [7]. First the case of the triangular container with a plane gas/liquid interface is considered, next the container having the shape of a circular sector with a curved interface is dealt with. The numerical results show the same behaviour as observed experimentally: convection caused by macroscale effects in the former, and microconvection in the latter case.