Numerical Simulation of Thermocapillary Convection with Evaporation Induced by Boundary Heating

The dynamics of a bilayer system filling a rectangular cuvette subjected to external heating is studied.The influence of two types of thermal exposure on the flow pattern and on the dynamic contact angle is analyzed.In particular,the cases of local heating from below and distributed thermal load from the lateral walls are considered.The simulation is carried out within the frame of a two-sided evaporative convection model based on the Boussinesq approximation.A benzine–air system is considered as reference system.The variation in time of the contact angle is described for both heating modes.Under lateral heating,near-wall boundary layers emerge together with strong convection,whereas the local thermal load from the lower wall results in the formation of multicellular motion in the entire volume of the fluids and the appearance of transition regimes followed by a steady-state mode.The results of the present study can aid the design of equipment for thermal coating or drying and the development of methods for the formation of patterns with required structure and morphology.

Bifurcation of thermocapillary convection in a shallow annular pool of silicon melt

In order to understand the nature of surface patterns on silicon melts in industrial Czochralski furnaces, we conducted a series of unsteady threedimensional numerical simulations of thermocapillary convections in thin silicon melt pools in an annular container. The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces are adiabatic. The results show that the flow is steady and axisymmetric at small temperature difference in the radial direction. When the temperature difference exceeds a certain threshold value, hydrothermal waves appear and bifurcation occurs. In this case, the flow is unsteady and there are two possible groups of hydrothermal waves with different number of waves, which are characterized by spoke patterns traveling in the clockwise and counter-clockwise directions. Details of the flow and temperature disturbances are discussed and number of waves and traveling velocity of the hydro- thermal wave are determined.

Thermocapillary migration and interaction of two nondeformable drops

A numerical study on the interaction of two spherical drops in the thermocapillary migration is presented in the microgravity environment. Finite-difference methods are adopted. The interfaces of the drops are captured by the front-tracking technique. It is found that the arrangement of the drops directly influences their migration and interaction, and the motion of one drop is mainly determined by the disturbed temperature field because of the existence of the other drop.

Instability in Two-Sided Thermocapillary-Buoyancy Convection with Interfacial

A new model of two-phase thermocapillary-buoyancy convection with phase change at gas-liquid interface in an enclosed cavity subjected to a horizontal temperature gradient is proposed,rather than the previous onesided model without phase change.We study the onset of multicellular convection and two inodes of convective instability,and find four different flow regimes.Their transition map is compared with the non-phase-change condition.Our numerical results show the stabilizing effect of interfacial phase change on the thermocapillarybuoyancy convection.

Velocity Field of Thermocapillary Convection in High Temperature Oxide Solution

We have investigated experimentally and theoretically the thermocapillary convective flow phenomena in a loop-shaped Pt wire heater of KNbO_(3)(20wt.%)and Li_(2)B_(4)O_(7) solution.Optical evaluations in connection with thermocouple measurements made it possible to get a new type of thermocapillary convective flow in the considered system.To study the kinematical behavior of thermocapillary convection,we have measured the stream velocities of flow.In a theoretical analysis,the flow velocity due to thermocapillary effect alone was estimated by balancing the surface tension forces by viscous forces.The velocity distribution in the solution near the margin of the heater was obtained,which is in agreement with the experimental result.

Radiative Heat Transfer and Thermocapillary Effects on the Structure of the Flow during Czochralski Growth of Oxide Crystals

A numerical study was carried out to describe the flow field structure of an oxide melt under 1) the effect of internal radiation through the melt (and the crystal), and 2) the impact of surface tension-driven forces during Czochralski growth process. Throughout the present Finite Volume Method calculations, the melt is a Boussinnesq fluid of Prandtl number 4.69 and the flow is assumed to be in a steady, axisymmetric state. Particular attention is paid to an undulating structure of buoyancy-driven flow that appears in optically thick oxide melts and persists over against forced convection flow caused by the externally imposed rotation of the crystal. In a such wavy pattern of the flow, particularly for a relatively higher Rayleigh number , a small secondary vortex appears nearby the crucible bottom. The structure of the vortex which has been observed experimentally is studied in some details. The present model analysis discloses that, though both of the mechanisms 1) and 2) end up in smearing out the undulating structure of the flow, the effect of thermocapillary forces on the flow pattern is distinguishably different. It is shown that for a given dynamic Bond number, the behavior of the melt is largely modified. The transition corresponds to a jump discontinuity in the magnitude of the flow stream function.

Asymptotic solution of thermocapillary convection in two immiscible liquid layers in a shallow annular cavity

The steady laminar two-dimensional thermocapillary convection of two superposed horizontal liquid layers in a shallow annular cavity was investigated using asymptotical analysis.The liquids were supposed to be immiscible with a nondeformable interface.The cavity was heated from the outer cylindrical wall and cooled at the inner wall.Bottom and top surfaces were rigid and adiabatic.Asymptotic solutions were obtained in the core region in the limit as the aspect ratio,which was defined as the ratio of the lower layer thickness to the gap width,and trended to zero.The numerical experiments were also carried out to compare with the asymptotic solution of the steady two-dimensional thermocapillary convection.It is found that the expressions of velocity and temperature fields in the core region are valid in the limit of the small aspect ratio.

Space experimental investigation on thermocapillary migration of bubbles

Results from a space experiment on bubble thermocapillary migration conducted on board the Chinese 22nd recoverable satellite were presented. Considering the temperature field in the cell was disturbed by the accumulated bubbles, the temperature gradient was corrected firstly with the help of the temperature measurement data at six points and numerical simulation. Marangoni number (Ma) of single bubble migrating in the space experiment ranged from 98.04 to 9288, exceeding that in the previous experiment data. The experiment data including the track and the velocity of two bubble thermocapillary migration showed that a smaller bubble would move slower as it was passed by a larger one, and the smaller one would even rest in a short time when the size ratio was large enough.

Space experiments of thermocapillary convection in two-liquid layers

In 1999, the space experiments on the Marangoni convection and thermocapillary convection in a system of two immiscible liquid layers in microgravity environment were conducted on board the Chinese scientific satellite SJ-5. A new system of two-layer liquids such as FC-70 liquid and paraffin was used successfully, with the paraffin melted in the space. Two different test-cells are subjected to a temperature gradient perpendicular or parallel to the interface to study the Marangoni convection and thermocapillary convection, respectively. The experimental data obtained in the first Chinese space experiment of fluid are presented. Two-dimensional numerical simulations of thermocapillary convections are carried out using SIMPLEC method .A reasonable agreement between the experimental investigation and the numerical results is obtained.

FREE SURFACE OSCILLATION OF THERMOCAPILLARY CONVECTION IN LIQUID BRIDGE OF HALFFLOATING-ZONE

Free surface deformations of thermocapillary convection in a small liquid bridge of half floating-zone are studied in the present paper. The relative displacement and phase difference of free surface oscillation are experimentally studied, and the features of free surface oscillation for various applied temperature differences are obtained. It is discovered that there is a sort of surface waves having the character of small perturbation, and having a wave mode of unusually large amplitude in one corner region of the liquid bridge.

Two-dimensional numerical simulation of thermocapillary convection in detached solidification

Numerical simulations of flow in the melt(CdZnTe) with different conditions are conducted using the finite-difference method.When the top surface of the melt is solid wall under microgravity condition,the thermocapillary convection is caused in the melt by the surface tension gradient on the free surface.As the Marangoni number is small,the flow is steady thermocapillary convection.As the Marangoni number exceeds the critical value,the steady flow transits into unstable thermocapillary convection.When the top surface of the melt is free surface under microgravity,two roll cells are observed in the melt,which are driven by both the surface tension gradients on the upper and lower free surfaces.When the top surface of the melt is free surface under gravity condition,the effect of the buoyancy on the flow is little as the Marangoni number is small.With the Marangoni number increasing,the effect of the buoyancy increases,which makes the upper roll cell weaken and the lower roll cell strengthen.

Numerical investigation of thermocapillary migration of the drop for large Marangoni numbers

An axisymmetric model is adopted to simulate the problem of unsteady drop thermocapillary motion for large Marangoni numbers. Front tracking methods are used in the investigation. It is found that the non-dimensional drop migration ve- locity will decrease with increasing Marangoni number. This agrees well with the experimental results obtained from the 4th Shen-Zhou space ship. In the meanwhile, this is also the first time for numerical simulations to verify the experimental phe- nomenon under large Marangoni numbers.

Experimental study of thermocapillary convection on a liquid bridge consisting of fluid with low Prandtl number in a floating half-zone

A device of mercury liquid bridge of floating half-zone is designed to experimentally explore thermo-capillary convection and its instability of a low Prandtl number liquid. Noncontacted diagnostic techniques were developed to monitor surface flow and surface deformation. The surface flow and the influence of a growing surface film (or skin) on the flow were observed It is shown that the film is a key factor in changing the behavior associated with the thermocapillary convection. The experiment indicates that the critical Marangoni number should be much higher than that expected by the numerical simulation. The condition and process of surface film growth are discussed. The surface oscillation of the mercury bridge wrapped with "dirt-film" was observed, and the characteristics and the frequency associated with this oscillation are given.

Rheological effect on thermocapillary flow of a liquid film jet painted on a moving boundary

In the present paper, a liquid (or melt) film of relatively high temperature ejected from a vessel and painted on the moving solid film is analyzed by using the second-order fluid model of the non-Newtonian fluid. The thermocapillary flow driven by the temperature gradient on the free surface of a Newtonian liquid film was discussed before. The effect of rheological fluid on thermocapillary flow is considered in the present paper. The analysis is based on the approximations of lubrication theory and perturbation theory. The equation of liquid height and the process of thermal hydrodynamics of the non-Newtonian liquid film are obtained, and the case of weak effect of the rheological fluid is solved in detail.

Period-doubling bifurcations of the thermocapillary convection in a floating half zone

This study experimentally explored the fine structures of the successive period-doubling bifurcations of the time-dependent thermocapillary convection in a floating half zone of 10 cSt silicone oil with the diameter d0=3.00 mm and the aspect ratio A=l/d0=0.72 in terrestrial conditions.The onset of time-dependent thermocapillary convection predominated in this experimental configuration and its subsequent evolution were experimentally detected through the local temperature measurements.The experimental results revealed a sequence of period-doubling bifurcations of the time-dependent thermocapillary convection,similar in some way to one of the routes to chaos for buoyant natural convection.The critical frequencies and the corresponding fractal frequencies were extracted through the real-time analysis of the frequency spectra by Fast-Fourier-Transfor-mation(FFT).The projections of the trajectory onto the reconstructed phase-space were also provided.Furthermore,the experimentally predicted Feigenbaum constants were quite close to the theoretical asymptotic value of 4.669 [Feigenbaum M J.Phys Lett A,1979,74:375-378].

Simulation of Thermocapillary Convection in a TwoLayer Immiscible Fluid System Using a Boundary Element Method

A boundary element method for simulating thermocapillary convection in a two-layer immiscible fluid system with flat and free interface has been developed.The divergence theorem is applied to the non-linear convective volume integral of the boundary element formulation with the pressure penalty function.Consequently,velocity gradients are eliminated and the complete formulation is written in terms of velocity.This avoids the difficulty of convective discretizations and provides considerable reductions in storage and computational requirements while improving accuracy.In this paper,we give the influence of different parameters(Marangoni number, Reynolds number)on thermocapillary convection in cavity with two-layer immiscible fluids.As shown by the numerical results,when the physical parameters between liquid encapsulant and melt are chosen appropriately, the detrimental flow in the bottom melt layer can be greatly suppressed.The influence of the free interface on thermocapillary convection is also shown.

Thermocapillary migration mechanism of molten silicon droplets on horizontal solid surfaces

Effective lubrication under extreme conditions such as high temperature is of considerable importance to ensure the reliability of a mechanical system. New lubricants that can endure high temperatures should be studied and employed as alternatives to traditional oil-based lubricant. In this paper, a thermocapillary model of a silicone-oil droplet is developed by solving the Navier–Stokes and energy equations to obtain the flow, pressure, and temperature fields. This is accomplished using a conservative microfluidic two-phase flow level set method designed to track the interface between two immiscible fluids. The numerical simulation accuracy is examined by comparing the numerical results with experimental results obtained for a silicone-oil droplet. Hence, the movement and deformation of molten silicon droplets on graphite and corundum are numerically simulated. The results show that a temperature gradient causes a tension gradient on the droplet surface, which in turn creates a thermocapillary vortex. As the vortex develops, the droplet migrates to the low-temperature zone. In the initial stage, the molten silicon droplet on the corundum substrate forms two opposite vortex cells, whereas two pairs of opposite vortices are formed in the silicone fluid on the graphite substrate. Multiple vortex cells gradually develop into a single vortex cell, and the migration velocity tends to be stable. The greater the basal temperature gradient, the stronger the internal thermocapillary convection of the molten silicon droplet has, which yields higher speeds.

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.

Instability of thermocapillary flow in liquid layers under microgravity

The instability of the thermocapillary flow in liquid layers is studied in the present paper using the linear stability analysis.Based on the two-dimensional steady flow state,the three-dimensional disturbance with a wave number in the spanwise direction is considered.The effects of the aspect ratio and free surface shape of the liquid layer on the flow instability are studied,and the results are compared with the case with the two-dimensional disturbance.

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.