Steady Thermocapillary Convection Research Articles

Thermocapillary instabilities in half zone liquid bridges of low Prandtl fluid with non-equal disks under microgravity

Thermocapillary convection and its stability are of particular importance during the floating zone crystal growth process. In this paper, three-dimensional (3D) transient numerical simulations are conducted to study the onset and general features of thermocapillary instabilities in half zone liquid bridges of low Prandtl fluid (Pr = 0.01) with non-equal disks under microgravity. The effect of disk radius ratio on the critical conditions is investigated. The results reveal that both the first critical Reynolds number Rec1 and the second critical Reynolds number Rec2 are sensitive to the disk radius ratio. However, for a liquid bridge with fixed length and volume, varying the disk radius ratio does not change the critical azimuthal wave number of the first bifurcation, as well as the critical oscillation mode of the second bifurcation. Characteristics of the 3D steady thermocapillary convection and the 3D oscillatory thermocapillary convection are presented. For the supercritical state at Re = 1.2Rec1 after the first bifurcation, the flow is 3D steady. The deviations between the amplitudes of azimuthal flow in liquid bridges with different disk radius ratios are insignificant. For the supercritical state at Re = 1.2Rec2 after the second bifurcation, the flow is 3D oscillatory. The oscillating frequency decreases with disk radius ratio. The oscillating frequency in a liquid bridge with a convex free surface is higher than that in one with a concave free surface.

Marangoni convection instability in a sessile droplet with low volatility on heated substrate

In order to investigate Marangoni convection in a sessile droplet on heated substrate, a series of three-dimensional numerical simulations are carried out in a wide range of Marangoni number. A low volatile silicone oil is adopted and the effect of evaporation is neglected. Bénard-Marangoni convection and thermocapillary convection may coexist in droplet because of its curve surface. When Marangoni number is low the flow is steady axisymmetric thermocapillary convection whereas the flow becomes to be asymmetric and loses its stability and Hopf bifurcation occurs when Marangoni number high. The oscillatory transition convection, steady Bénard-Marangoni convection and irregular oscillatory Bénard-Marangoni convection appear in sequence with increasing Marangoni number. The characteristics of each kind of convection modes are described and the critical Marangoni numbers for the incipience of them are determined in a wide range of contact angles. One of the most interesting phenomenon is that the shape of the Bénard-Marangoni convection cell in sessile droplet is circular, which is different from that of classical hexagonal cell in flat layer. With increasing contact angle, the critical Marangoni number for the onset of convection instabilities first decreases and then increases. Critical cell number of the steady Bénard-Marangoni convection keeps on only one for contact angle less than 15° whereas it increases sharply after angle lager than 15°. The numerical results are qualitatively verified by experiment.

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.

Some bifurcation routes to chaos of thermocapillary convection in two-dimensional liquid layers of finite extent

As a part of the preliminary studies for the future space experiment (Zona-K) in the Russian module of the International Space Station, some bifurcation routes to chaos of thermocapillary convection in two-dimensional liquid layers filled with 10 cSt silicone oil have been numerically studied in this paper. As the laterally applied temperature difference is raised, variations in the spatial structure and temporal evolution of the thermocapillary convection and a complex sequence of transitions are observed. The results show that the finite extent of the liquid layer significantly influences the tempo-spatial evolution of the thermocapillary convection. Moreover, the bifurcation route of the thermocapillary convection changes very sensitively by the aspect ratio of the liquid layer. With the increasing Reynolds number (applied temperature difference), the steady thermocapillary convection experiences two consecutive transitions from periodic oscillatory state to quasi-periodic oscillatory state with frequency-locking before emergence of chaotic convection in a liquid layer of aspect ratio 14.25, and the thermocapillary convection undergoes period-doubling cascades leading to chaotic convection in a liquid layer of aspect ratio 13.0.

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.

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.

Stability of Thermocapillary Convection in Rotating Shallow Annular Pool of Silicon Melt

In order to understand the effect of pool rotation on stability of thermocapillary convection, the critical conditions for the incipience of oscillatory flow in rotating shallow annular pools of silicon melt (Pr = 0.011) are investigated by means of linear stability analysis (LSA) under different pool rotation rates ranging from Ta = 0 to 1,513.9 and under microgravity condition. The results indicate that with increase of Ta numbers, the critical Marangoni for the incipience of hydrothermal wave (HTW) increases, i.e., pool rotation stabilizes the steady axisymmetric thermocapillary convection of silicon melt. The critical azimuthal wave number mc decreases linearly with increase of Ta number when Ta ≤ 378.5. However, when Ta > 378.5, mc almost keeps constant with mc = 34. When Ta 550.

Numerical Simulation of Steady Thermocapillary Convection in a Two-Layer System Using Level Set Method

Steady thermocapillary convection with deformable interface in a two-layer system is simulated by the second-order projection method combined with the level set method, in which the three-stage Runge–Kutta technique and second-order semi-implicit Crank–Nicholson technique are employed to temporally update the convective and diffusion terms, respectively. The level set approach is employed to implicitly capture the interface. The continuum surface force tension model is used to simulate the Marangoni effect. Simulations are conducted for both fixed angle and fixed points at the contact between the interface and the end walls. The numerical results show that, the interface bulges out near the hot wall and bulges in near the cold wall, due to the Marangoni effect. With Marangoni number increasing, the deformability of interface increases. The contact condition of interface with the end walls is important for the prediction of thermocapillary convection characteristics, and the contact points fixed condition is more close to real condition.

Numerical Study of Steady Thermocapillary Convection in a Cylindrical Half-Floating-Zone

Numerical Study of Steady Thermocapillary Convection in a Cylindrical Half-Floating-Zone

Asymptotic Solution of Thermocapillary Convection in a Differentially Heated Thin Annular Two-Layer Pool

The steady laminar two-dimensional thermocapillary convection of two immiscible liquid layers in a thin annular pool with one free surface, one liquid/liquid interface subjected to a radial temperature gradient was investigated using asymptotical analysis. The pool is heated from the inner cylindrical wall and cooled at the outer wall. Bottom and top surfaces are adiabatic. The asymptotic solution is obtained in the core region in the limit as the aspect ratio, which is defined as the ratio of the lower layer thickness to the gap width, trends to zero. The numerical experiments are also carried out to compare with the asymptotic solution of the steady two-dimensional thermocapillary convection. The results indicate that the expressions of velocity and temperature fields in the core region are valid in the limit of the small aspect ratio.

Steady parallel flow in an evaporating fluid heated from sidewalls

Evaporation is ubiquitous in nature, but very few attempts have been made in the past to couple the effects of evaporation with fluid flow behavior. In this theoretical paper we have discussed the effects of evaporation on the dynamics of steady state thermocapillary convection in a two-dimensional rectangular container. The liquid is heated by differentially heated sidewalls and mass loss from the interface due to evaporation is compensated by the liquid entering into the container through a lower inlet, thus keeping the thickness of the liquid layer constant. We show that for an evaporating liquid one can obtain a plane parallel base state profile which depends on the evaporative mass flux.

Asymptotic solution of thermocapillary convection in thin annular two-layer system with upper free surface

The steady laminar two-dimensional thermocapillary convection in the thin annular two superposed horizontal liquid layers with one free surface, one liquid/liquid interface subjected to a radial temperature gradient was investigated using asymptotical analysis. The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces are adiabatic. The asymptotic solution is obtained in the core region in the limit as the aspect ratio, which is defined as the ratio of the lower layer thickness to the gap width, goes to zero. The numerical experiments are also carried out to compare with the asymptotic solution of the steady two-dimensional thermocapillary convection. The asymptotic results indicate that the expressions of velocity and temperature fields in the core region are valid in the limit of the small aspect ratio.

Asymptotic solution of thermocapillary convection in a thin annular pool of silicon melt

The free surface of the silicon melt in a thin annular pool is subjected to a radial temperature gradient. Since the surface tension depends on the temperature, it will create a thermocapillary force on the free surface and, in turn, yield to thermocapillary convection in the bulk of the liquid by the viscous traction. This paper presents an investigation on the steady two-dimensional thermocapillary convection in a differentially heated annular pool of the silicon melt using the asymptotical way. The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces are adiabatic. The asymptotic solution is obtained in the core region in the limit as the aspect ratio, which is defined as the ratio of the pool height to the gap width, goes to zero. The numerical experiments are also carried out to compare to the asymptotic solution of the steady two-dimensional thermocapillary convection. The results indicate that the expressions of velocity and temperature fields in the core region from the asymptotic solution are found to be valid in the limit of small aspect ratio.

Effect of surface tension‐driven flow on BaB2O4 crystal growth from high temperature melt‐solution

AbstractThe surface tension driven‐flow in BaB2O4 (BBO) melt‐solution is visualized by differential interference microscope coupled with Schlieren technique, and the streamline of the steady thermocapillary convection is found to be in form of an axially symmetric pattern. Based on the observation of BBO crystal rotation caused by the convective vortex, the widths of interfacial concentration, heat and momentum boundary layer are calculated. The effect of thermocapillary convection on boundary layer thickness is also investigated. Results show that the width of boundary layer decreases linearly with the increasing of dimensionless Marangoni number. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Suppression of thermocapillary oscillations in sodium nitrate floating half-zones by high-frequency end-wall vibrations

Experiments to suppress thermocapillary oscillations by using high-frequency end-wall vibrations were carried out in sodium nitrate floating half-zones. Such a half-zone is formed by melting one end of a vertically held sodium nitrate crystal rod in contact with a hot surface at the top. Thermocapillary convection occurs in the melt due to the imposed temperature gradient at the free surface. When this temperature gradient is large enough, steady thermocapillary convection becomes unstable, and thermocapillary oscillations occur. In such a context, the bottom end of the crystal rod was vibrated at a high frequency to generate a streaming flow in a direction opposite to that of thermocapillary convection. It is observed that by generating a sufficiently strong streaming flow, the thermocapillary oscillations can be quenched everywhere in the melt zone.

Effect of liquid bridge form on oscillatory thermocapillary convection under 1 g and μg conditions

This study analyses the effect of temperature difference between hot and cool disk (Δ T), and non-dimensional liquid bridge volume ( V/ V o) on the transition process from steady thermocapillary convection to periodic or chaotic thermocapillary convection in a liquid bridge modeled after the floating zone method under normal gravity and microgravity conditions. From normal gravity and drop shaft experiments, the difference of the regime of the steady state and the oscillatory state was clarified on the Δ T− V/ V o plane under 1 g and μg conditions. A gap or stability region was observed in the specific V/ V o range under 1 g conditions. In the gap or stable region, after the gravity changed from 1 g to μg conditions, the temperature signals showed oscillation. From these results, the critical temperature difference under the μg conditions appeared to be smaller than that under the 1 g conditions. Temperature signals were defined as 6 different types of states. The various temperature oscillatory state regimes were obtained on a Δ T− V/ V o plane under 1 g and μg conditions. Under μg conditions, in these experimental conditions, all temperature oscillatory states exhibited only the Periodic state.

Effects of convective fluid motion upon oxide crystal growth in high temperature solution

For understanding of the influence of convective flow on crystal growth, space high temperature in situ observation instrument (SHITISOI) is dedicated to visualize and record the whole growth process of oxide crystals in high temperature up to 1,000°C. Model experiments using transparent liquids such as KNbO3 and a mixture of Li2B4O7+KNbO3 were chosen to investigate effects on ground and in space. On the earth, an investigation of growth kinetics of KNbO3 crystal related to two different states of convection: diffusive-advective flow and diffusive-convective flow, has been performed. The per unit length of a step e is calculated from the experimental data for two different states of convection. Analyses of these data show the effect of buoyancy convection is to enhance the sharpness of the interface. The growth of KNbO3 crystals from solution of KNbO3+Li2B4O7 was investigated in space. The streamlines of the steady thermocapillary convection in Li2B4O7 solvent was observed. Due to thermocapillary convection, KNbO3 crystal grains grew and filled the whole solution homogeneously. Earth-based quenching experiments are designed in order to study polyhedral instability of KNbO3 crystal, which is controlled by diffusion mechanism limitation. In all cases, when the crystal was nucleated near air/solution surface, it lost its polyhedral stability and varied from polyhedrons to dendrites. The thickness of diffusion mechanism limitation layer is about 60 µm.

Usefulness of experiments with model fluid for thermocapillary convection—effect of Prandtl number on two-dimensional thermocapillary convection

The floating zone technique is promising to realize large, high-quality crystals under microgravity. Thermocapillary flow, which is the dominant convection in a floating zone under microgravity, has been widely reported in the literature to be extremely sensitive to the Prandtl number Pr. In the present study, the effect of Pr on steady thermocapillary convection is re-evaluated based on both analytical and numerical solutions. The results indicate that an approximate similarity of temperature and velocity structures in a large Pr range is realized under suitable conditions, fixing the Marangoni number Ma, in steady two-dimensional regime.

Steady and oscillatory thermocapillary convection generated by a bubble

In this article, we study steady and oscillatory thermocapillary and natural convective flows generated by a bubble on a heated solid surface. The dynamic characteristics of the time-dependent convection are captured using a combined numerical-experimental approach. The index of refraction fringe distribution patterns constructed numerically by taking an inverse Abel transform of the computed temperature fields are compared directly to the experimental Wollaston prism (WP) interferograms for both steady-state and oscillatory convection. The agreement between numerical predictions and experimental measurements is excellent in all cases. It is shown that below the critical Marangoni number, steady-state conditions are attainable. With increasing Ma, there is a complete transition from steady state up to a final nonperiodic fluctuating flow regime through several complicated symmetric and asymmetric oscillatory states. The most prevalent oscillatory mode corresponds to a symmetric up and down fluctuation of the temperature and flow fields associated with an axially traveling wave. Careful examination of the numerical results reveals that the origin of this class of convective instability is closely related to an intricate temporal coupling between large-scale thermal structures which develop in the fluid in the form of the cold return flow and the temperature sensitive surface of the bubble. Gravity and natural convection play an important role in the formation of these thermal structures and the initiation of the oscillatory convection. Consequently, at low-g, the time evolution of the temperature and flow fields around the bubble are very different from their 1-g counterparts for all Marangoni numbers.

Linear stability of thermocapillary convection in cylindrical liquid bridges under axial magnetic fields

The stability of axisymmetric steady thermocapillary convection of electrically conducting fluids in half-zones under the influence of a static axial magnetic field is investigated numerically by linear stability theory. In addition, the energy transfer between the basic state and a disturbance is considered in order to elucidate the mechanics of the most unstable mode. Axial magnetic fields cause a concentration of the thermocapillary flow near the free surface of the liquid bridge. For the low Prandtl number fluids considered, the most dangerous disturbance is a non-axisymmetric steady mode. It is found that axial magnetic fields act to stabilize the basic state. The stabilizing effect increases with the Prandtl number and decreases with the zone height, the heat transfer rate at the free surface and buoyancy when the heating is from below. The magnetic field also influences the azimuthal symmetry of the most unstable mode.