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.     

Numerical Simulation of Thermocapillary Convection in a Shallow Rectangular Cavity Under the Action of Combining Horizontal Temperature Gradient with Vertical Heat Flux

In order to understand the influence of the vertical heat flux on thermocapillary convection, we conducted a series of unsteady two-dimensional numerical simulations of thermocapillary convection in a differently heated shallow rectangular cavity with vertical heat flux on the bottom by means of the finite volume method. The cavity was filled with the 1cSt silicone oil (Prandtl number Pr = 13.9) and aspect ratio is 30. It is found that a small vertical heat flux has slightly influence on the flow pattern of stable or unstable thermocapillary convection. However, the critical Marangoni number increases first, and then decreases with the increase of the heat flux. And the flow pattern of the oscillatory thermocapillary convection transits from a series of the rolls rotating clockwise and moving from the cold wall to the hot wall to the single roll near the hot wall and a series of rolls near the cold wall, further, two series of rolls moving from the hot wall and cold wall towards the hot spot with the maximum temperature. With the increase of the Marangoni number, the period and the wavelength of the oscillatory thermocapillary convection increase, but the wave speed decreases.     

Experiments on the transition from the steady to the oscillatory marangoni convection of a floating-zone under various cold wall temperatures and various ambient air temperature effects

The transition from the steady to the oscillatory Marangoni convection of a floating-zone under various cold wall temperatures and various ambient air temperature effects have been investigated experimentally by heating the sample from above (opposite direction of Marangoni convection and buoyant forces). The heat transfer takes place mainly through conduction as well as the natural convection of the air around the cylindrical liquid bridge. The ambient airflow in the present work is varied by varying the cold wall temperature and ambient air temperature. In this study, the transition from the steady to the oscillatory Marangoni convection flow of a high Prandtl number fluid in a floating half-zone is visualized by means of the already proven method of the “light-cut-technique”. The test fluid zone is held in ambient air at +4 °C, +10 °C, +16 °C, +23 °C, and +28 °C. The onset of oscillations, the oscillation level, and oscillation pattern are investigated under various conditions. It is found that the critical temperature difference (ΔT_Cr) varies substantially when the cold wall temperature and the ambient air temperature are varied.     

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.     

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.     

Bénard-Marangoni convection in two thin immiscible liquid layers with a uniform magnetic field

Summary The effect of a uniform magnetic field on Bénard-Marangoni convection in a shallow cavity, with differentially heated side walls, filld with two viscous, immiscible, incompressible an electrically conducting fluids is studied in the presence of a buoyancy force. The fluid-fluid interface and the free surface are assumed to be flat, and the driving forces for the flow are the thermocapillary and the buoyancy forces. Closed-form solutions, under thin layer approximation neglecting the side wall effects, are obtained for the stream function and the temperature. The solutions are obtained in various limiting cases, namely: absence of buoyancy force, absence of thermocapillary force and absence of magnetic field, and they coincide with the results existing in the literature. The velocity is calculated, and the resulting cell patterns are discussed for different values of , the ratio of the temperature gradients of surface tension at the interface and the free surface and the Hartmann number. There exist four different flow regimes depending on the values of but with reduced convection compared to the non-magnetic case for Marangoni convection. It is observed that it is possible to control the convection in the lower layer by a suitable choice of the magnetic field.     

MHD Effect on Interfacial Deformation of Thermocapillary Convection in Two-Layer System

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.     

Thermocapillary Convection in a Binary Mixture with Moderate Prandtl Number in a Shallow Annular Pool

This paper presented a series of numerical simulations on thermocapillary convection for mixed toluene/hexane solution with mass fraction of \(c_{0}=?26.27\)% in a shallow annular pool. The Prandtl number of the binary solution is 5.54. Results indicate that when the annular pool subjects to a radial temperature gradient, the solute shifts toward the inner wall under the Soret effect, which leads to a concentration gradient with the opposite direction to the temperature gradient. With the increase of surface heat dissipation, thermocapillary convection is enhanced and the flow is more prone to destabilization. Therefore, the critical thermocapillary Reynolds number of the flow destabilization and the corresponding critical oscillation frequency all decrease, but the critical wave number increases. After flow destabilization, the concentration fluctuation similar to the temperature fluctuation on the free surface appears. No matter the free surface is adiabatic or not, the flow always undergoes a transition from two-dimensional steady axisymmetric flow to the hydrothermal waves, and then to chaos with the increase of thermocapillary Reynolds number.     

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.     

表面张力对流对 BaB2O4晶体生长界面边界层的作用

利用高温光学实时观察方法，实时地观察了BaB2O4（BBO）高温熔体的表面张力对流效应以及BBO单晶的旋转生长过程，计算了固液界面附近的浓度、温度以及动量边界层厚度δc，δT和δv，并研究了热毛细对流对边界层厚度的影响．结果发现，浓度边界层厚度远远小于温度以及动量边界层厚度，说明晶体生长过程中，质量扩散在界面输运过程中起着主导性作用，同时发现，边界层厚度随体系无量纲Marangoni数的增大而线性地减小．     

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.     

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.     

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.     

Experimental investigation on hydrothermal wave in a shallow annular pool

An on-ground experiment was conducted for observation of pattern formation in thermocapillary convection in an annular pool (r_i=20 mm, r_o=40 mm, and depth d=1.4 mm) of silicone oil (0.65 cSt, Pr=10.1). By a shadowgraph technique, curved traveling hydrothermal waves (HTW) were observed in liquid pools differentially heated at the outer wall. The critical temperature differences (ΔT_c) for the incipience of HTW were measured and were semi-quantitatively consistent with our numerical analysis for d=1.0mm and Pr=6.7.     

Thermocapillary migration of a planar droplet at moderate and large Marangoni numbers

Thermocapillary migration of a planar non-deformable droplet in flow fields with two uniform temperature gradients at moderate and large Marangoni numbers is studied numerically by using the front-tracking method. It is observed that the thermocapillary motion of planar droplets in the uniform temperature gradients is steady at moderate Marangoni numbers, but unsteady at large Marangoni numbers. The instantaneous migration velocity at a fixed migration distance decreases with increasing Marangoni numbers. The simulation results of the thermocapillary droplet migration at large Marangoni numbers are found in qualitative agreement with those of experimental investigations. Moreover, the results concerned with steady and unsteady migration processes are further confirmed by comparing the variations of temperature fields inside and outside the droplet. It is evident that at large Marangoni numbers the weak transport of thermal energy from outside of the droplet into inside cannot satisfy the condition of a steady migration process, which implies that the advection around the droplet is a more significant mechanism for heat transfer across/around the droplet at large Ma numbers. Furthermore, from the condition of overall steady-state energy balance in the flow domain, the thermal flux across its surface is studied for a steady thermocapillary droplet migration in a flow field with uniform temperature gradient. By using the asymptotic expansion method, a non-conservative integral thermal flux across the surface is identified in the steady thermocapillary droplet migration at large Marangoni numbers. This non-conservative flux may well result from the invalid assumption of a quasi-steady state, which indicates that the thermocapillary droplet migration at large Marangoni numbers cannot reach a steady state and is thus an unsteady process.     

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.     

Evaluation of thermocapillary driving forces in the development of striations during the spin coating process

The evolution of a temperature gradient at the free surface of a coating solution during the spin coating process is examined. Solvent evaporation causes localized cooling at the top that can result in thermocapillary instability within the coating solution, and thereby driving convective flows that may result in non-uniform coatings. We examine the evolution of these temperature gradients by using a one dimensional finite difference model that simultaneously describes the thinning behavior (both by flow and by evaporation) and the temperature evolution within the solution. The entire system is initially isothermal but is subject to evaporation-driven cooling at the free surface of the gradually thinning fluid. The model is then used to determine the magnitude of the thermocapillary effects during the spin coating process. As test systems we simulate the spin coating of several pure alcohol solutions having different volatilities and therefore different evaporative-cooling powers. As the fluid thins, we calculate the instantaneous Marangoni (Mn) number, which signifies the magnitude of thermocapillary-driven convection. We compare these Mn values against their relevant threshold values, determined from prior reports in the literature, in order to deduce the magnitude of the instabilities they represent. If the Mn value is super-critical, then the instability that it represents will be sufficient for the onset of convection cells within a stagnant fluid layer of corresponding thickness. Because the radial outflow is fully laminar under normal conditions, super-critical Mn values imply that similar instabilities would arise within a spinning solution. Super-critical Mn values were observed under numerous conditions suggesting that thermocapillary instability may be responsible for striation features that develop in coatings made by spin coating. Trends related to spin-speed, solvent volatility, and initial solution thickness are discussed with the goal of improving the flatness of coatings that are made by this process.     

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.     

Thermocapillary flow in a liquid layer at minimum in surface tension

Summary In the present paper a class of similarity solutions for the two-dimensional Navier-Stokes and energy equations describing thermocapillary flows in a liquid layer of constant width and infinite extent is presented. The layer is bounded by a horizontal rigid plate from one side and opened to the ambient gas from the other one. The physical properties of the liquid are assumed to be constant except the surface tension which varies as a quadratic function with temperature. It is supposed that a constant temperature gradient exists along either the liquid free surface (case I) or the rigid boundary (case II).In both cases, by means of a similarity transformation, the equations of motion and energy are reduced to a system of three ordinary differential equations, one for the velocity and two for the temperature. The equation for the velocity can be solved separately from the other equations and its solution, found numerically, exists only for the Marangoni number less than a certain finite value. The solution of the whole system depends also on the Prandtl number. The solution of one of the temperature equations is presented in an analytical form and the other equation is solved numerically. Asymptotic formulas of the functions are also obtained for small and large Marangoni numbers. Flow pattern and temperature fields are presented. One convective roll exists in every semi-infinite layer. Fluid velocities at different points of the free surface are evaluated for an aqueous solution of n-heptanol and compared with those measured in the experiments.     

Influence of Buoyancy Force on Thermocapillary Convection Instability in the Differentially Heated Annular Pools of Silicon Melt

The influence of buoyancy force on the thermocapillary convection instability in the annular pools (R _i = 20 mm, R _o = 40 mm, and depth d ranging from 1 to 10 mm) of silicon melt (Pr = 0.011), differentially heated at the outer wall and cooled at the inner wall, is investigated numerically. The critical Marangoni numbers (Ma _c) for the incipience of oscillatory flow are determined by linear stability analysis (LSA) under both microgravity and normal gravity conditions. The results indicate that the buoyancy force destabilizes the thermocapillary convection under different liquid layer depths from 3 to 10 mm. With increasing the layer depth, the critical Ma number, critical azimuthal wave number and critical phase velocity decrease. Some of 3-D simulation results are compared with those of LSA. 3-D results are found consistent with the LSA results except for a case of D = 0.05 where 3-D simulation gives a stationary 3-D flow under a large Ma.