Thermocapillary Migration Research Articles

Direct numerical simulation of variable surface tension flows using a Volume-of-Fluid method

We develop a general methodology for the inclusion of a variable surface tension coefficient into a Volume-of-Fluid based Navier–Stokes solver. This new numerical model provides a robust and accurate method for computing the surface gradients directly by finding the tangent directions on the interface using height functions. The implementation is applicable to both temperature and concentration dependent surface tension coefficient, along with the setups involving a large jump in the temperature between the fluid and its surrounding, as well as the situations where the concentration should be strictly confined to the fluid domain, such as the mixing of fluids with different surface tension coefficients. We demonstrate the applicability of our method to the thermocapillary migration of bubbles and the coalescence of drops characterized by a different surface tension coefficient.

Study on multicomponent pseudo-potential model with large density ratio and heat transfer

Large density ratio multiphase flow is a persistent challenge within the field of computational fluid dynamics. This paper investigates improvements to the lattice Boltzmann based large density ratio multicomponent multiphase pseudo-potential model. The improvements include: the multiple-relaxation-time (MRT) collision operator; the exact difference method scheme; the Carnahan-Starling equation of state; and an addition of correction factor k to the equation of state. The improved model can be used for simulating large density ratio (O(1000)) multiphase flow with small spurious current and better numerical stability. The smaller spurious current can be obtained by decreasing k value, and yet interface thickness increases. The density ratio is 1284 for k=0.1 and the spurious current is reduced to 0.0069, which is much smaller than that of 0.033 in literature. The interfacial tension can be adjusted independently from density ratio by changing k value. A thermal multiphase flow model is developed based on the large density ratio pseudo-potential model. The model is validated by using static heat conduction and dynamic flow simulations. The result of the static heat conduction of the flat interface has smaller error with the theoretical solution than that of droplet. The result of thermocapillary migration is comparable with the theoretical prediction. Finally, the heat conduction melting is simulated by coupling the enthalpy-based method.

New insights in dehydration stress behavior of two maize hybrids using advanced distributed reactivity model (DRM). Responses to the impact of 24-epibrassinolide.

Proposed distributed reactivity model of dehydration for seedling parts of two various maize hybrids (ZP434, ZP704) was established. Dehydration stresses were induced thermally, which is also accompanied by response of hybrids to heat stress. It was found that an increased value of activation energy counterparts within radicle dehydration of ZP434, with a high concentration of 24-epibrassinolide (24-EBL) at elevated operating temperatures, probably causes activation of diffusion mechanisms in cutin network and may increases likelihood of formation of free volumes, large enough to accommodate diffusing molecule. Many small random effects were detected and can be correlated with micro-disturbing in a space filled with water caused by thermal gradients, increasing capillary phenomena, and which can induce thermo-capillary migration. The influence of seedling content of various sugars and minerals on dehydration was also examined. Estimated distributed reactivity models indicate a dependence of reactivity on structural arrangements, due to present interactions between water molecules and chemical species within the plant.

Numerical study of the thermocapillary droplet migration in a microchannel under a blocking effect from the heated upper wall

This study presents a numerical computation to investigate the transient thermocapillary migration of a small liquid droplet in a microchannel. When the upper wall of the microchannel is set to be a constant temperature higher than the lower wall, subjected to a uniform temperature gradient, the evolution of the temperature distribution both inside and outside the droplet affects the migration behavior of the droplet. During the initial stage, the smallest temperature on the free interface shifts from near the mid-plane to the advancing side. The imbalance in the surface tension along the free surface causes two pairs of thermocapillary vortices both inside and outside the liquid droplet which create a net thermocapillary momentum that pushes the droplet to migrate. With the passage of time, they decrease in magnitude due to a reduction of the net thermocapillary momentum. The highest temperature appears on the free interface after the lowest temperature reaches the advancing side. During this period, the net thermocapillary momentum turns to go against the droplet migration. As a consequence, the droplet first accelerates and then decelerates to zero velocity. With a higher upper wall temperature, the time taken for the migration of the droplet to stop is shorter. When the static contact angle is less than 90°, the migration of the droplet is assisted by the capillary force. Therefore, the time it takes to block the movement of the droplet becomes longer. On the other hand, the blocking time is shortened due to the negative effect of the capillary force when the static contact angle is greater than 90°.

Terminal thermocapillary migration of a droplet at small Reynolds numbers and large Marangoni numbers

In this paper, the overall steady-state momentum and energy balances in the thermocapillary migration of a droplet at small Reynolds numbers and large Marangoni numbers are investigated to confirm the quasi-steady state assumption of the system. The droplet is assumed to have a slight axisymmetric deformation from a sphere shape. It is shown that under the quasi-steady state assumption, the total momentum of the thermocapillary droplet migration system at small Reynolds numbers is conservative. The general solution of the steady momentum equations can be determined with its parameters depending on the temperature fields. However, a nonconservative integral thermal flux across the interface for the steady thermocapillary migration of the droplet at small Reynolds numbers and large Marangoni numbers is identified. The nonconservative integral thermal flux indicates that no solutions of the temperature fields exist for the steady energy equations. The terminal thermocapillary migration of the droplet at small Reynolds numbers and large Marangoni numbers cannot reach a steady state and is thus in an unsteady process.

On the migration of a droplet on an incline

A liquid droplet placed on a nonuniformly heated solid surface will migrate from a high temperature region to a low temperature region. The present study reports the results of an experimental investigation on the migration behavior of mineral oil droplets subjected to a thermal gradient on an inclined plane. A particular attention is paid to the relationship between the critical inclination angle and thermal gradients. It is shown that there exists a critical inclination angle at which the droplet migration is halted. This critical inclination angle can be readily predicted using analytical expressions derived in this paper. This study puts forward the understanding of the interface phenomenon of thermocapillary migration on an incline. The knowledge of the critical inclination is important in applications where the migration on an incline must be obstructed to retain adequate lubrication in the desired location.

Acoustothermal tweezer for droplet sorting in a disposable microfluidic chip.

Precise control over droplet position within a microchannel is fundamental to droplet microfluidic applications. This article proposes acoustothermal tweezer for the control of droplet position, which is based on thermocapillary droplet migration actuated by acoustothermal heating. The proposed system comprises an acoustothermal heater, which is composed of a slanted finger interdigital transducer patterned on a piezoelectric substrate and a thin PDMS membrane, and a PDMS microchannel. In the proposed system, droplets moving in a droplet microfluidic chip experience spatiotemporally varying thermal stimuli produced by acoustothermal heating and thus migrate laterally. In comparison to previous methods for droplet sorting, the acoustothermal tweezer offers significant advantages: first, the droplet position can be manipulated in two opposite directions, which enables bidirectional droplet sorting to one of three outlets downstream; second, precise control over the droplet position as well as improved droplet lateral displacement on the order of hundreds of micrometers can be achieved in a deterministic manner, thereby enabling multichannel droplet sorting; third, the PDMS microfluidic chip is disposable and thus can be easily replaced since it is attached to the substrate by reversible bonding, which allows the acoustothermal heater to be reused. Given these advantages, the proposed droplet sorting system is a promising droplet microfluidic lab-on-a-chip platform for tunable, on-demand droplet position control.

Thermally driven bubble evolution at a heater wire in water characterized by high-speed transmission electron microscopy.

This work investigates the early stage evolution of thermally nucleated microbubbles in water using in situ high-speed, 400 fps, transmission electron microscopy. A Pt wire Joule heater induced bubble nucleation and growth from air-saturated water at different levels of power. For all powers below Pt breakdown, the dissolved gas initiates bubble nucleation at the concave surface defects adjacent to the area of highest temperature. A combination of interfacial forces and stress relaxation drive rapid migration of the bubbles away from the nucleation site. Thermocapillary forces ultimately dominate and drive their return to the region of highest temperature. The dynamic response highlights the importance of this length and time domain, which has until now received limited direct study.

Terminal states of thermocapillary migration of a planar droplet at moderate and large Marangoni numbers

In this paper, thermocapillary migration of a planar droplet at moderate and large Marangoni numbers is investigated analytically and numerically. By using the dimension-analysis method, the thermal diffusion time scale is determined as the controlling one of the thermocapillary droplet migration system. During this time, the whole thermocapillary migration process is fully developed. By using the front-tracking method, the steady/unsteady states as the terminal ones at moderate/large Marangoni numbers are captured in a longer time scale than the thermal diffusion time scale. In the terminal states, the instantaneous velocity fields in the unsteady migration process at large Marangoni numbers have the forms of the steady ones at moderate Marangoni numbers. However, in view of the former instantaneous temperature fields, the surface tension of the top surface of the droplet gradually becomes the main component of the driving force on the droplet after the inflection point appears. It is different from that the surface tension of the bottom surface of the droplet is the main component of the driving force on the droplet for the latter ones. The physical mechanism of thermocapillary droplet migration can be described as the significance of the thermal convection around the droplet is higher than/just as the thermal conduction across the droplet at large/moderate Marangoni numbers.

Thermocapillary Flow and Coalescences of Heterogeneous Bubble Size Diameter in a Rotating Cylinder: 3D Study

Two dimensional axisymmetric and three-dimensional VOF simulations of gas/liquid transient flow were performed using a multiphase flow algorithm based on the finite-volume method. The results for motion of a multiple bubbles of a heterogeneous sizes aligned horizontally and perpendicular to a hot surface incorporating thermocapillary forces in a rotating liquid in a zero-gravity environment have been presented for the first time. No bubbles broke in any of the cases observed. The results also show that collision and agglomeration of bubbles of unequal sizes diameter are different from those of similar size diameters presented from earlier research work of Alhendal et al. Acta Astronaut. 117, 484–496 (2015). Different flow patterns such as thermocapillary bubble migration, collision, and stream function were observed and presented for the 2-D and 3-D models.

Wall effects on the thermocapillary migration of single gas bubbles in stagnant liquids

In this paper, the governing continuum conservation equations for two-phase flow are solved using the commercial software package (Ansys-Fluent 1) to investigate the thermocapillary movement of a single bubble in stagnant liquid under zero-gravity condition. The current results show that different temperature gradients lead to different bubble migration velocities, and bubble migration velocity varies linearly with the temperature gradient for the given conditions. Furthermore the inside column diameter was found to have a significant influence on the thermocapillary migration of the bubble. Calculation were made in columns with inside diameters Dr 15, 20, 30, 40, 60, 80, 100 and 120 mm. Reduction on bubble migration velocity only occurred when the ratio of the bubble diameter to the column diameter, db/Dr, is greater than 0.267 due to column wall effect. On the other hand, the influence of the column diameter on the rise velocity is negligible when db/Dr is equal to or smaller than 0.267. No bubble shape deformation were observed and the bubble were spherical in shape for all column width. Present investigation of the shape and trajectory of bubble motion driven by surface tension-gradient in different column width is a new area of study and aims to support research into space applications which can help to determine the new migration time and speed.

A lattice Boltzmann method for axisymmetric thermocapillary flows

In this work, we develop a two-phase lattice Boltzmann method (LBM) to simulate axisymmetric thermocapillary flows. This method simulates the immiscible axisymmetric two-phase flow by an improved color-gradient model, in which the single-phase collision, perturbation and recoloring operators are all presented with the axisymmetric effect taken into account in a simple and computational consistent manner. An additional lattice Boltzmann equation is introduced to describe the evolution of the axisymmetric temperature field, which is coupled to the hydrodynamic equations through an equation of state. This method is first validated by simulations of Rayleigh–Bénard convection in a vertical cylinder and thermocapillary migration of a deformable droplet at various Marangoni numbers. It is then used to simulate the thermocapillary migration of two spherical droplets in a constant applied temperature gradient along their line of centers, and the influence of the Marangoni number (Ca), initial distance between droplets (S0), and the radius ratio of the leading to trailing droplets (Λ) on the migration process is systematically studied. As Ma increases, the thermal wake behind the leading droplet strengthens, resulting in the transition of the droplet migration from coalescence to non-coalescence; and also, the final distance between droplets increases with Ma for the non-coalescence cases. The variation of S0 does not change the final state of the droplets although it has a direct impact on the migration process. In contrast, Λ can significantly influence the migration process of both droplets and their final state: at low Ma, decreasing Λ favors the coalescence of both droplets; at high Ma, the two droplets do not coalesce eventually but migrate with the same velocity for the small values of Λ, and decreasing Λ leads to a shorter equilibrium time and a faster migration velocity.

Thermocapillary Migration of Liquid Droplets Induced by a Unidirectional Thermal Gradient.

A liquid droplet placed on a nonuniformly heated solid surface will migrate from a high-temperature region to a low-temperature region. This study reports the development of a theoretical model and experimental investigation on the migration behavior of paraffin oil droplets induced by the unidirectional thermal gradient. Thin-film lubrication theory is employed to determine the migration velocity of droplets, and temperature dependence of viscosity is taken into account. Comparisons between experimental and numerical results are presented. An effective approach for estimating the thermocapillary migration velocity of droplets on lubrication is proposed.

Experimental research on thermocapillary migration of drops by using digital holographic interferometry

The thermocapillary migration of drops in a rectangular cell, with a heated top wall and a cooled bottom wall, was investigated experimentally on the ground. The rectangular test cell was 70 mm high, with a horizontal cross section of 40 mm × 40 mm. In the present experiment, 30 cSt silicon oil was used as the continuous phase, and a water–ethanol mixture was used as the drop phase, respectively. The drops ranged in size from 1.87 to 6.94 mm in diameter and were injected into the continuous phase, where the temperature gradients ranged from 0.193 to 0.484 °C mm−1. In order to measure the temperature distribution of the liquid, a digital holographic interferometry was used, which was non-contact, full-field, and in-situ. The holograms were recorded, and then the corresponding wrapped phase distributions images were numerically reconstructed. The temperature distribution of the continuous phase liquid in the cell had been obtained following the unwrapping. Also, through an algebra layer analysis, the temperature distribution around the drop during the thermocapillary migration was obtained. As a result, the drop was colder than the continuous phase liquid, and a thermal wake existed behind the drop. The influence of convective transport on the drop migration was also investigated for the Marangoni number in the range of 7–174. With the increasing of the Marangoni number, the dimensionless interface temperature difference decreased, which was caused by the convective transport enhanced results in the drop thermocapillary migration velocity becoming decreased. The data were compared with previous space experiments to explain the phenomena of the drop migration. Finally, with the increasing Marangoni numbers, the length of the thermal wake region increased, and the thermal wake region became extended.

Phase-field modeling of interfacial dynamics in emulsion flows: Nonequilibrium surface tension

A weakly nonlocal phase-field model is used to define the surface tension in liquid binary mixtures in terms of the composition gradient in the interfacial region so that, at equilibrium, it depends linearly on the characteristic length that defines the interfacial width. Contrary to previous works suggesting that the surface tension in a phase-field model is fixed, we define the surface tension for a curved interface and far-from-equilibrium conditions as the integral of the free energy excess (i.e., above the thermodynamic component of the free energy) across the interface profile in a direction parallel to the composition gradient. Consequently, the nonequilibrium surface tension can be widely different from its equilibrium value under dynamic conditions, while it reduces to its thermodynamic value for a flat interface at local equilibrium. In nonequilibrium conditions, the surface tension changes with time: during mixing, it decreases as the inverse square root of time, while in the linear regime of spinodal decomposition, it increases exponentially to its equilibrium value, as nonlinear effects saturate the exponential growth. In addition, since temperature gradients modify the steepness of the concentration profile in the interfacial region, they induce gradients in the nonequilibrium surface tension, leading to the Marangoni thermocapillary migration of an isolated drop. Similarly, Marangoni stresses are induced in a composition gradient, leading to diffusiophoresis. We also review results on the nonequilibrium surface tension for a wall-bound pendant drop near detachment, which help to explain a discrepancy between our numerically determined static contact angle dependence of the critical Bond number and its sharp-interface counterpart from a static stability analysis of equilibrium shapes after numerical integration of the Young-Laplace equation. Finally, we present new results from phase-field simulations of the motion of an isolated droplet down an incline in gravity, showing that dynamic contact angle hysteresis can be explained in terms of the nonequilibrium surface tension.

Thermocapillary migration of an isolated droplet and interaction of two droplets in zero gravity

Fluid transfer within a stagnant liquid presents a significant challenge in zero-gravity conditions due to the lack of buoyancy effects. This challenge can be overcome by the utilisation of the Marangoni effect, or more specifically thermocapillary migration. The thermocapillary migration of droplets is driven by temperature gradients within the multiphase system which bring about a surface tension gradient driving the flow from the cold to the hot region. The migration speed of the droplet is significantly impacted by the heat transfer both inside the droplet and in its surroundings.This paper presents the analysis of drop movement in a stagnant liquid using computational fluid dynamics (CFD). The commercial software package Ansys-Fluent v.13 [1] is used to solve the governing continuum conservation equations for two-phase flow using the Volume of Fluid (VOF) method to track the liquid/liquid interface in 2D domain. This approach has been shown to be a valuable tool for studying the phenomena of liquid–liquid interaction. A strong agreement has been found with experimental observations conducted in microgravity.The inherent velocity of drops has been found to decrease with increasing Marangoni number. This finding is in line with the previous space experiments of Xie et al. (2005) [2] and in contrast to the numerical results of Ma (1999) [3] using the same liquid for the droplet and the host liquid.Data obtained in the present numerical study has been used to derive an expression predicting the scaled droplet velocity as a function of Marangoni number. A numerical study of the interaction of two spherical droplets undergoing thermocapillary migration in microgravity is also presented. The temperature thrust from the leading droplet towards the trailing droplet was found to disturb its migration velocity, but the trailing droplet was found to have no influence on the migration of the leading droplet.

Micro-grooves design to modify the thermo-capillary migration of paraffin oil

Thermo-capillary migration is a phenomenon in which thermal gradients drive a liquid to flow without any external force. It is important to prevent the migration of liquid lubricants for moving mechanisms. This study aimed to investigate the influence of micro-grooves patterns on the migration of paraffin oil and obtain a design concept of micro-grooves patterns to obstruct the migration. Micro-grooves patterns with different orientations and geometric parameters were fabricated on the surface of SUS 316 stainless steel. Migration experiments of paraffin oils on each specimen were performed under various temperature gradients. The results indicated that micro-grooves patterns strongly modified the thermo-capillary migration. Micro-grooves perpendicular to the temperature gradient obstructed the migration effectively, while micro-grooves parallel to the temperature gradient accelerated the migration. The width and depth of micro-grooves influenced the migration behavior obviously, and the effects of these geometric parameters were discussed in detail.

Simulation on Thermocapillary-Driven Drop Coalescence by Hybrid Lattice Boltzmann Method

A hybrid two-phase model, incorporating lattice Boltzmann method (LBM) and finite difference method (FDM), was developed to investigate the coalescence of two drops during their thermocapillary migration. The lattice Boltzmann method with a multi-relaxation-time (MRT) collision model was applied to solve the flow field for incompressible binary fluids, and the method was implemented in an axisymmetric form. The deformation of the drop interface was captured with the phase-field theory, and the continuum surface force model (CSF) was adopted to introduce the surface tension, which depends on the temperature. Both phase-field equation and the energy equation were solved with the finite difference method. The effects of Marangoni number and Capillary numbers on the drop’s motion and coalescence were investigated.

Thermocapillary Migration of a Fluid Sphere in a Circular Tube

Thermocapillary Migration of a Fluid Sphere in a Circular Tube

Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow

In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface force (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter via Cahn–Hilliard equation which is solved in the frame work of LBE. The scalar convection–diffusion equation for temperature field is resolved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then thermocapillary migration of two/three dimensional deformable droplet are simulated. Numerical results show that the predictions of present LBE agreed with the analytical solution/other numerical results.