Thermocapillary Migration Research Articles

Thermocapillary migration of a compound drop inside a spherical cavity

Thermocapillary migration of a compound drop inside a spherical cavity

Thermocapillary migration of a compound drop in an arbitrary viscous flow

The thermocapillary migration of a concentric compound drop in an arbitrary viscous flow under the consideration of negligible Reynolds number is investigated. The thermocapillary effect refers to the migration of a drop under the influence of a temperature gradient. The thermal and hydrodynamic problems are examined. The thermal field is governed by the heat conduction equation whereas the hydrodynamic fluid velocities are governed by the linearized Navier–Stokes equations. Presence of temperature gradient results in variation of the interfacial tension which is assumed to depend on temperature linearly. Variation of interfacial gradient leads to the coupling of the hydrodynamic problem with the thermal problem through the boundary condition. A complete general solution of Stokes equations is utilized to obtain closed-form expressions for the velocity vector and pressure. The hydrodynamic forces acting on the compound drop are obtained and expressed in terms of Fax́en’s law. Some important asymptotic limiting cases of hydrodynamic drag are also derived. The hydrodynamic drag for cases of uniform flow, shear flow, and heat source with the known ambient flow are derived and it is found that in the case of shear flow, the hydrodynamic drag is contributed only by the thermal component and the shear flow has no effect on it. The obtained results for drag and torque in the limiting cases are in agreement with the existing results in the literature. Furthermore, the migration velocity of the compound drop is obtained by equating the hydrodynamic drag force to zero. The results obtained for migration velocity are explained with the aid of graphs. The migration velocity is found to be a monotonic function of the Marangoni number and the radius of the innermost drop.

A reduction-consistent phase field model for non-isothermal multiphase flows of N immiscible incompressible fluids

In this study, we developed a reduction-consistent phase field model for non-isothermal incompressible N-phase flows. The model is based on the high-order spectral element method. To account for thermocapillary effects, we reformulated the continuum surface force model specifically for N-phase flows. In order to enhance computational efficiency, time-independent coefficient matrices for all variables involved were reconstructed, and an unsymmetrized multifrontal LU factorization was employed to solve the linear algebraic equations, which is derived by discretization. To verify the model effectiveness in calculating surface tension and describing three-phase interfacial dynamics, we conducted several experiments, including the stationary two-droplet example, the three-phase droplet spreading, and the equilibrium morphology of double emulsion droplet. Through these experiments, both the reduction-consistency and robustness of our model were demonstrated. Moreover, we validated the proposed model's applicability to non-isothermal three-phase flows by investigating the thermocapillary migration of single droplet and two droplets of different phases. Notably, we explored the thermocapillary migration of three-phase double droplets, focusing particularly on how the encapsulation process affects overall thermocapillary motion. Our findings indicate that the interaction between the two vortices near the interfaces of the two droplets strongly influences their migration behavior.

A model for thermocapillary driven motion of liquid drops on an inclined plane

Thermocapillary drop migration is driven by surface tension gradient. On an inclined plane, gravity acts as an additional force that can have a promoting or resisting effect depending on the thermal gradient direction. The present 2-D model extends Brochard and Chebbi models for thermocapillary migration of small drops on horizontal plates by accounting for Marangoni and Poiseuille flow components, while including the effect of gravity, acting in the case of inclines. The present treatment is based on fundamentals of conservation of mass and momentum transport, and uses the lubrication approximation and extended boundary conditions at the receding and advancing contact lines. A model for the dynamics of migration is provided, and an analytical solution is derived for drop migration dynamics and the critical angle of inclination.

Thermocapillary migration of a droplet on a substrate with wettability difference: A comparison between slip and precursor film models in three dimensions

Moving contact line dynamics calculations include two models: precursor film models and slip models. The lubrication approximation method is used to establish a three-dimensional mathematical model to analyze the droplet thermocapillary migration behavior on a non-uniformly heated solid substrate with a wettability track. The contact line dynamics in the slip model and the disjoining pressure effect in the precursor model are proposed to regulate the substrate wettability. Both models are numerically implemented to investigate droplet spreading for three cases: free spreading on an isothermal substrate, thermocapillary migration on a uniform wettability substrate, and thermocapillary migration on a wettability-confined track. For the case of free spreading on an isothermal substrate, the three-dimensional results of the slip and precursor contact line models are essentially consistent with two-dimensional slip model results. For the case of thermocapillary migration on a uniform wettability substrate, the results of the two models essentially agree with the experimental results. Decreasing the thermal gradient reduces the discrepancies between the two models that result from the coordinate transformation method used in the slip model, which reduces the contact angles measured in the y-direction and enlarges the advancing contact angle in the migration direction. For the case of thermocapillary migration on a wettability-confined track, the slip model gradually shows a “dynamic-pinning” behavior with increasing equilibrium contact angle in the hydrophobic region. By contrast, the precursor film model maintains a stationary pinning behavior but separates a residual liquid outside the track. The precursor film model is preferred over the slip model in lubrication approximations for three-dimensional fluids when calculating complex moving contact dynamics caused by wettability differences. However, the precursor film model must be further optimized to prevent numerical instability.

INFLUENCE OF TEMPERATURE GRADIENTS AND FLUID VIBRATIONS ON THE THERMOCAPILLARY DROPLET BEHAVIOR IN A ROTATING CYLINDER

In this paper, a computational fluid dynamics (CFD) approach is used to analyze and numerically present the droplet movement in both stagnant and vibrated liquid in a rotating cylinder. The Ansys Fluent commercial software package is used to solve the governing continuum conservation equations for two-phase flow. The volume of fluid (VOF) method is used to track the liquid/liquid interface. The inherent velocity of droplets decreases with increasing Marangoni number, in agreement with previous space onboard experimental findings that have been documented in literature. Complex behaviors of droplets in zero gravity were observed due to small host fluid vibration and rotational motion, which are neglected in the presence of gravity. Therefore, the impact of varying vibration amplitude as well as angular velocity on bubble migration is presented. It has been found that thermocapillary migration slows down with high vibration amplitude (A), when the fluid is vibrating. For a static liquid inside a rotating cylinder, the velocity of thermocapillary droplet migration decreases with the angular velocity increase. Temperature gradient (∇T) increment also increases the droplet's migration speed inside the vibrating fluid and a rotating cylinder of different Marangoni numbers (Ma<sub>T</sub>). A flow pattern figure that illustrates the actual flow behind each modification is included with every result.

Thermocapillary migration of a drop with a thermally conducting stagnant cap

Hypothesis The thermocapillary migration of a spherical drop with a stagnant cap in the presence of a constant applied temperature gradient can be strongly affected by the finite thermal conductivity of the stagnant cap.Numerics The heat conduction of the stagnant cap is analytically modeled. The effects of the additional interfacial stresses generated by the disturbances to the local temperature field due to the presence of the cap at the fluid-fluid interface and the corresponding velocity of migration of the drop are evaluated by solving for the temperature and hydrodynamic field equations in and around the drop. An asymptotic model is derived to predict the terminal velocity in the presence of an infinitely conducting stagnant cap.Findings The effects of the surface conductivity and size of the stagnation region alongside the bulk thermal conductivities and viscosities of the drop and surrounding media are evaluated. The terminal velocity of the drop is shown to have a monotonic dependence on the conductivity of the stagnant cap. The bounds to the terminal velocity increment due to the stagnant cap are derived. These bounds can be of significance to multiphysics problems involving particle laden drops, Pickering emulsions and other multi-phase technologies where the conductivity of the surface adsorbents is non-negligible.

Phase-field-based lattice Boltzmann model for simulating thermocapillary flows.

This paper proposes a simple and accurate lattice Boltzmann model for simulating thermocapillary flows, which can deal with the contrast between thermodynamic parameters. In this model, two lattice Boltzmann equationsare utilized to solve the conservative Allen-Cahn equationand the incompressible Navier-Stokes equations, while another lattice Boltzmann equationis used for solving the temperature field, where the collision term is delicately designed such that the influence of the contrast between thermodynamic parameters is incorporated. In contrast to the previous lattice Boltzmann models for thermocapillary flows, the most distinct feature of the current model is that the forcing term used in the present thermal lattice Boltzmann equationis not needed to calculate space derivatives of the heat capacitance or the order parameter, making the scheme much more straightforward and able to retain the main merits of the lattice Boltzmann method. The developed model is first validated by considering the thermocapillary flows in a heated microchannel with two superimposed planar fluids. It is then used to simulate the thermocapillary migration of a two-dimensional deformable droplet, and its accuracy is consistent with the theoretical prediction when the Marangoni number approaches zero. Finally, we numerically study the motion of two recalcitrant bubbles in a two-dimensional channel where the relationship between surface tension and temperature is assumed to be a parabolic function. It is observed that due to the competition between the inertia and thermal effects, the bubbles can move against the liquid's bulk motion and towards areas with low surface tension.

The Mechanism of Droplet Thermocapillary Migration Coupled with Multi-Physical Fields

In this paper, the coupling effect of multiphysical fields of droplet migration is deeply studied by constructing a physical model of droplet migration with multiphysical fields. Digital holographic interferometry and particle image velocimetry are used to simultaneously measure the temperature and velocity fields of the mother liquor in the process of droplet migration for the first time. Due to the advancements of measuring, the zero-velocity region is also in the region where the thermal wake appears, four vortexes appear in the droplet migration and the off-axis behavior of double-droplet migration is found. The aim of this work is to analyze the coupling relationship of multiphysical fields, so as to reveal the physical laws of thermocapillary migration of single droplet and multiple droplets with the same phase and heterophase and to study the driving mechanism of the thermocapillary force and the flow of the mother liquor.

A fully explicit incompressible smoothed particle hydrodynamics approach for modeling transient heat transfer and thermo-capillary flows

This work presents a Lagrangian meshless method for modeling transient heat transfer and thermo-capillary effects at the interface of two-phase incompressible fluids. To simulate the thermo-capillary effect, we add the Marangoni force to the continuum surface force (CSF) model, by considering the forces caused by the surface tension gradient which acts tangentially to the interface position. In the current study, we extend our previously proposed model which uses a fully explicit incompressible smoothed particle hydrodynamics (EISPH) approach along with corrected SPH and VKF kernel function, to obtain stable and accurate results in non-isothermal single and multi-phase problems.The model is validated for several test cases, including transient heat conduction, natural convection heat transfer, and thermo-capillary droplet migration, against conventional numerical methods. The validated model is used to study the effects of several dimensionless parameters on thermo-capillary droplet migration. The results show that the proposed EISPH method can model accurately complex heat transfer problems such as thermo-capillary induced motion.

Dynamics of Thermocapillary-Driven Motion of Liquid Drops.

The thermocapillary migration of a drop placed on a solid plate is examined. The Brochard model using the lubrication approximation provides both Marangoni and Poiseuille flow components. The present 2D model extends Brochard analysis and provides a solution for the dynamics of drop migration using extended boundary conditions at the advancing and receding contact lines to account for both Marangoni and Poiseuille flow components, derived approximate drop profiles, and conservation of mass. The model is analytical, and the results are presented in a dimensionless form. The effects of the temperature gradient, surface tension coefficient to surface tension ratio, liquid viscosity, and static advancing and receding contact angles on migration dynamics are analyzed.

A linear mass and energy conserving numerical scheme for two-phase flows with thermocapillary effects

In this paper, a thermodynamically consistent phase-field model is employed to simulate the thermocapillary migration of a droplet. The model equations consist of a general Navier–Stokes equation for the two-phase flows, a Cahn–Hilliard equation for the diffuse interface, and a heat equation, and meanwhile satisfy the balance laws of mass, energy and entropy. In particular, the total energy of the system includes kinetic energy, potential energy and internal energy, which leads to a highly coupled and nonlinear equation system. We therefore develop a linear mass and energy conserving, semi-decoupled numerical method for the numerical simulations. As the model contains a heat (energy) equation, a simple error term introduced by the temporal discretization of the momentum equation can be absorbed into the heat equation, such that the numerical solutions satisfy the conservation laws of mass and energy exactly at the temporal discrete level. Several numerical tests are carried out to validate our numerical method.

Thermocapillary migration of a compound droplet on a substrate

We present a numerical study of the thermal migration of a compound droplet attached to a horizontal substrate. The different constant temperatures are applied to the left and right boundaries of the computational domain. The compound droplet contains one inner core. The problem is solved by using a front-tracking technique associated with the finite difference method. The investigated parameters include the radius ratio (Rio) of the inner droplet and the equivalent radius of the outer droplet (Rio varied in the range of 0.2 to 0.7), the viscosity ratio (μom) of the outer to middle fluids (μom varied in the range of 0.1 to 2.4), and the static contact angle θe (θe varied in the range of 60° to 120°). The results reveal that the inner droplet drives the compound droplet towards the hot region as its radius is sufficiently large (Rio≥ 0.6). The compound droplet moves towards the cold region as the value of Rio is less than 0.6. Increasing the viscosity of the outer fluid (by increasing μom) or θe causes the compound droplet to move towards the hot region. The trajectory of the centroid of the inner droplet is also investigated.

On the migration of a droplet driven by thermocapillary action in a wettability-confined track

We report a study of the thermocapillary migration of droplets under a radial thermal gradient and in a wettability-confined track. A three-dimensional mathematical model is established based on the lubrication approximation. By considering the contact-line dynamics, a method for determining the velocity of the contact line in different directions is proposed for a three-dimensional droplet. Numerical simulations are performed to investigate the variations in the droplet profile, contact angle, and contact line. Three substrate-wettability cases are considered: uniform, temperature-dependent, and track-dependent wettability. The results show that when the substrate wettability is uniform, the droplet height initially decreases rapidly, and its center becomes concave and then gradually evolves into a ring-like morphology. Reducing the temperature sensitivity of the liquid–gas interfacial tension or increasing the temperature sensitivity of the liquid–solid interfacial tension decreases the equilibrium contact angle and accelerates thermocapillary migration. When a droplet spreads in a wettability-confined track, a wave-like peak is formed on each side of the droplet along the track direction until it finally separates into two distinct parts. As the track width is decreased, the time taken for a droplet to split into two smaller droplets advances, and the separation time presents a linear relationship with the track width.

A hybrid axisymmetric conservative phase-field lattice Boltzmann method for hollow droplet migration

A hybrid axisymmetric conservative phase-field lattice Boltzmann method is applied to investigate the influence of Marangoni number (Ma), density ratio (ρr), and radius ratio (Rr) on thermocapillary migration of a deformable hollow droplet with difference in variable fluid properties, where ρr (Rr) is the density (radius) ratio of the hollow part of the droplet. The isotherms show that heat transfer around the hollow droplet is changed from conduction to convection with the increase in Ma. However, the temperature gradient across the hollow droplet decreases with Ma, which induces a small magnitude of migration velocity. When ρr is increased, the isotherms are accumulated around the hollow droplet front with a large temperature gradient, which enhances the hollow droplet migration, while the migration velocity is decreased with the increase in Rr. It is observed that thermocapillary migration of the hollow droplet finally becomes a pure droplet with the influence of aforementioned parameters, and it experiences interface breaking and coalescing, which causes a large transient variation in migration velocity. The magnitude of this transient variation in migration velocity is not obviously affected by Ma but significantly affected by ρr and Rr. The measured evolution of d (the dimensionless distance between inner and outer fronts of the hollow droplet) demonstrates that ρr has a significant influence on the reduction rate of d in comparison with the influence of Ma and Rr. Similar influences on the relative migration velocity between the fluid of the hollow part inside the droplet and the sealed fluid of the droplet are observed.

Thermocapillary flow‐induced core release from double‐emulsion droplets in microchannels

AbstractThe use of double emulsions (DEs), which represent colloidal structures composed of droplets nested within droplets, can provide for unparallel droplet manipulation in droplet‐based microfluidic technology due to their unique core–shell structures. The controlled release of cores in DEs is of particular interest. However, this process remains poorly explored. In this work, the thermocapillary flow induced by a temperature gradient is used as a driving force to control the core release and the impacts of different linear temperature gradients, core diameters, shell diameter, and core/shell diameter ratios on the thermocapillary flow and core release characteristics of DE droplets consisting of a water‐in‐n‐hexadecane‐in‐water system within a cylindrical microchannel are investigated. Most of the core and shell diameter conditions considered result in a double‐core release process, where the inner droplet volume is partially ejected before the remaining core is rewrapped by the outer droplet, and the remaining inner droplet volume is ejected later during a second core release event. However, relatively small core diameters of 50 and 75 μm produce conditions where the full inner droplet volume is ejected during a single‐core release process. In addition, we provide empirical relationships for accurately determining the time at which core release initially occurs under given DE parameters as well as for precisely determining whether the applied conditions will lead to single‐ or double‐core release processes. Therefore, the results of this study provide insights enabling the development of accurate inner droplet release technologies under thermocapillary migration.

Improved three-dimensional multiple-relaxation-time color-gradient lattice Boltzmann finite-difference model for thermocapillary flows

This study is devoted to developing a color-gradient lattice Boltzmann model capable of simulating thermocapillary flows with variable properties. To achieve the purpose, some modified work is conducted. The equilibrium distribution function for density is modified in the developed model. The scheme of multiple-relaxation-time is applied to deal with the single-phase and perturbation collision operators to enhance the algorithm stability, and a simple correction term is incorporated into the single-phase operator. In addition, the finite-difference method is adopted to solve the temperature field. The developed model is first used to simulate the layered two-phase flow in a horizontal channel to test the density ratio that can be achieved without temperature effect. Then two classical thermocapillary flow problems of thermocapillary-driven flow in a heated microchannel and thermocapillary migration of a deformable droplet or bubble are simulated, and the model is proven to successfully simulate the thermocapillary flows with density ratios up to 10.

Numerical study of thermocapillary migration of a droplet on an oleophilic track

The thermocapillary migration dynamics of a droplet with a non-circular footprint on an oleophilic track is studied numerically. Three-dimensional simulations showed that reducing the width of the oleophilic track improves the droplet migration velocity only upto a specific track width when compared to the change in velocity of a droplet with circular footprint. Further decrease in track width is seen to depreciate the enhancement in migration velocity. The spread length of the droplet on the oleophilic track is also observed to have a similar variation with track width. The two trends are related because the temperature difference in the droplet, which scales with spread length, determines the driving force governing the migration velocity characteristics. Further, a parametric study is performed to assess the migration behaviour on oleophilic tracks with change in substrate temperature gradient, droplet volume and contact angles within and outside the track. The influence of change in these parameters on the variation in enhancement in spread length and corresponding track width is also investigated. The study identified trends in the minimum track width that may be used to improve migration characteristics, as well as the order of enhancement that can be noticed with changes in each parameter.

Thermocapillary Bubble Oscillations and Migration in a Vibrating Cylinder in a Zero-Gravity Environment

Thermocapillary Bubble Oscillations and Migration in a Vibrating Cylinder in a Zero-Gravity Environment

Surface wave instability in the thermocapillary migration of a flat droplet

The surface wave instability (SWI) of thermocapillary migration is examined by linear stability analysis for a droplet on a unidirectional heated plane. Both a Newtonian fluid and an Oldroyd-B fluid are considered. The droplet, flattened by gravity, is susceptible to two kinds of instabilities: convective instability (CI), which is independent of surface deformation; and SWI, which occurs only when the Galileo number and the surface-tension number are not too large. The wavenumber of the latter is much smaller than that of the former, while the reverse is true for the wave speed. SWI is found at different Prandtl numbers (Pr), while its mode includes streamwise and oblique waves. Energy analysis suggests that the energy of the long-wave mode comes from the shear stress induced by the surface deformation, the energy source for the mode with finite wavelength is the work done by Marangoni forces, while the energy from the basic flow is only important in some cases at small Pr. For the Oldroyd-B fluid, a small elasticity slightly changes the critical Marangoni number of SWI, while larger elasticity changes the preferred mode from SWI to CI. The instability mechanism is discussed and comparisons are made with experimental results.