Thermocapillary Stresses Research Articles

Spreading of thin volatile liquid droplets on uniformly heated surfaces

We develop a mathematical model for the spreading of a thin volatile liquid droplet on a uniformly heated surface. The model accounts for the effects of surface tension, evaporation, thermocapillarity, gravity and disjoining pressure for both perfectly wetting and partially wetting liquids. Previous studies of non-isothermal spreading did not include the effects of disjoining pressure and therefore had to address the difficult issue of imposing proper boundary conditions at the contact line where the droplet surface touches the heated substrate. We avoid this difficulty by taking advantage of the fact that dry areas on the heated solid surface are typically covered by a microscopic adsorbed film where the disjoining pressure suppresses evaporation. We use a lubrication-type approach to derive a single partial differential equation capable of describing both the time-dependent macroscopic shape of the droplet and the microscopic adsorbed film; the contact line is then defined as the transition region between the two. In the framework of this model we find that both evaporation and thermocapillary stresses act to prevent surface-tension-driven spreading. Apparent contact angle, defined by the maximum interfacial slope in the contact-line region, decays in time as a droplet evaporates, but the rate of decay is different from that predicted in earlier studies of evaporating droplets. We attribute the difference to nonlinear coupling between different physical effects contributing to the value of the contact angle; previous studies used a linear superposition of these effects. We also discuss comparison of our results with experimental data available in the literature.

Effect of contact angle hysteresis on thermocapillary droplet actuation

Open microfluidic devices based on actuation techniques such as electrowetting, dielectrophoresis, or thermocapillary stresses require controlled motion of small liquid droplets on the surface of glass or silicon substrates. In this article we explore the physical mechanisms affecting thermocapillary migration of droplets generated by surface temperature gradients on the supporting substrate. Using a combination of experiment and modeling, we investigate the behavior of the threshold force required for droplet mobilization and the speed after depinning as a function of the droplet size, the applied thermal gradient and the liquid material parameters. The experimental results are well described by a hydrodynamic model based on earlier work by Ford and Nadim. The model describes the steady motion of a two-dimensional droplet driven by thermocapillary stresses including contact angle hysteresis. The results of this study highlight the critical role of chemical or mechanical hysteresis and the need to reduce this retentive force for minimizing power requirements in microfluidic devices.

Effects of the liquid polarity and the wall slip on the heat and mass transport characteristics of the micro-scale evaporating transition film

A mathematical model is developed to describe the micro-/nano-scale fluid flow and heat/mass transfer phenomena in an evaporating extended meniscus, focusing on the transition film region under non-isothermal interfacial conditions. The model incorporates polarity contributions to the working fluid field, a slip boundary condition on the solid wall, and thermocapillary stresses at the liquid–vapor interface. Two different disjoining pressure models, one polar and one non-polar, are considered for water as the working fluid so that the effect of polar interactions between the working fluid and solid surface can be exclusively examined on heat and mass transfer from the thin film. The polar effect is examined for the thin film established in a 20-μm diameter capillary pore. The effect of the slip boundary condition is separately examined for the thin film developed in a two-dimensional 20-μm slotted pore. The analytical results show that for a polar liquid, the transition region of the evaporating meniscus is longer than that of a non-polar liquid. In addition, the strong polar attraction with the solid wall acts to lower the evaporative heat transfer flux. The slip boundary condition, on the other hand, increases evaporative heat and mass flux and lowers the liquid pressure gradients and viscous drag at the wall. The slip effect shows a more pronounced enhancement as superheat increases. Another thing to note is that the slip effect of elongating the transition region can counteract the thermocapillary action of reducing the region and a potential delay of thermocapillary driven instability onset may be anticipated.

Viscous flow of a volatile liquid on an inclined heated surface.

We investigate the effects of evaporation on a gravity-driven flow of a viscous liquid on a heated solid surface. Vapor molecules are adsorbed on the dry areas of the solid and form a microscopic adsorbed film. The thickness of this film is calculated from the formulas for disjoining pressure and the principles of equilibrium thermodynamics. A lubrication-type approach is used to derive an evolution equation capable of describing both the macroscopic shape of the vapor-liquid interface and the adsorbed film on the vapor-solid interface. Under the conditions of negligible evaporation, the numerical solution of the evolution equation predicts translational motion and formation of capillary ridge, in agreement with previous investigations. Moderate evaporation is shown to slow down the flow and decrease the height of the capillary ridge, which implies a stabilizing effect of evaporation on the well-known instability observed in gravity-driven thin film flows. We also study the combined effects of evaporation and thermocapillary stresses and show that the latter act to reduce the velocity of the downward motion, but increase the height of the capillary ridge. Apparent contact angles are found from the solution and shown to increase with evaporation and contact line speed. For strong evaporation, steady state solutions are found such that evaporation balances the downward motion of the interface under the action of gravity.

Binary Fluid Mixture and Thermocapillary Effects on the Wetting Characteristics of a Heated Curved Meniscus

An investigation has been conducted into the interactions of binary fluid mixtures (pentane [C5H12] coolant and decane [C10H22] additive) and thermocapillary effects on a heated, evaporating meniscus formed in a vertical capillary pore system. The experimental results show that adding decane, the secondary fluid that creates the concentration gradient, actually decreases the meniscus height to a certain level, but did increase the sustainable temperature gradient for the liquid-vapor interface, so did the heat transfer rate, delaying the onset of meniscus instability. The results have demonstrated that interfacial thermocapillary stresses arising from liquid-vapor interfacial temperature gradients, which is known to degrade the ability of the liquid to wet the pore, can be counteracted by introducing naturally occurring concentration gradients associated with distillation in binary fluid mixtures. Also theoretical predictions are presented to determine the magnitudes of both the thermocapillary stresses and the distillation-driven capillary stresses, and to estimate the concentration gradients established as a result of the distillation in the heated pore.

Thermocapillary flow and rupture in films of molten metal on a substrate

We consider fluid flow in thin films of molten metal resulting from irradiation by a Gaussian laser beam. Surface tension gradients due to nonuniform heating induce a flow of the molten liquid away from the center of the irradiated area, leading to formation of dry areas on the substrate. We develop a mathematical model of the flow under the assumption of the large ratio of laser beam radius to film thickness. The model extends the standard lubrication-type analysis to include the highly nonlinear dependence of evaporative flux on local interfacial temperature, unsteady heat conduction in the substrate, and positive disjoining pressure due to unbalanced contributions from the kinetic energy of free electrons in the metal. The latter is proportional to the inverse square of the film thickness. We identify thermocapillary stresses as the main mechanism of rapid removal of liquid metal from the irradiated area. Characteristic times of the process, as well as shapes of the molten region surface, agree with experimental observations. We investigate rupture of the molten film and find two different rupture scenarios. The melt surface can either touch the substrate at a point (point rupture) or along a line at a certain radial distance away from the center of the irradiated area (ring rupture). Nondimensional criteria for these two mechanisms are identified. In particular, we show that positive disjoining pressure promotes ring rupture.

Thermocapillary suppression of the PlateauRayleigh instability: a model for long encapsulated liquid zones

A cylindrical liquid bridge is unstable when its length is longer than its circumference, the Plateau–Rayleigh limit. This capillary instability is modified by fluid motions adjacent to the interface, which can be induced by thermocapillary stress, among other means. A simple flow model with symmetry that mimics the situation in encapsulated floating zones is analysed. The interfacial balance equation is formulated as a bifurcation problem, appropriate when the flows are nearly rectilinear. This balance captures the competition between capillary stress and the flow-induced pressure. The fluid motions are shown to have a stabilizing effect; bridges much longer than the classical limit are stabilized. Numerical branch-tracing and the Lyapunov–Schmidt reduction methods provide the bifurcation structures of branching solutions. A normal-form analysis predicts standing-wave patterns due to mode–mode interaction. The model is proposed as an explanation for the extra long float zones observed in various spacelab experiments.

Thermocapillary migration of long bubbles in polygonal tubes. II. Experiments

We study experimentally the thermocapillary migration of a long gas bubble in a horizontal pipe of rectangular cross section. An imposed axial temperature gradient produces a gradient of surface tension leading to a steady migration of the bubble towards the hotter region. The bubble velocity is found to be independent of bubble length for sufficiently long bubbles, and to vary linearly with the temperature gradient. For pipes of small vertical dimension, gravity is negligible and the measurements of the bubble velocity are in good agreement with the zero gravity theory of Mazouchi and Homsy [Phys. Fluids 13, 1594 (2001)]. In larger pipes, gravity is no longer negligible and the bubble is observed to move faster than that theory predicts. A simple calculation taking into account the combined effect of buoyant rise and thermocapillary stress accounts for these results.

Dynamics of a vertically falling film in the presence of a first-order chemical reaction

The evolution of a vertically falling film in the presence of a simple first-order (exothermic or endothermic) chemical reaction is considered. The heat of reaction sets up surface tension gradients that induce thermocapillary stresses on the free-surface, thus affecting the evolution of the film. By using a long-wave expansion of the equations of motion and associated boundary conditions, we derive a nonlinear partial differential equation of the evolution type for the local film thickness. We demonstrate that, when the surface tension is an increasing function of temperature an exothermic reaction has a stabilizing effect on the free surface while an endothermic reaction is destabilizing. We construct bifurcation diagrams for permanent solitary waves and show that, in all cases the solution branches exhibit limit points and multiplicity with two branches, a lower branch and an upper branch. Time-dependent computations of the free-surface evolution equation show that the system always approaches a train of coherent structures that resemble the lower branch solitary waves. We also examine the absorption characteristics through the interface and we demonstrate that an endothermic reaction enhances absorption and mass transport. The opposite is true for an exothermic reaction.

Heating of compound samples in monoellipsoidal mirror furnaces

The heating of cylindrical compound samples in monoellipsoidal mirror furnaces is studied. A line-source model for the lamp (instead of a point source) is considered. The irradiation profiles obtained for this lamp model are analyzed; the case of defocussed lamps is studied analytically. The temperature field in the sample is obtained by means of a conduction-radiation model that includes the radiative exchange between the sample and the mirror. Graphite-silicon-graphite samples are considered; the melting of the silicon part and the surface temperature distribution in the melt are analyzed in dependence of lamp defocussing. By changing this parameter opposite surface temperature gradients in the melt can be obtained, which define opposite thermocapillary stresses.

Steady thermocapillary-buoyant flow in an unbounded liquid layer heated nonuniformly from above

Thermocapillary and buoyant flows induced by nonuniform heating of the free surface of a horizontally unbounded liquid layer over a cold solid bottom are studied numerically and by order of magnitude analyses for large Marangoni and Rayleigh numbers. The Prandtl number of the liquid is assumed to be of order unity or large, which are the cases of most interest in combustion. The asymptotic structures of plane and axisymmetric stationary flows are described qualitatively, showing that they consist of several horizontally spaced regions. Heat conduction and viscous forces are confined to thin boundary layers in a region around the heat source, while viscous forces extend to the whole liquid layer in a longer region where the flow is driven by the momentum imparted to the liquid by thermocapillary stresses around the source, in the case of plane thermocapillary flow; by this momentum and remaining thermocapillary stresses, in the case of axisymmetric thermocapillary flow; and by the horizontal gradient of a hydrostatic pressure distribution, in the case of buoyant flows. For large values of the Prandtl number, this region is followed by a region of viscosity-dominated flow which may be responsible for a large fraction of the heat loss to the bottom. A linear stability analysis of the surface boundary layer in the vicinity of the heat source gives values of the critical Marangoni number for the transition to oscillatory flow that are comparable to known experimental results.

The migration of a drop in a uniform temperature gradient at large Marangoni numbers

The steady thermocapillary motion of a spherical drop in a uniform temperature gradient is treated in the situation where convective transport of energy is predominant in the drop phase as well as in the continuous phase, i.e., when the Marangoni numbers are large. It is assumed that the Reynolds numbers in both phases are large as well; to leading order, the velocity fields are given by a potential flow field in the continuous phase and Hill’s vortex inside the drop. The migration velocity of the drop is obtained by equating the rate at which work is done by the thermocapillary stress to the rate of viscous dissipation of energy. The analysis deals with an asymptotic situation wherein convective transport of energy dominates with conduction playing a role only where essential. This leads to thin thermal boundary layers both outside and within the drop. The method of matched asymptotic expansions is employed to solve the conjugate heat transfer problem in the two phases. It is shown that the demand for energy within the drop, necessary to increase its temperature at a steady rate as it moves into warmer surroundings, results in a large temperature difference between the surface of the drop and its interior. The variation of temperature over the drop surface is large as well, and leads to a linear increase of the migration velocity of the drop with increasing Marangoni number. This result is strikingly different from that for the limiting case when the viscosity and thermal conductivity inside the drop become negligible compared to the corresponding properties in the continuous phase. This limit, which holds for a gas bubble, is recovered correctly from the analysis.

Thermocapillary migration of long bubbles in cylindrical capillary tubes

We consider the problem of the migration of a long bubble in a tube with a prescribed axial temperature gradient. The resulting thermocapillary stress moves the bubble toward hotter regions and we are interested in determining the speed of the bubble. Assuming small Peclet, Reynolds, Bond, and capillary numbers, Ca allows the uncoupling of the temperature field from the flow field, the use of creeping flow and lubrication theory, the assumption of axisymmetry, and the use of matched expansions in Ca, respectively. The structure of the solution is that of a constant thickness film bounded by constant curvature cap regions, with transition layers in between. A modified Landau–Levich equation governing the film profile in the transition regions is solved numerically, establishing the relationship between the unknown film thickness and the unknown bubble speed. A global mass conservation relation is then used to complete the solution and relate the bubble speed to the thermophysical properties. The solution is a function of a single dimensionless parameter, Δσ*=γTβa/σ where β is the temperature gradient, a the tube radius, σ the mean surface tension, and γT the temperature coefficient of surface tension. It is found that the speed increases with the temperature gradient as expected, and has a nonlinear dependence on Δσ* that is ultimately connected with the nonlinearity of the fluid dynamics in the transition regions.

Thermocapillary Effects on the Stability of a Heated, Curved Meniscus

An investigation of thermocapillary effects on heated menisci formed by volatile liquids in capillary pumped heat transfer devices has been conducted. This research was motivated by the importance of the evaporation process from porous or grooved media integral to the operation of capillary pumped heat transport devices such as heat pipes and capillary pumped loops. From analysis, a criteria was established which predicts the thermal conditions at which the destablizing influences of thermocapillary stresses near the contact line of a heated and evaporating meniscus cause the meniscus to become unstable. Experimentally, two different idealized models of capillary pumped phase change loops were investigated to assess the suitability of the predictions. Correspondence between theory and experiment was observed. Given the observed dry-out of the evaporator at higher heat inputs after the meniscus becomes unstable, the importance of predicting the conditions at the instability onset is made clear.

Thermocapillary Effects on the Wetting Characteristics of a Heated Curved Meniscus

An investigation of thermocapillary effects on a heated, evaporating meniscus, formed by a wetting liquid in a vertical capillary tube, has been conducted. Experiments were conducted to primarily observe how the wetting characteristics of the working fluid (pentane) are affected by the dynamics associated with the heating of and evaporation from a meniscus. The results have demonstrated that interfacial thermocapillary stresses arising from liquid-vapor interfacial temperature gradients can noticeably degrade the ability of the liquid to wet the pore.

Surfactant-induced retardation of the thermocapillary migration of a droplet

A neutrally buoyant droplet in a fluid possessing a temperature gradient migrates under the action of thermocapillarity. The drop pole in the high-temperature region has a reduced surface tension. The surface pulls away from this low-tension region, establishing a Marangoni stress which propels the droplet into the warmer fluid. Thermocapillary migration is retarded by the adsorption of surfactant: surfactant is swept to the trailing pole by surface convection, establishing a surfactant-induced Marangoni stress resisting the flow (Barton & Subramanian 1990).The impact of surfactant adsorption on drop thermocapillary motion is studied for two nonlinear adsorption frameworks in the sorption-controlled limit. The Langmuir adsorption framework accounts for the maximum surface concentration Γ′∞ that can be attained for monolayer adsorption; the Frumkin adsorption framework accounts for Γ′∞ and for non-ideal surfactant interactions. The compositional dependence of the surface tension alters both the thermocapillary stress which drives the flow and the surfactant-induced Marangoni stress which retards it. The competition between these stresses determines the terminal velocity U′, which is given by Young's velocity U′0 in the absence of surfactant adsorption. In the regime where adsorption–desorption and surface convection are of the same order, U′ initially decreases with surfactant concentration for the Langmuir model. A minimum is then attained, and U′ subsequently increases slightly with bulk concentration, but remains significantly less than U′0. For cohesive interactions in the Frumkin model, U′ decreases monotonically with surfactant concentration, asymptoting to a value less than the Langmuir velocity. For repulsive interactions, U′ is non-monotonic, initially decreasing with concentration, subsequently increasing for elevated concentrations. The implications of these results for using surfactants to control surface mobilities in thermocapillary migration are discussed.

The interaction of thermocapillary convection and low-frequency vibration in nearly-inviscid liquid bridges

The combined effect of thermocapillary stress and steady forcing due to vibrations of the disks in a model-half-zone axisymmetric liquid bridge is considered for low-viscosity liquids (i.e., with a large capillary Reynolds number), and low nondimensional vibration frequencies (i.e., small as compared to the capillary Reynolds number). An asymptotic model is derived for the slowly-varying streaming flow in the bulk (outside the oscillatory boundary layers) resulting from both effects that includes also buoyancy and other thermal expansion effects. This model is used to first analyze the steady streaming flow patterns in isothermal conditions and then to show that mechanical vibrations can annihilate almost completely thermocapillary flows of fairly large Reynolds numbers provided that: (i) the Prandtl number is appropriately small, (ii) both disks are vibrated, and (iii) the vibrating amplitudes, frequency and phases are appropriate (the counterbalancing effect depends crucially on the difference of the vibrating phases of both disks).

Droplet spindown in a high-temperature gas environment

The spindown and heating of a spherical droplet in an initially undisturbed infinite fluid is investigated by means of a numerical model based on finite-difference discretization techniques. The nonevaporating droplet enters the hot gas while rotating about a diameter and has no translational motion with respect to the suspending medium. Special attention is given to the transient secondary (nonrotational) motion developed as a result of shear interaction between the two phases. The results indicate that for droplet sizes and rotation frequencies representative of droplet combustion applications; i.e., Reynolds ∼ O(0.1), the secondary motion in both phases remains weak and heat transport is conduction-dominated. On the other hand, the secondary motion is strengthened with increased values of the rotational Reynolds number. The characteristic time for droplet spindown is found to be proportional to the square of the droplet radius. The results also show that the rotational deceleration time is of the same order of magnitude with the translational response time of the droplet. Finally, the thermocapillary stress effects on fluid dynamics and heat transfer are investigated in this flow configuration.

Thermocapillary flow effects on convective droplet evaporation

The thermocapillary convection effect on the dynamic behavior of a vaporizing-convecting hydrocarbon droplet is investigated numerically in a laminar, axisymmetric environment. The model predicts that surface-tension gradients along the gas-liquid interface strengthen droplet internal circulation and result in shorter evaporation lifetimes. The relative effect of thermocapillary stresses on overall droplet evaporation rates is nearly independent of Reynolds number. While Nusselt numbers on the droplet surface are little affected, a significant reduction of drag coefficient is observed when thermocapillary convection is taken into account. A substantial enhancement of the corresponding Sherwood numbers is also predicted. The results indicate that the Marangoni stresses cause temporal flow oscillations along the droplet interface at early stages of the droplet evaporation lifetime.

Evaporation from an extended meniscus for nonisothermal interfacial conditions

A study is presented to determine the effects of evaporation from the thin film region of a liquid-vapor meniscus within the micropores of a heat pipe porous or grooved wick on the interfacial shape, temperature distribution, and pressure distribution. Wayner's theoretical treatment of evaporating thin films is applied to the problem of an evaporating extended meniscus within circular or slotted pores. In this application of Wayner's model, the nondimensional momentum equation is uniquely scaled in terms of the capillary number, to justify the use of the static meniscus curvature as a boundary condition for the extended meniscus profile even for the dynamic evaporating conditions studied herein. This boundary condition for small capillary numbers is consistent with the observation of the nearly constant meniscus curvature with evaporation rate that the thin film must asymptotically approach. From these basic tenets, the mechanical and thermal behavior of a stably, evaporating, nonisothermal extended meniscus is predicted. The resulting predictions are qualitatively consistent with the experimental findings of Wayner and the previous theoretical studies. They further support the claims that for the cases studied herein, both thermocapillary stresses and vapor recoil stresses at the liquid-vapor interface are negligible. However, scaling arguments are presented that identify the conditions necessary for these terms to be important.