Vladimir S. Ajaev

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.

Dynamics of volatile liquid droplets on heated surfaces: theory versus experiment

We consider the evaporation of volatile liquid droplets deposited on a heated substrate in a pure saturated vapour environment. A mathematical model is developed that incorporates the effects of surface tension, evaporation, thermocapillarity, gravity, disjoining pressure, as well as unsteady heat conduction in the solid substrate. The apparent contact line is treated mathematically as a transition region between the macroscopic droplet shape and the adsorbed film of liquid on the heated substrate. Theoretical parametric studies are conducted to clarify the effects of thermocapillarity and wetting properties on the droplet dynamics. An experimental study is conducted in a closed container with de-ionized water droplets on a stainless steel foil heated by an electric current. The interface shapes are recorded together with the temperature profiles under the droplets, measured using thermochromic liquid crystals. Experiment and theory are in very good agreement as long as the conditions of applicability of our lubrication-type mathematical model are satisfied.

Static and dynamic contact angles of evaporating liquids on heated surfaces

We studied both static and dynamic values of the apparent contact angle for gravity-driven flow of a volatile liquid down a heated inclined plane. The apparent contact line is modeled as the transition region between the macroscopic film and ultra-thin adsorbed film dominated by disjoining pressure effects. Four commonly used disjoining pressure models are investigated. The static contact angle is shown to increase with heater temperature, in qualitative agreement with experimental observations. The angle is less sensitive to the details of the disjoining pressure curves than in the isothermal regime. A generalization of the classical Frumkin-Derjaguin theory is proposed to explain this observation. The dynamic contact angle follows the Tanners law remarkably well over a range of evaporation conditions. However, deviations from the predictions based on the Tanners law are found when interface shape changes rapidly in response to rapid changes of the heater temperature. The Marangoni stresses are shown to result in increase of the values of apparent contact angles.

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 ex...

Steady flow and evaporation of a volatile liquid in a wedge

We develop a lubrication-type model of a liquid flow in a wedge in the limit of small capillary numbers and negligible gravity. Liquid flows under the action of capillary pressure gradients and thermocapillary stresses, and evaporates due to heating from the solid walls on which a constant axial temperature gradient is imposed. Steady vapor-liquid interface shapes are found for different wedge angles and material properties of the liquid. In the limit of weak evaporation (e.g., in the adiabatic region of a heat pipe) and negligible Marangoni number, the flow rate is the same in all cross sections and can be controlled by changing the wedge angle. We find the wedge angle that results in the maximum value of the flow rate for a given contact angle. For finite evaporation rates, both the flow rate and the amount of liquid in each cross section along the wedge decrease until the point of dry-out is reached. The location of the dry-out point is studied as a function of evaporation conditions. Somewhat counteri...

The effect of evaporation on fingering instabilities

We investigate the flow of evaporating thin films of viscous liquid on inclined solid substrates under the influence of gravity. A lubrication-type approach is used to develop a three-dimensional model of the flow including physical effects such as capillarity, gravity, Marangoni stresses, disjoining pressure, and evaporation. Numerical simulations are then carried out based on the model. The effect of evaporation on the so-called fingering instability that develops along the contact line in the transverse direction of the flow is studied. It is found that evaporation acts to suppress the instability if the evaporation number, a nondimensional measure of the mass flow rate across the interface, is above a critical value. The critical value decreases as the inclination angle is decreased. For the values of evaporation number below the critical one, the fingers grow initially but then saturate at a length that depends on the evaporation conditions. It is also shown that thermocapillarity acts to enhance the...

HEAT TRANSFER, PHASE CHANGE, AND THERMOCAPILLARY FLOW IN FILMS OF MOLTEN METAL ON A SUBSTRATE

A mathematical model has been developed for heat transfer and fluid flow in thin films of molten metal during nanosecond pulsed laser irradiation. Heat conduction in the substrate is modeled using the finite-difference approach, while description of heat transfer and viscous flow in the film is based on the assumption of the large ratio of laser beam radius to film thickness and involves numerical solution of a partial differential equation for the thickness. The model includes the highly nonlinear dependence of evaporative flux on local interfacial temperature and positive disjoining pressure due to free electrons in the metal. Thermo-capillary stresses which result from radially nonuniform heating are identified as the main mechanism of removal of liquid metal from the irradiated area. Characteristic times of the process, as well as shapes of the molten surface, agree with experimental observations.

Application of Floquet theory to the stability of liquid films on structured surfaces

We consider instability of a liquid film on a substrate structured by an array of gas-filled grooves. The instability is driven by disjoining pressure, while the effect of structuring on viscous flow in the film is modeled by a square-wave variation of the slip length along the substrate. Linear stability criteria are established analytically using Floquet theory and compared with the predictions of a straightforward numerical approach, all in the framework of a lubrication-type model. Then, stability is analyzed for a more general model based on Stokes flow approximation; validity of the lubrication-type approach is discussed. The structuring is found to enhance the instability for a wide range of conditions. Resonant interaction between the interfacial deformations and the substrate structuring pattern leads to discontinuities in the dispersion curves, a situation analogous to appearance of gaps in the energy spectra seen in the applications of Floquet theory in solid state physics.

Boundary-integral simulations of containerless solidification

We carry out boundary-integral simulations of a two-dimensional liquid droplet surrounded by air and solidified from a cool point on the boundary. There are three interfaces in the problem: solid-liquid, air-liquid, and air-solid. All three evolve in time in such a way that certain tri-junction conditions must be satisfied. Our numerical method describes the quasi-steady evolution of the interfaces in the limit of zero surface energy on the solidification front. A new iterative technique is developed to describe the interface evolution when mass and total energy are conserved and the local trijunction conditions are satisfied at every instant in time. A method is also developed for efficient numerical integration over the interfaces by taking advantage of analytical formulas for Greens functions. We start the simulations by studying the case of equal densities of the solid and liquid. This allows us to verify the numerical method and obtain some estimates of the speed of the solidification front. Solid-liquid interface flattening is observed at the intermediate stages of solidification. When the densities of the two phases are different, elongated solidified particles are observed when the solid density is smaller than the liquid density. At the final stages of solidification, a corner is formed in agreement with observations in related experiments.

On drag reduction in a two-phase flow

Bubbles collected on a local hydrophobic surface with nanocoating in a two-phase flow in a minichannel have been detected experimentally. It has been proposed to use the effect of concentration of gas bubbles on hydrophobic segments of the surface of the channel with contrast wettability for ensuring drag reduction. A two-dimensional flow model with the Navier slip condition in the region of the bubble layer gives criteria of drag reduction, depending on the slip length, dimension of bubbles, and dimension of the segment with nanocoating. The presence of the bubble layer on half of the surface of the channel can increase the flow rate of a liquid flowing through the channel by 40% at a fixed pressure gradient.