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The evaporation process of drops on a solid surface is widely applied to daily life and industrial fields. Both contact angle hysteresis and the sensitivity of gas-liquid interfacial tension to temperature are important factors affecting the drop evaporation reflected in the contact line and contact angle. To investigate the internal mechanism, according to the lubrication theory and slip boundary conditions, we establish a mathematical model of the drop evaporation on a uniformly heated solid wall with considering the effect of contact angle hysteresis. This model is numerically solved by using a coordinate transformation method and Freefem++14.3, a highly efficient solver. The accuracy of the numerical calculation method is verified by comparing the numerical results with experimental results, and the grid independence is validated. The effect of contact angle hysteresis on the dynamics of evaporating drops is discussed, and the evaporation characteristics of drops with different tension sensitivities of the air-liquid interface to temperature are further investigated. The results show that the contact angle hysteresis has an apparent influence on the drop evaporation process which includes drop spreading stage, contact line pinning stage, and depinning stage. In the drop spreading stage, the increase in the hysteresis angle shortens the spreading time, and reduces the spreading velocity and radius, while in the contact line pinning stage, the pinning time is prolonged and the reduction of drop mass is significantly increased with hysteresis angle increasing. In the contact line depinning stage, the contact angle hysteresis reduces the contact angle, and a flatter shape emerges, thereby enhancing the ability to transfer heat and accelerating evaporation as well as shortening the depinning time. In addition, the large hysteresis angle leads to a large advancing contact angle and a small receding contact angle. The reduction in receding contact angle is more notable than the increment of advancing contact angle. The temperature sensitivity coefficient of the gas-liquid interfacial tension can be increased by reducing the receding contact angle, thereby improving the wettability of the drops on the wall enhancing the heat transfer and accelerating the evaporation. Regulating the contact angle hysteresis and the sensitivity of the interfacial tension to temperature can realize the manipulation of the drop movement, thus controlling the evaporation process.

The dynamics of evaporating sessile drop on a uniformly heated, horizontal, solid substrate is considered. On the basis of lubrication theory and Navier slip condition, an evolution equation for the height of the two-dimensional drop is established. The numerical results show that the drop evolution is governed by capillary force, gravity, thermal capillary force and evaporation. Gravity exerts a promoting effect on drop spreading, while capillary force and thermal capillary force inhibit drop spreading. The typical dynamic features including contact line pinning or partial pinning modes during the drop evaporation are illustrated by changing the temperature-sensitive coefficient in the present model, and the drop lifetime of contact pinning mode is found to be shorter than that of contact line partial pinning mode. Under the same temperature-sensitive coefficient of three interfaces, the drop evolution is indicated with three typical stages: 1) spreading stage, 2) contact line pinning stage, and 3) both contact line and contact angle decreasing stage. As interface tension of liquid-gas or liquid-solid is more sensitive to temperature, the drop evolution is divided into two typical stages: 1) spreading stage and 2) contact line pinning stage. The equilibrium contact angle tends to be smaller and the substrate wettability is improved, leading to the increased spreading speed, the prolonged time of the contact line to reach pinning: the faster the evaporation rate, the shorter the lifetime of drop is. Additionally, the same effect of sensitivity of liquid-gas and liquid-solid interface tension to temperature on the wettability of substrate is observed. When the interface tension of solid-gas is more sensitive to temperature, the drop evolution is characterized in four typical stages: 1) spreading stage, 2) contact line pinning stage, 3) contact line de-pinning and constant contact angle stage, and 4) both contact line and contact angle decreasing stage. The equilibrium contact angle tends to be greater and the substrate wettability is deteriorated, leading the spreading speed to decrease. Hence, it is more effective to manipulate the drop movement in the presence of evaporation by regulating the temperature-sensitive coefficient of the solid-gas interface.

The aim of the present paper is to investigate the gravity-driven draining process containing insoluble surfactants, with the coupling effects of surface elasticity and disjoining pressure taken into consideration. A set of evolution equations including liquid film thickness, surface velocity and surfactant concentration, is established based on the lubrication theory. Assuming that the top of the liquid film is attached to the wireframe and the bottom is connected to the reservoir, the drainage stability is simulated with the FreeFem software. The characteristics of film evolution under the coupled effects of surface elasticity and disjoining pressure are examined, respectively. The simulated results show that the surface elasticity and the disjoining pressure have significant influences on the vertical thin film draining process. Under the effect of the surface elasticity alone, the initial film thickness increases with the elasticity increasing and the black film only forms on the top of the liquid film, but cannot stably exist and breaks quickly. The addition of the surface elasticity can increase the liquid film thickness and the drainage time, reduce the surface velocity, and rigidify the interface. When the disjoining pressure is applied merely, the surfactant flows into the reservoir continuously; hardly can the liquid film form a surface tension gradient and thus cannot form a countercurrent phenomenon. Under the coupling effect of the surface elasticity and disjoining pressure, a more stable liquid film forms. In the early stage of drainage, surface elasticity increases the film thickness, reduces the surface speed and generates the liquid countercurrent to slow the drainage process. When the black film appears, the electrostatic repulsion of the disjoining pressure is notable and makes the black film stable. The results obtained in the paper are in agreement with some of the experimental results in the literature. However, the elasticity-related surface tension and surfactant concentration model used is a simplified model. The nonlinear relationship between surface tension and surfactant concentration should be further considered in future theoretical models.

The aim of the present paper is to investigate the gravity-driven draining process containing soluble surfactant when considering the coupling effects of surface elasticity and surfactant solubility. A nonlinear coupling evolution equation including liquid film thickness, surface velocity and surfactant concentration (both on the surface and in the bulk) is established based on the lubrication theory. Assuming that the top of liquid film is attached to the wireframe and the bottom is connected to a reservoir, the drainage evolution is simulated with the software called FreeFem. The effects of surface elasticity and solubility on liquid film draining are discussed under their coupling. The simulation results show that the surface elasticity is an indispensable factor in the process of liquid film drainage with soluble surfactant, and the surfactant solubility also has an important influence on the process. At the initial stage of liquid draining, the initial thickness of liquid film increases with increasing surface elasticity, and the surface tends to be more rigid; with the drainage proceeding, the liquid film with high and low elasticity illustrate different notable draining features:in the case of low surface elasticity, the distribution of surfactant forms a surface tension gradient from top to bottom on the film surface, leading to positive Marangoni effect that counteracts gravity. However, in the case of high elasticity, the film surface presents a surface tension gradient from bottom to top, resulting in a reverse Marangoni effect, which accelerates the draining and makes the film more susceptible to instability. The solubility of surfactant dominates the number of adsorbent molecules on the film surface, which affects the surface elasticity. When the solubility of the surfactant is great (β → 0), the film is extremely unstable, and it breaks down quickly. As the solubility decreases (namely, β increases), the stability of the film increases, and the initial surface elasticity also rises. The surface elasticity gradually approaches to the limiting dilational elasticity modulus due to the film being thinner.

The thermocapillary migration characteristics of a self-wetting drop on the non-uniformly heated, horizontal, solid substrate are investigagted by numerical simulation. Based on the lubrication theory, an evolution equation for the height of the two-dimensional drop is established. The substrate underlying the drop is subjected to a temperature gradient which induces surface tension gradient-driven drop deformation and migration. The self-rewetting fluid has non-monotonic dependence of the surface tension on temperature with a well-defined minimum, and the position of the minimum corresponding to the temperature on the substrate is called the critical point. The effect of the relationship between the critical point and the drop position on drop dynamics is analyzed. With the temperature sensitivity coefficient of three interfaces under the same condition, the substrate is illustrated with constant wettability. The direction of drop migration will alter as the initial drop location moves to the left relative to the critical point position, resulting from the variation of the interplay among thermocapillary, gravity, and capillarity forces within the drop. But the drop always migrates toward the high interfacial tension region due to the thermocapillary force. In the presence of substrate wettability variations, the drop migrates toward the low temperature region no matter where the drop is placed relative to the critical point. This is due to the fact that the deterioration of substrate wettability on the right side of the drop prevents the drop from migrating toward the hot region. Under the critical point being on the left or within the drop, as the initial drop location moves to the left relative to the critical point position, the enhancement of the thermocapillary force toward the left leads to increased moving speed of the left contact line and increased spreading area. When the critical point is positioned on the outer right side of the drop, the speed of the left contact line sharply decreases at t=6103, caused by the suddenly deteriorating substrate wettability. Hence, it is effective to manipulate the self-wetting drop movement by regulating the relationship between the critical point and the initial drop location. To inhibit the migration of the drop toward the cold region, the drop should be placed on the right side of the critical point.

When a liquid film is under the evaporating or condensing condition, the flow stability is clearly different from that under an isothermal condition due to the thermal non-equilibrium effect at the interface, especially under a lower Reynolds number. Based on Prandtl's boundary layer theory and complete boundary conditions, the universal temporal and spatial stability formulations are established using the collocation method for two-dimensional surface waves of the evaporating or condensing and isothermal liquid films draining down an inclined wall. The evolution equations indicate that the flow stability is closely related to the Reynolds number, thermocapillarity, inclination angle, liquid property, and evaporation, isothermal, or condensation actions. The effects of the above factors are investigated with neutral stability curves at different Reynolds numbers, and stability characteristics are fully indicated in theory for evaporating or condensing films. Results show that the evaporation process destabilizes the film flow and condensation process stabilizes the film flow. Thermocapillarity has a stabilizing effect in an evaporation condition and an adverse effect in the condensation condition. For a lower Reynolds number, the vapor recoil and thermocapillary take dominant effects when compared to the inertia force in determining flow stability. At a higher Reynolds number, the flow stability is controlled by the inertia force. Present study indicates that the disturbance increases with an increase of the Reynolds number and inclination angle, and decreases with increase of Ka numbers. Furthermore, the effects of liquid properties and inclination angle are always significant. © 2005 Wiley Periodicals, Inc. Heat Trans Asian Res, 34(4): 243–257, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20062

A mathematical model is established to investigate the gravity-driven draining process of a vertical thin liquid film containing insoluble surfactants when considering the synergistic effect of surface viscosity and disjoining pressure. Lubrication theory is used to derive a coupled equation set describing the evolution of the film thickness, surfactant concentration and surface velocity. The equation set is solved numerically by the FreeFem program based on the finite element method. The film is assumed to be supported by the wire frame at both the top and bottom, thus the mass of liquid and the mass of total surfactants are conserved in the simulation. The characteristics of film evolution under the constant and variable surface viscosity are examined. Simulation results show that the surface viscosity is a crucial factor affecting the film drainage process. When neglecting the effect of surface viscosity, the film surface exhibits the mobile mode, while the film surface presents the rigid mode in the case of the surface viscosity considered. Increasing the surface viscosity, the rate of film drainage is slowed down significantly, leading to a reduction of the film thinning and enhancement of film stability, which is consistent with the results obtained by Naire et al. The disjoining pressure is a key factor in the formation of black film. When the disjoining pressure is only involved in the model, the length of the black film region is relatively short, nevertheless, if the effect of surface viscosity is only considered, a stable black film does not form. Under the synergistic effect of the disjoining pressure and surface viscosity, a very long and thin but stable black film is found in the middle segment of the film. Additionally, the thickness of black film increases and the appearance time postpones with the increase of surface viscosity. Considering the influence of concentration-dependent surface viscosity, the drainage rate is greatly affected. In the early stage, due to the smaller overall surface viscosity, the surface velocity is relatively large. With increasing surface viscosity at the bottom of film, the strength of the film surface tends to be enhanced, and then the anti-perturbation ability of the film is promoted and the film thinning is retarded. There is no significant difference in the length nor the appearance time of black film except that the thickness of black film with concentration-dependent surface viscosity is lower than that with the constant viscosity, thus the flow stability is weaker than that with the constant viscosity. In addition, the presence of the disjoining pressure slows down the thinning of blackest portion of the film and the surfactant concentration at this position. In the numerical results of the variable surface viscosity given by Braun et al. it is observed that the concentration of surfactant could almost be swept to clean in the top part of film. That is possibly because the effect of the disjoining pressure is neglected by Braun et al. It should be pointed out that the surface elasticity plays an important role in the stability of film. Therefore, it is necessary to consider the effect of surface elasticity in the future investigation.

For the drainage under the gravity of a vertical foam film containing insoluble surfactant, an improved concentration-dependent disjoining pressure model is formulated based on the published experimental results. The lubrication theory is used to establish the evolution equations of the film thickness, the surface concentration of insoluble surfactant, and the surface velocity, and the evolution characteristics of the film under different disjoining pressures are simulated numerically. The results show that the drainage process of a vertical liquid film generally undergoes two stages:the first stage is the thick film stage and the gravity plays a leading role in the drainage process; the subsequent stage is the thin film stage, the effects of capillary pressure and disjoining pressure increase gradually, and the disjoining pressure dominates the evolution of the film. The disjoining pressure effect is closely related to surfactant type and the correlation strength between the surfactant concentration and electrostatic repulsion force of disjoining pressure. For the ionic surfactant, electrostatic repulsion force increases with the increase of the surfactant concentration, but it is opposite for the nonionic surfactant. It is likely that the free hydroxide ions, which are considered to render the surface negatively charged, are partly adsorbed by the nonionic surfactant. So the surface charge of the foam film decreases as the concentration of the nonionic surfactant increases, resulting in a decrease in electrostatic repulsion. Therefore, some ionic surfactants can improve the stability of liquid film drainage and slow down the drainage process, while the effects of some nonionic surfactants are opposite. When the disjoining pressure is positively correlated with surfactant concentration, with the increase of correlation strength coefficient α, the thinning and drainaging processes of the film tend to slow down, hence the stability of the film is enhanced. When the disjoining pressure is negatively correlated with surfactant concentration, with the increase of the absolute value of α, the drainage process of the film is accelerated and the risk of film rupture is augmented. The results obtained in this paper are consistent with some of the experimental results, indicating that the concentration-dependent disjoining pressure is indeed an important factor in maintaining the stability of foam film containing some certain anionic or nonionic surfactants. The improved concentration-dependent disjoining pressure model established in this paper could not explain the phenomena of parts of cationic nor non-ionic surfactant film in drainage experiments. It can be inferred that the structure of surfactant molecule, the more detailed disjoining pressure model and the coupling of the disjoining pressure and surface elasticity should be considered in the future work.

The spreading characteristics of a droplet on a heated substrate have direct influences on its spreading area and heat transfer, so the exploration in this aspect is of important significance for cooling electronic and aerospace equipments. In the present paper, the evolution model of a droplet on a heated solid substrate is established based on the lubrication theory, and spreading processes are simulated respectively when the wall temperature is uniform and decreases exponentially from the center to both sides. A method of assessing the heat flux and heat transfer capacity of a two-dimensional liquid droplet is proposed. Influences of spreading characteristics and heat convective condition at the liquid-gas interface on heat transfer feature of the droplet are examined, and the results are in good agreement with the published ones in the literature. The simulated results show that in the case of uniform wall temperature, the evolution of the droplet is dominated mainly by gravity and illustrates symmetrical spreading characteristics, and the thickness profile presents a single-peak feature of which the value diminishes with time. The heat flux across the droplet surface decreases from both sides to the center, and the surface area of the droplet increases with time slightly, so the performance of heat transfer is strengthened to a certain extent. When the wall temperature decreases exponentially from the center to both sides, the spreading process of the droplet manifests three obvious stages, in which a single-peak feature of thickness profile gradually evolutes into a double-peak feature after surviving for a short period of time, and the peak values of the double-peak first increase firstly and then decrease, resulting from the complex game of gravity and thermocapillary force and their alternative dominance in the evolution. The variations of the dynamic contact angle and travelling speed of the contact line with time can also reflect the above characteristics. The heat flux in the center of the droplet increases, while its values at the double-peak and contact lines decrease with time. In addition, the heat flux at the contact line has a distinct jump feature compared with that at the adjacent position. The droplet surface area expands significantly with time, so the heat transfer capability is improved apparently. Enhancing heat convective condition at the liquid-gas interface, namely greater Biot number, slows the droplet spreading process, which inhibits the expansion of the droplet surface area. However, it enables the droplet to stay in a higher temperature region, resulting in the enhancement of heat dissipation of the droplet. Therefore, the comprehensive interactions of the above aspects strengthen the heat transfer capability, and this phenomenon tends to be increasingly significant over time. Greater Biot number delays the variations of the dynamic contact angle and the travelling speed of the contact line, without changing their general characteristics.

In most of researches about the droplet spreading on a substrate, one adopts aprecursor layer to relieve the stress singularity near the contact line without considering wall properties, which, however, is inapplicable for studying the relationship of the wettability with wall temperature. In this paper, the spreading of a heated droplet on the solid substrate, under the action of the three-phase contact line, is simulated. The influences of the wall temperature on wettability and droplet spreading are examined from the viewpoint of equilibrium contact angle. The simulated results show that when the wall temperature is uniform, the evolution of droplet spreading is dominated only by the gravity, illustrating symmetrical spreading characteristics. When the temperature gradient is applied to the wall, the combination of thermocapillary force and gravity drives the droplet into spreading, therefore the main part of the droplet migrates toward the low temperature region due to the Marangoni effect. The left contact line continually moves toward the left side while the right contact line first moves toward the right side, then turns to the left side after the receding time. The spreading range of the droplet is changed notably because of different travelling speeds of the contact line on both sides. With the increase of the temperature gradient, the Marangoni effect is promoted, resulting in a faster migration toward the low temperature region. A thin film is formed between the contact line in the hotter region and the bulk of the droplet, where the gravity and thermocapillary force dominate the spreading successively. The present simulation shows that the surface wettability is not only dependent on its chemical composition and geometrical morphology, but also closely related to wall temperature. When the sensitivities of the liquid-solid, liquid-gas and solid-gas interfacial tensions to temperature are all identical, the equilibrium contact angle between the droplet and the wall keeps constant, leading to a uniform wettability on the wall. When the liquid-solid interfacial tension or the liquid-gas interfacial tension is more sensitive to temperature than the other two interfaces, the equilibrium contact angle increases and the wettability tends to be worse, presenting a more hydrophobic substrate, which decelerates the spreading of the droplet with the contact line moving to the colder region. As the solid-gas interfacial tension is more sensitive to temperature than the other two interfaces, the equilibrium contact angle tends to lessen, and the contact line feels a more hydrophilic substrate (the droplet wets perfectly when the equilibrium contact angle decreases to zero), hence the spreading is enhanced. The present results indicate that the equilibrium contact angle plays a key role in the evolution of a heated droplet on a horizontal plate. The simulation conclusions can provide a theoretical basis for relevant experimental findings, which promotes the understanding of the relationship between wall temperature and its wettability.

The spreading of a drop of soluble surfactant over thin aqueous layer in the presence of isothermal heating is considered. Use of lubrication theory yields a coupled pair of partial differential equations for the film thickness, surface and bulk surfactant concentrations. The numerical results by the PDECOL procedure show that droplet patterns are obtained under the isothermal conditions of underlying substrate. The effects of variation of Marangoni parameter and air-liquid Biot number are examined. Decreasing Marangoni parameter leads to flatter drop peak and longer time to rupture. Decreasing the air-liquid Biot number flattens the film thickness and surfactant concentration profile of the drop. In all situations the surfactant is found to rigidify the air-liquid interface, while the thermal capillary acts to destabilize the film.

To analyze the effect of disjoining pressure for the soluble surfactant spreading process of a non-uniform thin film, the present paper establishes a 3-equations mathematical model and derives nonlinear evolution equations of the film thickness, surface monomers concentration and bulk monomers concentrations in the lubrication approximation. Additionally, this paper investigates a candidate mechanism that could create and drive this instability, disjoining pressure rupture, via numerical simulations coupled with analytical techniques. The evolutional profiles and transport characteristics of a drop spreading are investigated.

Prion diseases, also known as the transmissible spongiform encephalopathies (TSEs), are a group of slowly developing neurodegenerations occurring in human and animals. Prion diseases can be transferred between animals, humans, from humans to animals, and from animals to humans. As a result, the central nervous system is attacked, resulting in microglia activation, astrocytosis, prion plaque deposition, and neuronal degeneration. Prion also targets on the eye and brain visual system. In scrapie-infected sheep, chronic wasting disease (CWD)-infected mule deer, and experimental animals infected with scrapie, transmissible mink encephalopathy (TME), and Creutzfeldt-Jakob disease (CJD), damage has been found in the outer and inner nuclear layers of the retina, brain stem, optic nerve, optic tract, optic radiation and visual cortex. This article reviews the prion agent and infectivity in the eye and brain visual system, and the visual and oculomotor pathology in animal prion diseases. Effects of PrP genotypes and PrPSc types on visual and oculomotor disorders will be discussed.

For the spreading of a droplet covered with insoluble surfactant over an inclined random heterogeneous substrate, the base state and disturbance evolution equations for the film thickness and surfactant concentration are derived using the lubrication theory. Stability of the droplet spreading on the heterogeneous substrates, and effects of dimensionless parameters as well as the internal mechanism are investigated based on the non-modal stability theory. Results show that the disturbances of film thickness and surfactant concentration exihibit a double-hump shape and the crests lie in thinning regions at the bottom of the droplet. With increasing disturbance wave number, the stability of the droplet spreading is enhanced, but the maximal disturbance transfers from the right-side to the left-side of the droplet bottom. Increasing inclined angle leads to severe instability distinctly. Improving topographical height or increasing wave number of the substrate leads the spreading stability to firstly enhance and then to weaken. The droplet evolution displays an unstable spreading under smaller capillary number, with the maximum disturbance to be increasing before decreasing.

Targeting the flowing of insoluble surfactant-laden film over topography substrate, the lubrication theory is adopted to derive the evolution equations of thin liquid film thickness and interfacial surfactant concentration. The flowing characteristics of the film on topography surfaces, and the influence of topography structure are examined based on the numerical simulation with PDECOL code. The results show that when the thin film of insoluble surfactant flows over periodic grooving topography, the depression appears at the negative step, while the ridge is shaped at the positive step, both of which increase gradually with time going by. Compared with the case of the flat base, the surfactant-laden film spreading speed is enhanced. Increasing the groove depth or reducing the groove steepness leads to the increase of the rupture possibility of the film. Improving the groove width promotes the film flowing. The decrease of the steepness can cause the film to form a ridge feature before entering into the first groove. Gravity has the opposite effects on the up-hilling and down-hilling processes of liquid film flow, which causes the flow stability to deteriorate. Conjoining pressure induced by intermolecular forces can accelerate the film, giving rise to a notable dewetting phenomenon, while disjoining pressure has an adverse effect.

For the flow of an insoluble surfactant-laden thin film and droplet on inclined random topography, the lubrication theory is used to derive the evolution equations of thin liquid film thickness and interfacial surfactant concentration. Characteristics of thin film flow and droplet spreading, as well as the influence of topography structure are numerically simulated with PDECOL code. Results show that under the action of gravitational component and Marangoni effects, the thin film flow and droplet spreading is accelerated; the capillary ridge emerges at the thin film edge and the droplet center; and at the bottom of the thin film and droplet, the depression is generated. While the deformation of liquid film free surface is more significant due to the effect of random topography. The increasing θ has a role of enhancing gravitational component and Marangoni effects, leading to the enhancement of the capillary ridge and depression. The increase of D promotes the thin film flow and droplet spreading, but causes the deformation amplified; and the increased k0 can induce the evolutions of thin film flow and droplet spreading to slow down and inhibit the formation of capillary ridge and depression. In addition, compared with the thin film flow, the impact of D and k0 on the speed of droplet spreading is relatively weak.