Analytical solutions for partially wetting two-dimensional droplets.

We present a new analytical solution for the static shape of a two-dimensional droplet in equilibrium with a surrounding thin film on a solid substrate. The modeling includes the effects of capillarity and disjoining-conjoining pressure accounting for intermolecular forces between the solid and the liquid. We derive new analytical solutions for the shape of the droplet, the cross-sectional area, the half-width, and the maximum curvature and inflection points. We study the effects of the size of the droplet on the apparent contact angle. The shape of the droplet in the contact line region is compared with profiles obtained by employing approximations suggested in the literature, and the observed differences are discussed. Finally, we present the time evolution to the steady state to show how the whole profile, including the thin film, evolves to the corresponding stationary configuration.

Experimental study of vapor bubbles in small-sized channels.

This paper investigates the effect of evaporation on the shape of liquid/vapor interfaces in small-scale systems. Vapor bubbles are generated due to localized heating in a small-sized channel (with an inner dimension of 3x3x200 mm) filled with pentane, for which heat fluxes and temperature distributions are simultaneously measured. The length of the resulting vapor bubble is studied as a function of the power input and heater temperature, and is found to be not only repeatable but nonhysteretic. The bubble length depends nearly linearly on the power input, in qualitative agreement with an approximate theory of Ajaev and Homsy [V.S. Ajaev, G.M. Homsy, J. Colloid Interface Sci. 244 (2001) 180]. In addition, it is found that vapor bubbles oscillate very slowly due to the effect of thermal relaxation.

Linear stability of a draining film squeezed between two approaching droplets.

In the present paper we analyze the effect of infinitesimal non-axisymmetric perturbations in determining the critical gap thickness at which a draining, finite radius thin-film becomes unstable. The film is part of the suspending fluid trapped between two approaching deformable drops under the action of a flow field. We carry out a linear stability analysis in the context of a quasi-static approximation where the rate of growth of the disturbances is assumed to be much faster than the rate of film drainage. An analytical solution is derived for the model in the special case of a uniformly thick film, for two types of perturbation: fixed-end and free-end. It is shown, for this special case, when the hydrodynamic force pushing the drops together from the external flow is constant, that the four most unstable disturbances are of the free-end kind, associated with the lowest frequency modes of azimuthal variation in the film thickness. Higher modes are stabilized by surface tension. Our analysis also shows that adopting the unretarded form of the van der Waals disjoining pressure yields results similar to the analysis when electromagnetic retardation effects are included in the calculation. A second case is analyzed where the film is also of uniform thickness but its lateral extent and the gap thickness are both time-dependent. This case was included to extend the predictions to glancing drop-collisions where the external hydrodynamic force is time-dependent. We find that there is a maximum capillary number below which the film becomes unstable, and that there is range of angles in the trajectory where the film becomes unstable, but that outside this range the film is stable.

Dynamic response of geometrically constrained vapor bubbles.

We consider a two-dimensional model of a vapor bubble between two horizontal parallel boundaries held at different temperatures. When the temperatures are constant, a steady state can be achieved such that evaporation near the contact lines at the hot bottom plate is balanced by condensation in colder areas of the interface near the top. The dynamic response of the bubble is probed by treating the case of time-dependent wall temperatures. For periodic modulations of the wall temperature the bubble oscillates about the steady state. In order to describe such time-dependent behavior we consider the limit of small capillary number, in which the effects of heat and mass transfer are significant only near the contact lines at the bottom plate and in a small region near the top. When the bottom temperature is modulated and the top temperature is held fixed, the amplitude of forced oscillations is constant for low-frequency modulations and then rapidly decays in the high-frequency regime. When the top temperature is modulated with fixed bottom temperature, the dynamic-response curve is flat in the low-frequency regime as well, but it also flattens out when the frequency is increased. This shape of the response curve is shown to be the result of the nonmonotonic behavior of the thickness of the liquid film between the bubble interface and the top plate: when the temperature is decreased, the film thickness increases rapidly, but then slowly decays to a value which is smaller than the initial thickness. The dynamic response is also studied as a function of the forcing amplitude.

Steady Vapor Bubbles in Rectangular Microchannels.

We consider vapor bubbles in microchannels in which the vapor is produced by a heater element and condenses in cooler parts of the interface. The free boundary problem is formulated for a long steady-state bubble in a rectangular channel with a heated bottom. Lubrication-type equations are derived for the shape of the liquid-vapor interface in a cross-sectional plane and in the regime for which the vapor phase fills most of the cross section. These equations are then solved numerically over a range of parameter values with given temperature profiles in the walls and subject to a global integral condition requiring evaporation near the heater to balance condensation in colder areas of the interface. Our results show that depending on the temperature, the side walls can be either dry or covered with a liquid film and we identify criteria for these two different regimes. The asymptotic method breaks down in the limit when capillary condensation becomes important near the bubble top and a different approach is used to determine the shape of the bubble in this limit. Solutions here involve localized regions of large mass fluxes, which are asymptotically matched to capillary-statics regions where the heat transfer is negligible. Copyright 2001 Academic Press.