Truncated versus extended microfilms at a vapor-liquid contact line on a heated substrate

The microstructure of a contact line formed by a liquid and its pure vapor on a perfectly wetted superheated smooth substrate, with the disjoining pressure most often in the form of a positive inverse cubic law (nonpolar case), is routinely considered to end up in a microfilm extended over adjacent "dry" parts of the solid surface. Invoking the spreading coefficient as an additional independent parameter within this framework, we argue however that a regime with a truncated microfilm is chosen instead if the spreading coefficient is decreased below a positive (still perfect wetting) critical value dependent upon the superheat, in which case the extended-microfilm thickness is surpassed by that of the "pancake" introduced by de Gennes and co-workers. Conversely, for a given positive spreading coefficient, there is a critical superheat above which the microfilm gets truncated, whereas for a negative one (partial wetting) the truncated regime should be preferred at any superheat. A parametric study of the apparent contact angle (a nonlinear eigenvalue of the steady microstructure problem) versus the spreading coefficient is carried out. When the latter is negative, Young's law is asymptotically recovered. Microfilm fronts on a bare surface are shown to be advancing or receding in accordance with the selected regime. A slightly more general class of disjoining pressures is also touched upon. The analysis is based in part upon thermodynamic considerations and in part upon a standard one-sided model of an evaporating liquid layer in the lubrication approximation.     

Cnoidal wave trains and solitary waves in a dissipation-modified Korteweg-de Vries equation

A generalization of the Korteweg-de Vries equation incorporating an energy input-output balance, hence a dissipation-modified KdV equation is considered. The equation is relevant to describe, for instance, nonlinear Marangoni-Bénard oscillatory instability in a liquid layer heated from above. Cnoidal waves and solitary waves of this equation are obtained both asymptotically and numerically.     

Effect of internal heat release on the capillary force acting on a droplet and on the interaction of droplets with each other and with a wall

In the Stokes approximation and on the assumption that the Péclet numbers are small expressions are derived for the capillary force acting on a droplet and for the steady drift velocity of the droplet with a given arbitrary surface distribution of the surface tension coefficient in the presence of uniform internal heat release. The thermocapillary interaction of two droplets in an unbounded external fluid and of a single droplet with a wall, due to internal heat release, is considered. Quasistationary expressions are obtained for the velocity of the droplets in such processes in the absence of gravitation. Certain limitations on the applicability of the results obtained are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 124–132, September–October, 1990.     

On integrable singularities and apparent contact angles within a classical paradigm Partial and complete wetting regimes with or without phase change

Far from claiming any ultimate resolution of the contact line paradoxes, we draw the reader’s attention to the fact that a somewhat controversial paradigm, originally employed by de Gennes and collaborators, actually appears both to be quite reasonable at its foundations and to lead to physically consistent final results in a wide variety of situations. Curiously enough, while containing a singularity in itself, the approach nonetheless renders the classical contact-line singularities — both hydrodynamic and thermal — integrable, in particular as far as several quantities of interest are concerned. It is also readily applicable to quite a few situations: from equilibrium shapes and moving contact lines of a non-volatile liquid, to cases with evaporation into (and even condensation from, although it is not studied here) either a pure-vapor or an inert-gas atmosphere. The paradigm actually consists in an approach involving both the (positive or negative) spreading coefficient and the disjoining pressure in the form of a positive inverse cubic law, a conceptual framework that most notably describes structures with truncated precursor films on a macroscopically bare solid surface. Whether or not the remaining integrable divergences at molecular scale can truly be considered as “benign” has to be discussed on the basis of more involved mesoscopic or microscopic approaches, quite outside the scope of the present study.     

Steady streaming around a spherical drop displaced from the velocity antinode in an acoustic levitation field

The steady (acoustic) streaming associated with a sphericaldrop displaced from the velocity antinode of a standing waveis studied. The ratio of the particle size to the acoustic wavelengthis treated as small but non-zero, and the solution is developedin the form of a two-term expansion in terms of the correspondingsmallness parameter. The drop viscosity is assumed to be muchhigher than that of the surrounding fluid, which is the casefor a drop in a gas medium. There are essentially three distinctregions where the steady streaming flow is analysed: insidethe drop (internal circulation), in the Stokes shear-wave layerat the surface on the gas side, and the gas outside the Stokeslayer (the outer streaming region). Solutions for the internalcirculation and the outer streaming are obtained in the limitof small Reynolds number. Despite the gas-to-liquid viscosity ratio being small, the outerstreaming may be dramatically affected by the fact that thesphere is liquid as opposed to solid. The parameter that measuresthe effect of liquidity is essentially the viscosity ratio dividedby the relative (to the particle size) thickness of the Stokeslayer. The case of a solid sphere is recovered by letting thisparameter go to zero.     

Nonstationary motion of drops under the action of capillary and mass forces

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 28–35, July–August, 1991.     

Bénard instabilities in a binary-liquid layer evaporating into an inert gas: Stability of quasi-stationary and time-dependent reference profiles

This study treats an evaporating horizontal binary-liquid layer in contact with the air with an imposed transfer distance. The liquid is an aqueous solution of ethanol (10% wt). Due to evaporation, the ethanol mass fraction can change and a cooling occurs at the liquid-gas interface. This can trigger solutal and thermal Rayleigh-Bénard-Marangoni instabilities in the system, the modes of which corresponding to an undeformable interface form the subject of the present work. The decrease of the liquid-layer thickness is assumed to be slow on the diffusive time scales (quasi-stationarity). First we analyse the stability of quasi-stationary reference profiles for a model case within which the mass fraction of ethanol is assumed to be fixed at the bottom of the liquid. Then this consideration is generalized by letting the diffusive reference profile for the mass fraction in the liquid be transient (starting from a uniform state), while following the frozen-time approach for perturbations. The critical liquid thickness below which the system is stable at all times quite expectedly corresponds to the one obtained for the quasi-stationary profile. As a next step, a more realistic, zero-flux condition is used at the bottom in lieu of the fixed-concentration one. The critical thickness is found not to change much between these two cases. At larger thicknesses, the critical time at which the instability first appears proves, as can be expected, to be independent of the type of the concentration condition at the bottom. It is shown that solvent (water) evaporation plays a stabilizing role as compared to the case of a non-volatile solvent. At last, an effective approximate Pearson-like model is invoked making use in particular of the fact that the solutal Marangoni is by far the strongest as an instability mechanism here.