Analytical solutions for the profile of two-dimensional droplets with finite-length precursor films.

By means of the lubrication approximation we obtain the full family of static bidimensional profiles of a liquid resting on a substrate under partial-wetting conditions imposed by a disjoining-conjoining pressure. We show that for a set of quite general disjoining-conjoining pressure potentials, the free surface can adopt only five nontrivial static patterns; in particular, we find solutions when the height goes to zero which describe satisfactorily the complete free surface for a finite amount of fluid deposited on a substrate. To test the extension of the applicability of our solutions, we compare them with those obtained when the lubrication approximations are not employed and under conditions where the lubrication hypothesis are not strictly valid, and also with axisymmetric solutions. For a given disjoining-conjoining potential, we report a new analytical solution that accounts for all the five possible solutions.

Final state of a perturbed liquid film inside a container under the effect of solid-liquid molecular forces and gravity.

We investigate theoretically the possible final stationary configurations that can be reached by a laterally confined uniform liquid film inside a container. The liquid is under the action of gravity, surface tension, and the molecular interaction with the solid substrate. We study the case when the container is in an upright position as well as when it is turned upside down. The governing parameters of the problem are the initial thickness of the film, the size of the recipient that contains the liquid, and a dimensionless number that quantifies the relative strength of gravity with respect to the molecular interaction. The uniform film is always a possible final state and depending on the value of the parameters, up to three different additional final states may exist, each one consisting in a droplet surrounded by a thin film. We derive analytical expressions for the energy of these possible final configurations and from these we analyze which state is indeed reached. A uniform thin film may show three different behaviors after a perturbation: The system recovers its initial shape after any perturbation, the fluid evolves towards a drop (if more than one is possible, it tends toward that with the thinnest precursor film) for any perturbation, or the system ends as a uniform film or a drop depending on the details of the perturbation.

Closed-form expression for the profile of partially wetting two-dimensional droplets under gravity.

Analytical solutions for the shape of both hanging and sitting droplets under the effects of gravity and surface tension are presented. The modeling also includes the action of molecular forces arising between the liquid and the substrate, which are responsible for the formation of a stable nanometric film in the region close to the droplet contact line. The shape of the droplet is completely described by an analytical solution that also accounts for the pancake-shaped droplets as a limiting case. We find expressions that relate microscopic and nanoscopic aspects, such as the strengths of the molecular forces and the thickness of the nanometric film, to macroscopic quantities, such as the cross-sectional area and the width of the droplet. We study the effect of gravity on the contact angle and find that for small droplets the contact angle follows a power law with the droplet's size. For sitting droplets we find that the there is an upper limit for the value of the gravity.

Thin-film flows with moving contact lines: An approach to reducing computing time.

A numerical method to reduce the computing times of thin-film flows with moving contact lines is presented. The flows of a film and a droplet are calculated in a frame that moves with a nonconstant velocity U(t). The criterion employed to define this velocity is to reduce the maximum height change in the flow's most critical zone. The efficiency of the algorithm in reducing the CPU time is tested in gravity-driven flows, where the computing time is reduced by up to a factor of 13 depending on the parameters of the problem.

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.

Stability study of a constant-volume thin film flow.

We study the stability of a constant volume of fluid spreading down an incline. In contrast to the commonly considered flow characterized by constant fluid flux, in the present problem the base flow is time dependent. We present a method to carry out consistently linear stability analysis, based on simultaneously solving the time evolution of the base flow and of the perturbations. The analysis is performed numerically by using a finite-difference method supplemented with an integral method developed here. The computations show that, after a short transient stage, imposed perturbations travel with the same velocity as the leading contact line. The spectral analysis of the modes evolution shows that their growth rates are, in general, time dependent. The wavelength of maximum amplitude, lambda_{max} , decreases with time until it reaches an asymptotic value which is in good agreement with experimental results. We also explore the dependence of lambda_{max} on the cross sectional fluid area A , and on the inclination angle alpha of the substrate. For considered small A 's, corresponding to small Bond numbers, we find that the dependence of lambda_{max} on A is in good agreement with experimental data. This dependence differs significantly from the one observed for the films characterized by much larger A 's and Bond numbers. We also predict the dependence of lambda_{max} on the inclination angle alpha .

Spreading of a micrometric fluid strip down a plane under controlled initial conditions.

We experimentally study the spreading of a small volume of silicon oil down a vertical plane with small Bond number. The initial condition is characterized by a horizontal long fluid strip with cross sectional area A and width w(0). We find that the experiments are characterized by a unique nondimensional parameter, R proportional w4(0)/(a2A), where a is the capillary length. An empirical criterium to estimate the onset of the contact line instability is established. The later rivulet formation at the contact line leads to a pattern which is characterized by a dominant wavelength. We find that this wavelength is approximately proportional to R(-1/4).