Analysis of the Spreading Radius in Droplet Impact: The Two-Dimensional Case

We study droplet-impact problems in a three-dimensional cylindrical or equivalent two-dimensional Cartesian geometry. Such structures do have an approximate experimental realization, and they are often simulated a test-bed for computational methods. We focus on droplet impact on a smooth homogeneous surface as well as head-on collision of two droplets. We perform an energy-budget analysis and introduce a correlation which predicts the maximum spreading radius as a function of Reynolds number and Weber number. We show how the dissipation term in this analysis can be decomposed into boundary-layer dissipation in the droplet lamella (where applicable), and head loss. We use existing results in the literature (simulations and experiments), as well as our own simulation results to validate the correlation. Dissipation by head loss is a key term in the analysis: only by modeling it accurately can one obtain good agreement between the simulations and the theory.Comment: 22 pages, 16 figure

Nonlinear dynamics of phase separation in thin films

We present a long-wavelength approximation to the Navier-Stokes Cahn-Hilliard equations to describe phase separation in thin films. The equations we derive underscore the coupled behaviour of free-surface variations and phase separation. We introduce a repulsive substrate-film interaction potential and analyse the resulting fourth-order equations by constructing a Lyapunov functional, which, combined with the regularizing repulsive potential, gives rise to a positive lower bound for the free-surface height. The value of this lower bound depends on the parameters of the problem, a result which we compare with numerical simulations. While the theoretical lower bound is an obstacle to the rupture of a film that initially is everywhere of finite height, it is not sufficiently sharp to represent accurately the parametric dependence of the observed dips or `valleys' in free-surface height. We observe these valleys across zones where the concentration of the binary mixture changes sharply, indicating the formation of bubbles. Finally, we carry out numerical simulations without the repulsive interaction, and find that the film ruptures in finite time, while the gradient of the Cahn--Hilliard concentration develops a singularity.Comment: 26 pages, 20 figures, PDFLaTeX with RevTeX4 macros. A thorough analysis of the equations is presented in arXiv:0805.103

Flow stability in shallow droplets subject to localized heating of the bottom plate

We investigate theoretically the stability of thermo-capillary convection within a droplet when subject to localized heating from below. To model the droplet, we use a mathematical model based on lubrication theory. We formulate a base-state droplet profile, and we examine its stability with respect to small-amplitude perturbations in the azimuthal direction. Such linear stability analysis reveals that the base state is stable across a wide parameter space. We carry out transient simulations in three spatial dimensions: the simulations reveal that when the heating is slightly off-centered with respect to the droplet center, vortices develop within the droplet. The vortices persist when the contact line is pinned. These findings are consistent with experimental studies of locally heated sessile droplets

Symmetry-Breaking in Point-Heated Droplets

We investigate theoretically the stability of thermo-capillary convection within a droplet when heated by a point source from below. To model the droplet, we use a mathematical model based on lubrication theory. We formulate a base-state droplet profile, and we examine its respect to small-amplitude perturbations in the azimuthal direction. Such linear stability analysis reveals that the base state is stable across a wide parameter space. We carry out transient simulations in three spatial dimensions: the simulations reveal that when the heating is slightly off-centered with respect to the droplet center, vortices develop within the droplet. The vortices persist when the contact line is pinned. These findings are consistent with experimental studies of point-heated sessile droplets.Comment: 15 figure