27 research outputs found
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