The simultaneous effect of small deformation and short-range
van der Waals attraction on the coalescence efficiency of two different-sized
slowly sedimenting drops is
considered. For spherical drops, it has been shown previously that the
tangential
mobility of drop surfaces makes collision possible even without van der
Waals
attraction; on the other hand, even a small amount of deformation precludes
drops from
coming into contact unless van der Waals attraction is accounted for. In
the present
work, the conditions are delineated when these two small-scale factors,
acting in
opposite directions, have a considerable combined effect on the coalescence
efficiency.
The problem is solved by matched asymptotic expansions valid for small
capillary
numbers (Ca). The outer solution, for two spherical drops moving
in
apparent contact without van der Waals attraction, determines the
contact force as a function of
time. This force is used as the driving force for the inner solution of
the relevant
integro-differential thin-film equations (coupling the flow in
the small-gap region to
that inside the drops) to determine whether coalescence occurs during the
apparent
contact motion. The initial gap profile for the inner solution
is provided by matching
with the outer trajectory for spherical drops approaching contact.
The analysis shows that, for Ca[Lt ]1, the near-contact
deformation is mainly
axisymmetric, greatly simplifying the inner solution; nevertheless, determination
of
the critical horizontal offsets leading to coalescence and the
parametric analysis are
computationally very intensive. To facilitate these tasks, a substantially
new, highly efficient, and absolutely stable numerical method for solving
stiff
thin-film equations is
developed. Unlike for spherical drops, when the upstream intersection
area is a circle,
the existence of a second coalescence zone for deformable drops is found
over much
of the parameter space. Results are mapped out for a range of four dimensionless
parameters (capillary number, size and drop-to-medium viscosity ratios,
dimensionless
Hamaker parameter). As a physical application, predicted coalescence efficiencies
are
shown for a system of ethyl salicylate drops in diethylene glycol.
The present solution extends the range of drop sizes where the coalescence
efficiencies are known theoretically and can be used in drop population
dynamics.
Comparison with full three-dimensional boundary-integral calculations for
deformable
drops without van der Waals attraction is also made to demonstrate that,
when
the drop-to-medium viscosity ratio is of the order of unity, the present
asymptotic
approach is valid in a wide range of small and moderately small capillary
numbers.