The
coalescence of drops is important in industrial as well as natural processes
involving emulsions or dispersions of small drops of one fluid in a second ambient
fluid. Coalescence is of common occurrence in applications as diverse as
petroleum production, foodstuffs (e.g. milk, mayonnaise, and salad dressing), dense
sprays, and solvent extraction. Coalescence is also prevalent in the
production of advanced materials by sintering, growth of raindrops, and life
sciences (e.g. membrane and cell fusion). In many processes in the
petrochemical industries, it is necessary to remove a dispersed water phase
from a continuous oil phase. In the petroleum industry, water-in-oil emulsions
are formed during the production of crude oil. If the oil is not dehydrated or
demulsified, the presence of water in the oil can result in corrosion of pipes,
deactivation of catalysts, and increased costs in transporting the unwanted
water. As surface-active species are present in many applications in industry,
the goal of this research is to advance the fundamental understanding of
coalescence of surfactant-laden drops.
In the
specific problem of interest here, two drops are slowly brought together and
allowed to touch. Upon contact, a small liquid neck forms between the drops. The
expansion of the neck is controlled by the Laplace or capillary pressure which
diverges when the curvature of the interface is infinite at the point where the
drops first touch. The major objective of the research is to probe the nature
of the singularity in the dynamics when the drops are covered with a monolayer
of an insoluble surfactant.
Here, the
system being considered is isothermal and consists of two identical spherical
drops of radii A. The drop fluid is an incompressible Newtonian fluid of
constant density d and constant viscosity m. The drops are surrounded by a
dynamically passive gas that exerts a constant pressure on the drops. The
surface tension of the clean liquid-gas interface is T. Moreover, both drops
are assumed prior to contact to be covered uniformly with a monolayer of an
insoluble surfactant. In the simulations, the drops are brought together
quasi-statically and a small neck is formed between them. To date, our primary
goal has been to analyze by simulation the flows and the topological changes in
the interface shapes that occur in the vicinity of the singularity and to
uncover the various regimes of coalescence.
The flow and
surfactant transport problems are governed by the continuity and the fully
nonlinear Navier-Stokes equations for the velocity and the pressure, and the
nonlinear surface convection-diffusion equation for surfactant concentration.
The effect of surfactant concentration on surface tension is governed by a
nonlinear equation of state. These equations are solved subject to the
traction and kinematic boundary conditions along the free surfaces and symmetry
boundary conditions along the axis of symmetry. The initial condition is such
that the fluid within the just joined drops is quiescent and the concentration
of surfactant is uniform along the liquid-gas interface.
During the
first year, the computational algorithm, and computer code, to solve the free
boundary problem comprised of the aforementioned equations was developed and
tested. Our approach is based on a method of lines algorithm that utilizes an
implicit, adaptive finite difference time integrator and a Galerkin/finite
element method for spatial discretization. To deal with the free boundary
nature of the problem, the domain is discretized with an adaptive elliptic mesh
generation algorithm.
The new code
was benchmarked against another code that had been developed by a more senior graduate
student whose research is focused on coalescence in the absence of
surfactants. All of the tests carried out to date have shown that the new code
predicts results that are identical to the other, older code. Two key
dimensionless groups in the coalescence problem are the Ohnesorge number Oh (the
viscosity m divided by the square root of the product of density d, radius A,
and surface tension T) and the Peclet number Pe (a ratio that measures the
importance of convection to diffusion of surfactant).
Among other
things, we have devoted a great deal of attention this year to exploring the
effect of surfactant on the scaling law(s) for the dependence on time t of the
radius r(t) of the small bridge that connects the two drops. In the absence of
surfactant, we had previously shown that the initial regime of coalescence is
the inertially-limited viscous (ILV) regime where inertial and viscous forces
compete with surface tension force to determine the dynamics. As time
advances, the dynamics then transitions when Oh < 1 (> 1) to an inertial
(viscous) regime where inertial (viscous) force alone balances surface tension
force. Without surfactant, the inertial and the viscous regimes do not share a
phase boundary. This year, we have uncovered quite surprisingly that the
dynamics can sample all three regimes in succession if surfactant is present on
the interface. This discovery was facilitated by the ability to tune Oh and Pe
in the simulations to accentuate the role of certain forces relative to others.
The attached figure, which shows the variation with time t of the minimum neck
radius r_{min}, highlights the occurrence of the ILV (where r_{min} ~ t),
viscous or Stokes (where r_{min} ~ t ln t), and inertial (where r_{min} ~
t^{1/2}) regimes during the coalescence of two surfactant-covered drops of
Oh=0.1 when Pe=300 in a single coalescence event.
This ACS/PRF
grant already has had a tremendous positive effect on the PI’s career and the
graduate students involved in the project. After just receiving this grant,
the PI was able to recruit one of the top first year students to his group. He
was then able to recruit a second top student the next year to work in this
general area. The PI now has three graduate students who are working on different
aspects of drop coalescence and the group is well on its way to establish
itself as a leader in the field of drop coalescence.
printer friendly
