Partial Coalescence at Liquid Interfaces

1
Partial Coalescence at Liquid Interfaces
François Blanchette Terry P. Bigioni
James Franck Institute, University of Chicago
time
1.5 mm
t 0ms
1.2ms
2.9ms
4.1ms
Coalescence from rest of a drop of ethanol
(radius R 0.5mm) with a reservoir of ethanol.
The daughter drop bounces, then comes to rest
before undergoing the same process.
Context
Previous work
Numerical model
Validation
Governing Equations
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3
4
5
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R Drop radius mi inner viscosity
mo outer viscosity s surface tension
ri inner density ro outer density
gravity
1- Under gravity, a drop slowly comes into
contact with a reservoir of the same fluid.

• Before pinch off, 256 points ensure
• numerical convergence
• mass conservation
• energy conservation

Charles Mason (1960) observed multiple
coalescence. Thoroddsen Takehara (2000) found
t (r R3 / s) ½ as the relevant time scale.
Pikhitsa Tsargorodskaya (2000) suggested a
mechanism relying on surface elasticity due to
surfactant. Many groups work on coalescence,
bouncing Couder et al., Leal et al. etc.
Popinet Zaleski (1999)
Replace the free surface by forcing term. Track
the position of the interface (S) with markers.
Introduce the volume of inner fluid, outer
fluid C 0, inner fluid C 1 0 C 1
2- The drop coalesces with the lower fluid.
Comparison with experiments
Incompressible Navier-Stokes equations. On the
interface Equal tangential stresses. Normal
stresses balanced by surface tension. Initial
conditions Both fluids at rest.
Connected drop and reservoir. Boundary
conditions Assume rotational symmetry.
Other boundaries are far away.
density viscosity r C
(1-C) / a m C (1-C) / l
3- The mother drop pinches off and leaves
behind a daughter drop.
Scales Time t v ri R3 / s, length R,
density ri
Fundamental (unanswered) questions
Bo gravity g R2 (ri ro) / s
surface tension
Under what conditions does partial coalescence
occur?
Oh viscosity mi / v ri R s
surface tension
Top experiment Middle vertical velocity (blue
down, red up) Bottom horizontal velocity (blue
in, red out)
4-The daughter drop bounces and the process
starts over.
(multiple coalescence)
Ratios a mi / mo l ri/ ro
R 0.5mm, Bo 0.09, Oh 0.01, l 50, a
50 time is in millisecond.
What is the mechanism?
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10
Summary
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Pinch off mechanism
Scaling argument
Liquid-liquid systems
Other observations
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8
Vertical displacement of the top of the
drop. Converging waves stretch the drop vertically
Time evolution of a drop of ethanol
Daughter drop velocity depends on Bo and Oh.

• Rayleigh-Plateau instability does not
• cause pinch off.
• Pinch off is determined by competition
• between horizontal and vertical collapses.
• If capillary waves delay vertical collapse,
• pinch off may occur.
• We found a general criterion to determine
• whether or not pinch off occurs.

Capillary waves stretch the drop and allow pinch
off to occur.
Partial coalescence is not truly self-similar
Damping rate D 2 k2 mi / ri
k wave number
Traveling time tw p R / v s k / ri
Saggy drops (Bo gt 0.2) form satellite droplets
Very saggy drops (Bo gt 0.5) eject tiny droplets
Amplitude fraction left Exp(-D tw) D tw (k
R)3/2 2p mi / v s ri R (k R)3/2 2p
Oh
No pinch off resulted!!
Denser outer fluids are favorable to pinch off
as they carry waves more effectively

• Setting all velocities to 0 at most elongated
  states yields no pinch off Rayleigh-Plateau
  instability does not cause pinch off.

Neglecting gravity, pinch off occurs if
No pinch off if D tw gt 1. (or Oh gt Ohc)
Liquid drops in air
Drop-drop partial coalescence also occurs
Rather

• Horizontal and vertical collapse are competing.
• Capillary waves are generated early on.
• Waves converge at the drops summit.
• Drop is stretched by the waves.
• Vertical collapse is delayed.
• The horizontal collapse reaches completion if
• the delay is sufficient.

1
mi /(risR)1/2
No pinch off
B
Pinch off
B gt 1.6 is required for partial coalescence
Viscous outer fluids can also damp capillary wave
and dissipate energy
For more, ask to see the movies!!
g (ri - ro) R2 / s
Acknowledgements Wendy Zhang, Eric Corwin,
Heinrich Jaeger, NSF-MRSEC DMR-213745
Black circles follow the evolution of a single
drop.
1.5mm
0.9ms
2.6ms
3.4ms
t0ms
Simulations of the same drop of ethanol shown
above. Here the Bond number is Bo 0.1 and the
Ohnesorge number is Oh 0.01.
time