Partial Coalescence at Liquid InterfacesFrançois Blanchette & Terry P. Bigioni
James Franck Institute, University of Chicago
4-The daughter drop bounces and the process starts over.
t = 0ms 1.5 mm 1.2ms 2.9ms 4.1ms
Context1- Under gravity, a drop slowly comes into contact with a reservoir of the same fluid.
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.
2- The drop coalesces with the lower fluid.
3- The mother drop pinches off and leaves behind a daughter drop.
Charles & Mason (1960) observed multiple coalescence.
Thoroddsen & Takehara (2000) found t ~ ( R3 / ) 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.
Previous work
Fundamental (unanswered) questions:
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.
Oh = viscosity = i / √ i R surface tension
Scales: Time: = √ i R3 / , length: R, density: i
Bo = gravity = g R2 (i – o) / surface tension
Ratios: = i / o = i/ o
Numerical model
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
density viscosity = C + (1-C) / = C + (1-C) /
Validation
Before pinch off, 256 points ensure• numerical convergence• mass conservation• energy conservation
Top: experiment Middle: vertical velocity (blue down, red up) Bottom: horizontal velocity (blue in, red out)
R = 0.5mm, Bo = 0.09, Oh = 0.01, = 50, = 50time is in millisecond.
Comparison with experiments
R = Drop radiusi = inner viscosityo = outer viscosity = surface tensioni = inner densityo = outer density
gravity
(multiple coalescence)
t=0ms 0.9ms 2.6ms 3.4ms1.5mm
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.
• Horizontal and vertical collapse are competing.
• Capillary waves are generated early on.
• Waves converge at the drop’s summit.
• Drop is stretched by the waves.
• Vertical collapse is delayed.
• The horizontal collapse reaches completion if the delay is sufficient.
Pinch off mechanism
k = wave number
Damping rate: D = 2 k2 i / i
Traveling time: tw = R / √ k / i
Amplitude fraction left ~ Exp(-D tw):D tw = (k R)3/2 2 i / √ i R = (k R)3/2 2 Oh
No pinch off if D tw > 1. (or Oh > Ohc)
Capillary waves stretch the drop and allow pinch off to occur.
Scaling argument
= g (-o R2 /
Liquid-liquid systems
Denser outer fluids are favorable to pinch off as they carry waves more effectively
Neglecting gravity, pinch off occurs if:
Summary Other observationsDaughter drop velocity depends on Bo and Oh.
Saggy drops (Bo > 0.2) form satellite droplets
Very saggy drops (Bo > 0.5) eject tiny droplets
For more, ask to see the movies!!
No pinch off resulted!!
Time evolution of a drop of ethanolVertical displacement of the top of the drop.Converging waves stretch the drop vertically
• Setting all velocities to 0 at most elongated states yields no pinch off Rayleigh-Plateau instability does not cause pinch off.
No pinch off
Pinch off
Black circles follow the evolution of a single drop.
=
i /(
iR
)1/2
Liquid drops in air
B
1
B > 1.6 is required for partial coalescence
Partial coalescence is not truly self-similar
time
time
Acknowledgements: Wendy Zhang, Eric Corwin, Heinrich Jaeger, NSF-MRSEC #DMR-213745
Governing Equations
Under what conditions doespartial coalescence occur?
What is the mechanism?
(numerical fit)
i + 0.53 o
((i+1.9 o)R)1/2
< 0.026
1 2 3 4 5
6 7 8 9 10
Rather:
• 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.
Viscous outer fluids can also dampcapillary wave and dissipate energy
Drop-drop partial coalescence also occurs:
Popinet & Zaleski (1999)
