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)