1. Partial Coalescence at Liquid Interfaces François 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 1.2ms2.9ms 4.1ms Context 1- 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 ~ (  R 3 /  ) 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 R 3 / , length: R, density:  i Bo = gravity = g R 2 (  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,  = 50 time is in millisecond. Comparison with experiments R = Drop radius  i = inner viscosity  o = outer viscosity  = surface tension  i = inner density  o = outer density gravity ( multiple coalescence ) t=0ms 0.9ms2.6ms3.4ms 1.5mm 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 k 2  i /  i Traveling time: t w =  R / √  k /  i Amplitude fraction left ~ Exp(-D t w ): D t w = (k R) 3/2 2   i / √   i R = (k R) 3/2 2  Oh No pinch off if D t w > 1. (or Oh > Oh c ) Capillary waves stretch the drop and allow pinch off to occur. Scaling argument = g (    -  o  R 2 /  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 observations Daughter 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 ethanol Vertical 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 /(  i  R) 1/2 Liquid drops in air B 1 B > 1.6 is required for partial coalescence Partial coalescence is not truly self-similar time Acknowledgements: Wendy Zhang, Eric Corwin, Heinrich Jaeger, NSF-MRSEC #DMR-213745 Governing Equations Under what conditions does partial coalescence occur? What is the mechanism? (numerical fit)  i + 0.53  o ((  i +1.9  o )  R) 1/2 < 0.026 1 2345 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 damp capillary wave and dissipate energy Drop-drop partial coalescence also occurs: Popinet & Zaleski (1999)