Formation and detachment of a drop at the tip of a capillary. The drop and matrix fluids are both Newtonian, with density ratio and viscosity ratio . Other dimensionless parameters are , , and . (a) Early stage of drop growth at dimensionless time . The left half shows the finite-element grid and the right half plots the interface and streamlines. (b) Shortly after the pinch-off, .
Comparison of the drop size between our simulation and Wilkes et al. (Ref. 29). In our calculations, , . is varied by using different fluid densities and interfacial tension; varies between and , and between and . Wilkes et al. (Ref. 29) treated the drop surface as a free surface such that , . and are of the same order of magnitude as ours.
Variation of the minimum neck radius on the thread with time for the simulation in Fig. 2. is scaled by the nozzle radius and by the flow time .
Geometric setup for simulating drop formation in a flow-focusing device. Shown is the meridian plane of the axisymmetric device, and the computational domain is the top half. The characteristic length is the radius of the nozzle .
Snapshots of drop formation in the dripping regime at , . The time is made dimensionless by . In dimensional terms, the frequency of drop formation is roughly (enhanced online). [URL: http://dx.doi.org/10.1063/1.2353116.1]10.1063/1.2353116.1
The radius of the neck, measured at its thinnest part, oscillates in time before dropping to zero. is scaled by and by . The period of oscillation, converted using dimensional parameters in Anna et al.’s experiment, is about .
Development of the capillary wave inside a long nozzle. All other parameters are the same as Fig. 5. (a) Geometry of the domain; (b) radius of the thread as a function of the axial distance from the inlet of the nozzle to the outlet. The radius and the position are scaled by the nozzle radius .
Dependence of the drop radius on the flow-rate ratio . (a) Effect of varying one flow rate while keeping the other fixed. The intersection of the two curves, at , , and for the inner fluid and and for the outer fluid, is near the transition point; decreasing either or causes dripping while increasing either leads to jetting. (b) Comparison between two fixed values while varies: lower with , and higher with , . The former experiences the transition at while the latter is entirely in the jetting regime.
The drop radius increases with the radius of the downstream collection tube . , , , , and . A linear fit to the numerical data is .
The velocity field near the drop at (a) and (b) shortly before pinch-off. The parameters are the same as in Fig. 5. The maximum velocity (dark) is nearly and the minimum (white) is below .
A cycle of drop formation in the jetting regime at , . The time is made dimensionless by . In dimensional terms, the frequency of drop formation is roughly (enhanced online). [URL: http://dx.doi.org/10.1063/1.2353116.2]10.1063/1.2353116.2
Tipstreaming at , , , , .
Geometric setup for simulating the formation of compound drops. Shown is the meridian plane of an axisymmetric device, and the computational domain is the top half.
Snapshots showing the different stages of a successful encapsulation process. The times are made dimensionless by the flow time . , (enhanced online). [URL: http://dx.doi.org/10.1063/1.2353116.3]10.1063/1.2353116.3
Compound drop formation is sensitive to the flow rates. (a) 25% flow rate decreases from (b), (c) 25% of flow rate increases from (b). Conditions for (b) are identical to those in Fig. 14. The flow-rate ratios are kept fixed at 3:6:40, and the three streams have equal density and viscosity. Time is scaled by of (b).
Phase diagram for compound drop formation. The circles indicate formation of compound drops with one or two inner drops, and the crosses failure of encapsulation. The inner and the outer fluids are identical, and the viscosity ratio is between the middle and the outer fluid. is defined for the flow in the inner tube. The inner-middle-outer flow-rate ratios are fixed at 3:6:40.
Simple drop formation in the dripping regime when the inner fluid is (a) viscoelastic and (b) Newtonian. The snapshots are taken shortly before the drop detaches. , . In (a), , and the gray-scale contours depict the polymer tensile stress along the flow direction. The drop size is 7.5% larger than in (b), while the critical jet length, from the orifice to the point of pinch-off, is 10.1% longer.
Simple drop formation in the jetting regime when the inner fluid is (a) viscoelastic and (b) Newtonian. , . In (a), , and the gray-scale contours depict the polymer tensile stress along the flow direction, with a maximum , scaled by . Compared with the Newtonian case, the jet length is 9.0% shorter for the viscoelastic case and the drops are 4.0% smaller in diameter.
Viscoelastic effects on compound drop formation. All three fluids are Newtonian in (a). The middle fluid is viscoelastic in (b) and (c), two snapshots following pinch-off of the inner jet. , , and for (b) and (c). The times are made dimensionless by the flow time . The gray-scale contours indicate the level of polymer tensile stress in the middle fluid, with a maximum on the order of , scaled by .
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