6.4 Results and discussion
6.4.2 Capillary waves and pinch-off dynamics
Traveling capillary waves have been reported earlier following the work of Blanchette and Bigioni [140] in the context of coalescence of a drop onto a liquid pool, where they reported that the partial coalescence critically depends on the ability of the capillary waves to vertically stretch the drop and delay the vertical collapse. The traveling capillary waves carry momentum along with it. Later on, these propagating capillary waves are studied extensively [107, 108, 116] and it has been observed that the surface tension and the curvature change the pressure in the liquid, because of which traveling capillary wave is engendered. The capillary waves propagate along the interface of the drop and converge at its apex. The converged capillary waves exert a vertical pull and stretch the drop upwards. This upward stretching can be seen in Fig. 6.4, where the dotted line indicates the position of the drop apex at the time of start of the coalescence. Owing to the capillary pull, the drop is stretched
6.4 Results and discussion 113 upward and crosses the dotted line, which can be seen at τ = 0.66 and τ = 0.82 of Fig. 6.4.
= 0.023 = 0.396 = 0.66 = 0.82
Figure 6.4: The coalescence sequence of two drops at different instants of time. The dotted line represents the position of the mother drop tip at the start of coalescence.
The upward pull exerted by the converging capillary waves vertically stretch the drop, which can be seen at τ = 0.66 and τ = 0.82. The dimensionless parameters are Oh1 = 0.005, Oh2 = 0.001, Bo= 0.131 and A = 0.8.
z*
0 0.2 0.4 0.6 0.8 1
5.2 5.3 5.4 5.5 5.6 5.7 5.8
Df / Dm = 1.0 Df / Dm = 1.5 Df / Dm = 2.0 Df / Dm = 2.5 Df / Dm = 3.0 Df / Dm = 4.0
Figure 6.5: The temporal variation of the drop tip for different values of diameter ratio Df/Dm. The dimensionless parameters are Oh1 = 0.005, Oh2 = 0.001, Bo= 0.131 and A = 0.8.
Capillary wave is produced by the expanding neck because of the high curvature generated near the contact line. This high curvature is generated at the initial stage of merging and decreases with the expansion of the neck. With the increased value of Df/Dm, the flatness of the father drop surface increases. Owing to the higher
flatness of the father drop surface, a higher curvature is generated in the contact region. Higher curvature produces stronger capillary waves [166, 167]. This effect of sharpness of the neck region on the generated capillary waves can be seen in Fig. 6.5, where the drop tip position is plotted with time for different values ofDf/Dm. The vertical stretching of the mother drop decreases continuously with the decrease in Df/Dm because of the decrease in the strength of the propagating capillary waves.
This local curvature contributes to the limiting condition of critical diameter ratio, below which satellite formation is inhibited. Earlier, Gilet et al. [165] also observed the similar motion of the drop tip owing to the effect of capillary waves during the coalescence of a drop on a liquid pool. Zhang and Thoroddsen [168] plotted the tip position of a bubble with time during the coalescence of two unequal sized bubbles.
They observed higher fluctuation of the bubble tip compared to the coalescence of drops and concluded that during the coalescence of bubbles, several wave crests reach the apex before the daughter bubble is pinched off.
(a) (b)
z*
0 0.2 0.4 0.6 0.8 1
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8
Oh1 = 0.0001 Oh1 = 0.001 Oh1 = 0.005 Oh1 = 0.01 Oh1 = 0.02 Oh1 = 0.03
z*
0 0.2 0.4 0.6 0.8 1
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8
Oh2 = 0.0001 Oh2 = 0.001 Oh2 = 0.005 Oh2 = 0.01 Oh2 = 0.02 Oh2 = 0.03
Figure 6.6: The temporal variation of the drop tip for different values of Ohnesorge number. In (a) the drop tip position is plotted for different drop liquid viscosity keeping the surrounding fluid viscosity constant (Oh2 = 0.001) while in (b) sur- rounding fluid viscosity is varied keeping the drop fluid same (Oh1 = 0.005). The other dimensionless parameters are Bo= 0.131 and A = 0.8.
The propagation of the capillary wave is damped by viscosity, which reduces the vertical pull on the drop. Moreover, the increased viscosity also reduces the amplitude of the capillary waves, because increased viscosity reduces the curvature change during neck expansion [159]. This viscous damping effect of the capillary
6.4 Results and discussion 115 waves can be seen in Fig. 6.6, which shows the position of the upper drop apex with time for different values of Ohnesorge number. In Fig. 6.6(a), the drop tip position is plotted for different values ofOh1. The increased values ofOh1 decreases the vertical stretching of the drop, because the propagation of the capillary waves are damped by the increased viscosity of the drop fluid. The same effect is observed when the viscosity of the surrounding fluid is increased, keeping the viscosity of the drop fluid same. This effect of the surrounding fluid viscosity can be observed in Fig. 6.6 (b), which shows the temporal variation of the drop tip for different Oh2. Both the drop fluid and the surrounding fluid viscosity affect the capillary wave propagation. It has been observed that the propagation speed of the capillary waves is dominated by the high viscosity fluid.
During the coalescence of a drop on a flat liquid pool, capillary waves also travel along the flat interface, away from the point of contact of the drop and the pool interface. The capillary waves traveling along the liquid pool interface have no effect other than deforming the liquid pool interface. In a similar manner, during the coalescence of two unequal sized drops, the capillary waves travel along the interface of the father drop in the downward direction. However, during the coalescence of two unequal sized drops, not only the capillary waves traveling through the interface of the mother drop, but also the waves moving through the interface of the father drop in the downward direction affect the coalescence dynamics. The capillary wave motion through the interface of the father drop is shown in Figures 6.7(a)and 6.8(a).
The capillary waves, which travel through the interface of the lower father drop converge at its bottom and impart a downward pull on the interface. The capillary waves pull the drop bottom in the downward direction and stretch it to elongate the drop, in the similar way it pulls the upper drop in the upward direction (and forms the column structure). Similar stretching was observed earlier by Menchaca-Rocha et al. [161] in their experiments of coalescence of two mercury drops. The stretching is more prominent with the decrease in the size of the father drop.
In the case of bigger father drops, the capillary waves need to travel a longer distance to reach the base of the father drops. The capillary wave motion is continu- ously damped out by viscosity, because of which, the strength of the capillary waves continuously decreases and makes the downward capillary pull weaker. As a result, the deformation of the drop decreases as the size of the drop increases and finally becomes unaffected by the capillary wave motion once it crosses a critical limit.
However, in the case of the smaller father drops, the capillary waves reach the
(a)
(b)
(c)
Figure 6.7: (a) The movement of the capillary wave is indicated in the streamline plot, during the coalescence of two drops having diameter ratio Df/Dm = 1.8. The capillary waves reach the bottom of the lower drop before satellite formation and restrict the pinch-off. The profiles from left to right are at timeτ = 0.16, 0.36, 0.53 and 0.82 respectively. (b)u−velocity and (c)v−velocity contour at the same instants of time. The axis labels are in meter. The other non-dimensional parameters are Oh1 = 0.005, Oh2 = 0.001, A = 0.8, and Bo= 0.131.
6.4 Results and discussion 117
(a)
(b)
(c)
Figure 6.8: (a) The movement of the capillary wave is indicated in the streamline plot, during the coalescence of two drops having diameter ratio Df/Dm = 2.72. The pinch-off takes place before the capillary waves reach the bottom of the lower drop.
The profiles from left to right are at timeτ = 0.16, 0.36, 0.53 and 0.96 respectively.
(b) u−velocity and (c) v−velocity contour at the same instants of time. The axis labels are in meter. The other non-dimensional parameters are Oh1 = 0.005, Oh2
= 0.001, A= 0.8, and Bo = 0.131.
drop bottom early, and the capillary pull affects the drop shape and the coalescence dynamics. The capillary wave motion on the lower drop gives a critical condition for the pinch-off of the daughter drop. The downward pull exerted by the capillary waves stretches the father drop and hinder the inward movement of the neck (Fig.
6.7) which finally restricts the pinch-off. Because of this reason, the capillary waves must not reach the bottom of the father drop before the neck merges, for pinch-off to occur. With the increase in diameter ratio of the two drops (Df/Dm), the time taken by the capillary waves to reach the bottom of the father drop increases. With further increase in the diameter ratio, the neck merges before the capillary waves reach the drop bottom (Fig. 6.8) and culminate the pinch-off.
The criterion of coalescence, whether it is partial or complete, is determined by the competition between the horizontal and vertical rate of collapse [108, 140]. The upward pull generated by the capillary waves stretches the mother drop upward and retards the vertical rate of collapse. This convergence of capillary waves can stretch the drop upwards by as much as 30% of its initial radius [140]. This can be seen in Fig. 6.4, which shows the axial position of the drop tip at different time instants.
Figures 6.7 (b) and 6.8 (b) show the u−velocity contour, and, Figures 6.7 (c) and 6.8 (c) show the v−velocity contour in the coalescence regime at different time instants. The horizontal and vertical movement inside the drop can be seen in the u−velocity and v−velocity contour. The upward momentum, exerted by the converged capillary waves, can be seen in Fig. 6.7 (c) and Fig. 6.8 (c). This upward pull stretches the drop and delays the vertical collapse of the drop. Because of this delay in vertical collapse, the horizontal collapse gets sufficient time for completion and the neck pinches off at the centerline to form the daughter drop.