Major findings

The bubble shape during the growth of the bubbles, the merger event and the departure of the merged bubble is shown in Figure 5.20 for a contact angle ofθ =40^◦. Both bubbles are shown although only one of them is actually simulated as mentioned above. From the moment of nucleation to around 5.7 ms the bubbles grow without having any visible influence on each other. However, if the results are analyzed in detail, one can see that there is a small influence. Rather than growing directly above the nucleation site, the bubble centers move slightly away from each other. At 5.7 ms the bubbles have grown big enough to touch each other and they immediately merge by generating a small circular vapor bridge from one bubble to the other. Due to the very high curvature of the liquid-vapor interface at the perimeter of the vapor bridge, the circular opening between the bubbles grows rapidely (see Figure 5.21). In a very short time of less than 0.5 ms the vapor bridge grows to a considerable size reaching almost down to the heater surface. In the following images of the sequence, capillary waves can be seen which travel along the liquid-vapor interface. These waves originate from the position where the liquid-vapor interfaces merge and cause a large excitation of the interface motion. Once the waves have run all the way along the liquid-vapor interface, they are reflected at the outer part of the bubble and move back inwards.

The barrel-like shape and the lemon-like shape that can be seen at 7.9 ms and 10.2 ms, respectively, are typical for bubble or droplet mergers. Siedel and co-workers [97] observed very similar bubble shapes and capillary wave motion during experimental investigations of lateral bubble mergers. Ata [2]

investigated the dynamics of merging air bubbles with and without particles deposited on the liquid-gas interface and also observed very similar bubble shapes and capillary wave motion. As mentioned above, the vapor bridge between the bubbles does not reach all the way down to the heater surface. A small amount of liquid is trapped underneath the vapor bridge. At 10.2 ms one can see that the shape of this trapped liquid volume corresponds almost to a truncated cylinder. Although the bubbles have already merged, their vapor patches on the heater surface are still separated from each other by the trapped liquid. In principal, the trapped liquid is pushed sideways into the bulk liquid by surface tension forces (see Figure 5.22). However, this process does not seem to be fast enough. The trapped liquid volume becomes unstable and pinches of the bulk liquid. At the moment when the vapor patches merge (between 11.2 and 11.7 ms) a considerable amount of trapped liquid remains within the merged vapor patches. Thus, the generation of the liquid droplet sitting inside the merged bubble which was observed

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Figure 5.20:Bubble dynamics during a lateral bubble merger. Contact angle isθ =40^◦.

Figure 5.21:Detailed view of the merging process and the growth of the vapor bridge between the bubbles. Contact angle isθ =40^◦.

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Figure 5.22:Detailed view of the trapped liquid, the elevated pressure (in Pa) within the trapped liquid and the resulting flow into the bulk liquid at 10.2 ms. Contact angle isθ =40^◦.

experimentally, is nicely predicted by the simulation. After the formation of the droplet, the merged bubble starts to detach from the heater surface due to buoyancy. The droplet has only little effect on the detachment of the merged bubble. At 16 ms one can see that the droplet only causes a separation of the dry vapor patch on the heater surface into two vapor patches just before the bubble lifts off.

The formation of the droplet inside the merged bubble is a result of the hydrodynamics and the wetting behavior. In principal, there are two competing effects that govern the process. The first effect results from surface tension and leads to an elevated pressure inside the trapped liquid which pushes the liquid sideways into the bulk liquid. This elevated pressure inside the trapped liquid and the resulting flow are shown in Figure 5.22. The magnitude of the pressure elevation depends on the surface tension coefficient and the curvature of the liquid-vapor interface of the trapped liquid. As mentioned above, the volume of the trapped liquid corresponds to a truncated cylinder. The curvature of the truncated cylinder depends on its size, but also on the contact angle. The second effect is the inertia of the trapped liquid which limits the transport of the liquid from the trapped position into the bulk. If the overall merging process is slow enough or if the surface tension forces are strong enough, the trapped liquid might be able to flow completely into the bulk liquid. However, if the merging process is very fast, the trapped liquid does not have enough time to escape. Consequently, the trapped liquid pinches of the bulk liquid and forms a droplet. These considerations are supported by the fact that Mukherjee and Dhir [80] did not observe the formation of a droplet when they simulated lateral mergers of two bubbles. Their simulations were performed with water at a pressure of 1 bar which has a significantly higher surface tension coefficient and a smaller liquid density than HFE-7100. Under these circumstances the pressure within the trapped liquid is higher while the inertia of the liquid is smaller. Thus, the trapped liquid can more quickly escape to the bulk liquid and no droplet formation can be observed.

It is obvious that the bubble dynamics during a bubble merger strongly depend on the contact angle.

In order to learn more about the bubble dynamics and the droplet formation, a small parameter study for the contact angle was accomplished. The contact angle is assumed to be independent of the local evaporation rate and of the speed of the 3-phase contact line. In addition to the simulation with a contact angle of θ = 40^◦ which has been discussed above, simulations with larger and smaller contact angles are performed. The results of the simulation with a contact angle ofθ =60^◦ are shown in Figure 5.23.

There is not much qualitative difference compared to the simulation with a contact angle ofθ = 40^◦. The growth time until the bubbles are grown big enough to first touch each other is almost identical.

After the bubbles merge, the vapor bridge between the bubbles grows rapidely and causes some capillary waves that run along the liquid-vapor interface. The volume which is occupied by the trapped liquid is rather slim compared to the case with a contact angle ofθ =40^◦, which results in an earlier formation of the droplet (around 8.4 ms compared to 11.5 ms). The fact that the trapped liquid volume is rather

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Figure 5.23:Bubble dynamics during a lateral bubble merger. Contact angle isθ =60^◦.

slim is due to the shape of a truncated cylinder. If the volume of the trapped liquid is the same, a higher contact angle leads to a slimer cylinder shape. While the droplet formation happens earlier, the bubble detachment process takes longer compared to the case with a contact angle of θ = 40^◦. The merged bubble lifts off from the heater surface at around 22 ms compared to 16.5 ms. The longer contact to the heater surface is expected as the attaching forces at the 3-phase contact line are higher in the case of a higher contact angle. The effect has also been mentioned during the discussion of the simulation results for the single bubble configuration (see section 5.1).

The results of the simulation with a contact angle of θ = 20^◦ are shown in Figure 5.24. While an increase of the contact angle compared to θ = 40^◦ did not change the results qualitatively, the small contact angle ofθ =20^◦leads to significantly different bubble dynamics. As in the simulations discussed before, the bubbles grow without much interaction until around 5.8 ms. However, here the formation of the vapor bridge between the bubbles leads to a complete lift-off of the merged bubble from the heater surface. As discussed above, the vapor bridge forms and grows rapidely. The liquid underneath the vapor bridge is pushed down towards the heater. It can either escape to the side into the bulk liquid or flow towards the contact line region and push the 3-phase contact line of the trapped liquid outwards. The situation is shown in Figure 5.25 for the simulation with a contact angle of θ = 40^◦. The downward motion of the liquid-vapor interface pushes the liquid towards the stagnation point underneath the vapor bridge. The resulting outward motion of the 3-phase contact line of the trapped liquid can clearly be seen. This leads to a decreasing size of the vapor patches and in the case of a contact angle ofθ =20^◦ to a bubble lift-off at around 8.9 ms. Interestingly, the dynamics of the upper part of the bubble does not seem to be much affected by the lift-off. Again, a barrel-like shape and a lemon-like shape can be

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Figure 5.24:Bubble dynamics during a lateral bubble merger. Contact angle isθ =20^◦.

Figure 5.25:Detailed view of the flow underneath the liquid bridge at 6.2 ms. The flow towards the stagnation point and the outward motion of the 3-phase contact line of the trapped liquid can clearly be seen. Contact angle isθ =40^◦.

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observed at 8.1 ms and 9.9 ms, respectively. After the lift-off, the bubble remains oscillating in a very short distance to the heater and starts to rise only slowly. The slow rise and the oscillations lead to a re-attachment of the bubble at around 13.8 ms. However, the re-attachment lasts only for a very short time as the bubble is now rising with a higher velocity and lifts off completely from the heater surface.