Film Control to Study Contributions of Waves to Droplet Impact Dynamics on Thin Flowing Liquid Films

Essentially, two categories of impacts were studied; the first was for drops with low inertia (i.e., drop Weber number, (We_d= ρdu²/σ) ranging from 3.1 to 24.0 while the second was for drops with high inertia (i.e.,We_d 94 to 539) resulting in a splash outcome. The same experimental procedure, however, was followed for both studies. Other related dimensionless quantities used in the study include the film Reynolds number (Re = ρq/wµ, ranging between 55.5 and 333), the film Weber number (We = ρh_Nu_N²/σ, ranging between 0.1061 and 2.1024), the drop Ohnesorge number (Oh = µ/(ρσd)^1/2, ranging between 0.0018 and 0.0025) and the Kapitza number (Ka = σρ^1/3/g^1/3µ^4/3, which was calculated to be 3363 for water). The Nusselt film thickness (h_N = [(3µ²Re)/(ρ²gsinβ)]^1/3) was found to range from 4.034 x 10^-4 to 7.328 x 10^-4 m, while the Nusselt film velocity (u_N = ρgsinβh_N²/3µ) was found to range from 0.1376 to 0.4545 m/s. For all above equations, q is the film flow rate, varying between 0.001667 and 0.01 m³/s; β is the substrate inclination angle, fixed at 15˚ to the horizontal; µ and ρ are the viscosity and density, respectively, of water estimated at 0.001 Pa s and 1000 kg/m³; σ is the surface tension force (0.072 N/m); and g is the gravitational force (9.81 m/s²).

In the low inertia impacts, the trends observed, though a little similar (Figure 4), exhibited a number of distinctly spottable differences. First, it was generally noticed that the size of the satellite drop produced on the wave hump region was always bigger compared to other regions of impact. In retrospect, the opposite was found true on the capillary wave region. The satellite drops were always very small. This occurs because the radial wave produced by the impacting drop becomes suppressed by the existent capillary ripples. As a result, further wave propagation to vertically elongate the drop is inhibited, which results in the drop losing its potential to develop a sufficiently long vertical column, thereby leading to the ejection of only tiny secondary drops from the slender columns formed. It was also observed that the tendency of a cascade was much reduced on the wave hump compared to other regions. In all the cases examined, the product of partial coalescence, hardly experienced another partial coalescence, while on a flat film, up to three to four are observed. The column height was also observed to be higher and most tilted in the flow direction on the wave hump region in comparison with other regions.

On the flat film region in comparison with other regions of impact, there is an increase in the tendency of a bouncing outcome. This occurs due to the strong lubrication force exerted on the drop by this thin flat film, which slows down the drainage/thinning of the intervening air layer between the drop and the film, thereby preventing the merger. This then results in the observed drop deformation as well as the eventual lift-off. In comparison, impacts on the wave hump are more prone to partial coalescence, partly due to the thickness of the film, the absence of pre-existing waves (as found in the capillary wave region), and finally the reduced lubrication force caused by flow recirculation in this region. These cumulatively result in the generation of rather longer columns than those produced on other regions.

With an increase in liquid film flow rate (i.e., film Re); impacts on the capillary waves often resulted in a gentle sliding of the drop of the capillary wave without merger (see Figure 5a-5h). This rolling drop (Figure 5d-5f) later then climbs the on-coming solitary hump (Figure 5g and 5h) where it experiences a partial coalescence (not shown). However, the outcome of impact on the flat film region changes from a steady partial coalescence to favor the bouncing mode. In the case of the impact on the capillary wave, the increase in film Re led to more closely peaked capillary waves which then acted as a "cushion" on which the drop "rode", hence the observed sliding of the drops. At the least Re, a very quick pinching off of drop is usually observed on the flat film region (of size 90% of the initial drop), with this drop experiencing some "dancing" mode before it later merges and results in a normal partial coalescence. This is, however, not observed on other regions of the controlled film.

With an increase in drop We_d, it was observed that the column height increased both on the flat film region and the wave hump but reduced on the capillary wave region.

Finally, with an increase in drop size, longer and wider columns were observed on the flat film region, which in turn gave rise to a bigger satellite drop. However, on the wave hump, this was not observed, instead, a transition to total coalescence was observed. On the capillary wave, the increase in drop size led to reduced sliding of the drop and a transition to partial coalescence. The biggest drop, however, yielded almost immediately to total coalescence. A summary of these results is presented in Table 1.

Beyond droplet velocity 1.70 ± 0.03 m/s, a splash outcome is observed in all three regions on the film surface (Figure 6). However, though a similar outcome is observed as well in this regime, striking differences are observed in the morphology of crown formed-its height, diameter, wall thickness, tilt angle, coalescing time as well as number and size-distribution of ejected secondary droplets.

In the 'wave hump region', the crown structure is different from that in the 'capillary' and 'flat film regions', as its shape is more regular. It also possesses a thicker crown wall and the crown height is higher than those observed in the 'capillary' and 'flat film regions'. There are also fewer secondary droplets ejected from its rim in comparison with the crowns formed in the other regions. Finally, a longer coalescence time is observed before the crown is swept away by the flowing film.

In the 'capillary wave' and 'flat film region', the crowns formed are also quite different based on a number of features. First, it was observed that the rear height of the crown is affected by the capillary humps as well as the flow reversal dynamics in this 'capillary wave region', hence causing the crown formed to appear more upright. This flow reversal results in the transport of liquid mass backward which augments the rear height of the crown formed. This, however, is not observed on the flat films: the crown is naturally tilted in the liquid flow direction and tilts even further with increasing Re. This tilt can be observed in both the upstream and downstream ends of the crown. In comparison, on the capillary waves, as the film Re is increased, the rear side of the crown appears to become more 'upright' in a manner quite opposite to that observed on flat films. The crown height on the flat film is however, higher than that on the capillary waves due to the confinement of the substrate. There is also a more rapid onset of secondary droplet ejection from the crown rim, on the capillary waves in comparison with that on flat films. Finally, more secondary droplets are ejected on the rim of the crown on flat films than that on capillary waves.

Temporal evolution of the crown shows a weak dependence of the crown diameter on film Re in all regions of the flow. The weakest dependence on Re is observed in the 'wave hump region'. In the 'flat film region', the crown height is observed to increase with Re as expected, since larger Re are associated with thicker films. The degree of crown inclination towards the flow direction is also higher with increasing Re in the 'flat film', and 'wave hump' regions; this effect, however, appears to be less pronounced in the 'capillary wave region'.

In the 'wave hump region', there are fewer secondary droplets ejected with increasing Re. There appears to be a somewhat weak dependence of the crown height on Re, while there is a decrease in the crown coalescing time with increasing Re, which is the result of the increased speed of the flowing film upon which the impact occurs, which quickly sweeps the coalescing crown away from the original impact point. There is also a change in the inclination of the crown in the 'wave hump region' depending on the competition between the inertia of the impacting drop and that of the flowing film. At lower Re, the crown faces the downstream direction, while at higher Re values, it faces the upstream (Figure 7). This trend is not observed in the 'capillary wave' and 'flat film regions'.

In the 'capillary wave region', more secondary droplets are observed at lower Re. There is also an increase in the overall crown height with Re, and, at lower Re, droplet ejection is mainly towards the streamwise direction (with the crown rim higher at the front than at the rear and also tilted more towards the streamwise direction). The height becomes more symmetric at higher Re, which is believed to be as a result of the balancing effect of the higher humps which capillary waves possess at their rear, thereby balancing-off the crown rim height at the back.

With the drop Weber effect, it can be observed that the crown diameter increases at a greater rate with increasing We_d; the largest rate is associated with the 'wave hump region'. Further differences observed in the number and size distribution of the ejected secondary droplet in this splashing regime are shown in Figure 8 and Figure 9, respectively. A summary of these results is presented in Table 2.

Figure 1: Experimental rig. (A) Schematic representation of the experimental rig, consisting of the falling film unit for the flow of liquid film on an inclined glass substrate; a film control unit (consisting of a solenoid valve connected across a non-latching relay via data acquisition card and a function generator which sends automated signal controlling the opening and closing of the solenoid valve); a syringe pump used for the generation of droplets of controlled sizes from calculated heights above the film surface, and a high-speed camera for digital imaging. The obtained results are analyzed on the computer system. Reproduced from Adebayo & Matar 2017¹⁵ by permission of The Royal Society of Chemistry. (B) A pictorial view of the rig. (C) – (D) Pictorial description of lighting arrangement. Please click here to view a larger version of this figure.

Figure 2: Effect of film control on wave evolution dynamics on a flowing liquid film. (A) Shadowgraph image of film surface prior to film control. The film is characterized by the presence of naturally evolving waves which are stochastic in nature and exhibit irregular spatiotemporal dynamics. (B) Shadowgraph image of the film surface after forcing. The waves are spatiotemporally regular and predictable, rendering contributions from the spatial structure to drop impact easy to study. (C) Solitary wave formation on a controlled flowing liquid film highlighting the different regions on the film surface viz. capillary wave, flat film and wave hump regions. (D) Magnified view of a singular wave structure showing the flow profile in each zone. Reproduced from Adebayo & Matar 2017¹⁵ by permission of The Royal Society of Chemistry. Please click here to view a larger version of this figure.

Figure 3: Spatial resolution at 5000 fps. With a substrate inclination angle of 15˚, the spatial resolution is calculated to be 67.5 µm/pixel and 46.6 µm/pixel in the streamwise and spanwise directions, respectively. Please click here to view a larger version of this figure.

Figure 4: Effect of film control on the outcome of low inertia drops impacting different regions of a controlled flowing film, contrasted against an uncontrolled film. The droplet fall-height is 0.005 m, drop size is 3.3 mm, film speed is 5 x 10^-3 m³/s, forcing frequency is 2 Hz, corresponding to film Re 166.5, drop We 3.134 and Oh 0.0021. The drop approaches the film surface (a) and on contact (b), triggers the drainage of the intervening air layer between it and the film. These results in the deformation of the drop shape and a radial spread of capillary ripples on the film surface, initiated at impact point (c-d). Once the air layer is ruptured, a merger of the liquid drop with the liquid film is observed (e) and a vertical growth of cylindrical liquid column (in a partial/total coalescence case). This is followed by a run-up of capillary waves on the column formed, which elongates it. Finally, a pinch-off of a satellite drop is observed (g-h), in a partial coalescence case, which is of smaller size to the initial mother drop. A repeat of the coalescence process is seen as well (i-j). Qualitative differences are seen in the outcomes observed (either bouncing or sliding or partial coalescence) and the presence of a cascade; while quantitative differences are observed in the pinch-off time, the size (height and width) of the liquid column formed, size of ejected satellite drop, and the cascade points. Please click here to view a larger version of this figure.

Figure 5: Droplet sliding on the capillary wave region of a controlled flowing film. The droplet diameter is 2.3 mm, with a fall height of 0.008 m while the film flow rate is 10 x 10^-3 m³/s, corresponding to Oh = 0.0024, We_d = 5.014, and film Re = 333, respectively. Forcing was carried out at 2 Hz. (a) Approach. (b) Contact. (c-f) Rolling drop. (g-h) Climbing the oncoming solitary hump. Please click here to view a larger version of this figure.

Figure 6: Effect of film control on splashing phenomena on the different regions of impact on a controlled flowing film, contrasted against an uncontrolled film. The droplet diameter is 3.3 mm, with a fall height of 0.25 m while the film flow rate is 5 x 10^-3 m³/s, corresponding to Oh = 0.0021, We_d = 224.8, and film Re = 166.5, respectively. Forcing was carried out at 2 Hz. The liquid drop approaches the film surface (a) and immediately upon contact (b), develops an ejecta sheet which grows into a crown (c). The growing crown (d-e) later yields to a Rayleigh-Plateau instability which leads to the ejection of smaller droplets from its rim (f-j). The crown collapses afterward and coalesces with the film (k), being transported away by the oncoming flow. The unique differences in the impact outcome on the individual regions of impact are seen in the size (height and diameter) of the crown formed, number and size distribution of ejected secondary drops, the degree of crown tilt, wall thickness, crown facing direction and final coalescence time. Reproduced from Adebayo & Matar 2017¹⁵ by permission of The Royal Society of Chemistry. Please click here to view a larger version of this figure.

Figure 7: Effect of film Reynolds and drop Weber on crown propagation in the 'wave hump region'. The droplet size is 3.3 mm, corresponding to Oh = 0.0021 and the drop fall heights was varied from 0.20 to 0.35 m (corresponding to We_d = 179.8 – 314.7) while Re is in the range of 55.5 to 333. The red diamonds depict outcomes with the crown facing the downstream direction while the blue diamonds show upstream-facing crown outcomes. The crown inclination is affected by the competition between the inertia of the impacting drop and that of the flowing film. Specifically, at low Re, the crown is inclined towards the streamwise direction but as the inertia of the flowing film gains in importance, the direction changes and faces upstream. This crown-upstream-facing direction is maintained beyond a Re value of approximately 250 regardless of the magnitude of We_d. Reproduced from Adebayo & Matar 2017¹⁵ by permission of The Royal Society of Chemistry. Please click here to view a larger version of this figure.

Figure 8: Variation of the number of secondary droplets ejected from the crown rim in the different impact regions of a controlled film (viz 'capillary wave', 'flat film', and 'wave hump' regions, shown from left to right, respectively) contrasted against an uncontrolled film. The droplet size is 3.3 mm corresponding to Oh = 0.0021, and the drop heights have been varied from 0.20 to 0.35, resulting in impact velocities within the range 1.981 – 2.621 m/s (corresponding to We_d = 179.8 – 314.7). The red rectangles depict drop fall height of 0.35 m, the green diamonds 0.3 m, the blue circles 0.25 m, and the orange squares 0.2 m, respectively. The number of ejected secondary drops increase with drop We in all regions while an uneven trend is observed with film Re increase: On the wave hump, there is a decrease in the number of ejected secondary drops while on both capillary wave and flat film regions, there is a slight increase. A dip is noticed around film Re 166.5 for the capillary wave, which occurs as a result of the competition between the tangential velocities of the drop and that of the film. The disproportionate trend observed on the uncontrolled films is believed to occur as a result of the stochastic nature of the waves on the film surface. Reproduced from Adebayo & Matar 2017¹⁵ by permission of The Royal Society of Chemistry. Please click here to view a larger version of this figure.

Figure 9: Effect of impact region on the size distribution of ejected secondary droplet on a controlled film contrasted against an uncontrolled film. The drop size is 3.3 mm while the film flow rate is 5 x 10^-3 m³/s corresponding to a film Re of 166.5 and drop Oh 0.0021. The drop's fall heights are 0.2, 0.25, 0.3 and 0.35 m corresponding to We_d 179.8, 224.8, 269.8 and 314.7 respectively. On the capillary wave, the shape of the distribution is largely unaltered with Weber number increase but a noticeable increase in the number of drops of the range 0.5 to 1.0 mm. On the flat films, however, the size distribution is observed to vary from 0 to 2.0 mm, and a shift is observed towards the 0 to 0.5 mm sized drops as the Weber number is increased. This increase in the number of small drops ejected clearly differentiates the flat film region from the other regions. On the wave hump, the size distribution shows that large drops in the range (1.0 to 2.0 mm) are ejected even for the smallest Weber number examined. In contrast to the above, the drop size distributions associated with an uncontrolled film do not exhibit a discernibly distinct shape owing to the stochastic nature of waves on such films. Reproduced from Adebayo & Matar 2017¹⁵ by permission of The Royal Society of Chemistry. Please click here to view a larger version of this figure.

Table 1. Parametric differences on low-inertia droplet impact dynamics on different regions of a controlled flowing film.

Table 2. Parametric differences on high-inertia droplet impact dynamics on different regions of a controlled flowing film (the splashing regime).