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

Droplet impact is a very common phenomenon in nature and attracts attention due to its aesthetic fascination and wide-ranging applications. Previous studies on flowing liquid films have neglected the contributions of spatial structures of waves to the impact outcome, while this has recently been shown to have a significant influence on the drop impact dynamics. In this report, we outline a step-by-step procedure to investigate the effect of periodic inlet forcing of a flowing liquid film leading to the production of spatiotemporally regular wave structures on drop impact dynamics. A function generator in connection with a solenoid valve is used to excite these spatiotemporally regular wave structures on the film surface while the impact dynamics of uniform-sized droplets are captured using a high-speed camera. Three distinct regions are then studied; viz. the capillary wave region preceding the large wave peak, the flat film region, and the wave hump region. The effects of important dimensionless quantities such as film Reynolds, drop Weber and Ohnesorge numbers parameterized by the film flow rate, drop speed, and drop size are also examined. Our results show interesting, hitherto undiscovered dynamics brought about by this application of film inlet forcing of the flowing film for both low and high inertia drops.


Introduction
Droplet impact is a very common phenomenon in nature and attracts attention from any curious observer 1 . It constitutes an active research area due to its numerous applications including spray-cooling, fire-suppression, inkjet-printing, spray-coating, deposition of solder bumps on printed circuit boards, the design of internal combustion engines, surface-cleaning, and cell-printing 2 . Its application extends also to agriculture, e.g., sprinkling irrigation and crop spraying 3,4 . Pioneering work dates back to the 19 th century, with the work of Worthington 5 , while major advances have only been recently made due to the emergence of high-speed imaging 6 . Since then, several studies have been carried out; using different types of impact surfaces ranging from solids 7,8 , shallow, 9 and deep liquid pools 10,11 to thin films 12,13 .
However, despite the large volume of research on droplet impact on liquid surfaces (i.e., shallow and deep pools and quiescent films), impact on flowing thin liquid films has not received as much attention. In addition, hitherto, studies have neglected the contributions of spatial structures of waves to droplet impact dynamics.
In this report, we present a detailed experimental procedure to investigate the droplet impact process on flowing films whose dynamics are influenced by inlet-forcing of the liquid flow rate; below, we refer to them as 'controlled' films. We find that these have numerous applications in multiphase industries (e.g., in cooling towers, in distillation columns, and also in the annular flow regime observed in two-phase flows), especially as film control has become an important step in the intensification of both heat and mass transfer in many process industries 14 . The interested reader is referred to our previous work 15 for more details on the results of our research efforts on this.
This application of frequency oscillations of the inlet flow rate results in the formation of regular waves on the film surface. We focus on the solitary wave family, which is essentially characterized by widely-separated narrow peaks and is preceded by a series of front-running capillary waves 16,17,18 . We study the outcome of the impacts associated with the three main parts of the solitary wave structure: the 'flat film', 'wave hump', and front-running 'capillary wave' regions. We also contrast these results with those associated with uncontrolled flowing films. Our results show that the stochastic nature of wave appearance on the uncontrolled film markedly affects the outcome of drop impact, with the separate regions of the controlled film also showing new mechanisms, which we have detailed both qualitatively and quantitatively.
In the previous paper 15 , using the same procedure, we studied the effect of film control on droplet impact dynamics in the splashing regime. The obtained results showed both quantitative and qualitative differences in the crown morphology (height, diameter, wall thickness, tilt angle, and direction) as well as the number and size distribution of the ejected secondary droplets.
1. Once film flow is established on the rig, start the syringe pump and observe the impact of the dripping drops on the film surface. 2. Start the function generator and observe the production of spatiotemporally regular waves on the film surface. 3. Ensure successive drops are impacting the different regions of the controlled film surface. 4. Observe the post-triggering frame number and set this to approximately half of the video length to adequately capture the impact. 5. Power on the light source and trigger the image capturing once an impact occurs. 6. Power off the light source once image capturing is complete to avoid overheating of the liquid film. 7. Visually analyze the obtained snapshot on the computer screen. Check to see if the impact has occurred on one of the flat film, capillary wave, or wave hump regions. 8. Trim down the video to the portion showing the impact process and save the frame range in a video/image format. 9. Repeat 3.5 -3.8 and record individual impact on all regions on the film surface, viz. solitary hump, capillary waves, and flat film.

Image Post-processing and Analysis
1. Place a ruler in the field of view and calculate the spatial resolution by counting how many pixels fit across 1 cm. Using the calibration image, obtain a scale factor for image dimension measurement. 2. Compare the outcomes of the impact process on the different impact regions from the high-speed images. Check to see notable differences. 3. Using a suitable MATLAB image-processing routine, measure the characteristics features of the product of the impact process: viz. in the splashing mode, measure the crown height, diameter, wall thickness, tilt angle, crown-facing direction, number and size distribution of ejected secondary droplets. 4. Carry out similar quantitative analyses as 4.3 above for the low-Weber impacts. Count the pinch-off time of satellite drops from the timeframed images and measure the apex length and width of the column formed in partial coalescence before pinch-off of secondary drops. Measure the size of ejected secondary drops. Count the number of cascade in a repeated pinch-off process. 5. Observe all qualitative differences in each region.

Representative Results
Essentially, two categories of impacts were studied; the first was for drops with low inertia (i.e., drop Weber number, (We d= ρdu 2 /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 N u N 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 liftoff. In comparison, impacts on the wave hump are more prone to partial coalescence, partly due to the thickness of the film, the absence of preexisting 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.
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. 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.

Discussion
In this section, we provide a few tips necessary to ensure qualitative results are obtained from the protocol. First, the glass substrate on which the liquid film flows must be kept completely dirt-free to ensure the properties of the liquid film are kept uncompromised. This is achievable by regular cleaning (probably using a suitable detergent, and wiped off over a tray to avoid dissolution into the system). Similarly, there should be a regular replacement of the whole test-liquid after some experimental rounds, to guarantee accurate results.
Secondly, the fluid-distribution chamber must be well-meshed and also kept air-tight to ensure the outflowing liquid film is uniform. This can be done by manually siphoning air out of the distribution box before each experiment. The use of micrometer-steps at the film inlet is also advised to set the gap-height at the film inlet to the exact film thickness predicted by the Nusselt estimate of the film flow at the corresponding Reynolds number. This will prevent a hydraulic jump or backflow at the inlet.
The operation of the solenoid valve must also always be checked and ascertained properly. This is because an appropriate pulsation of the flow is required to ensure the production of the forced waves. This could be checked from the regular clicking sound of the solenoid valve as well as a perceived pulsation along the connection pipes. The liquid flow rate into the syringe pump must also be set carefully to ensure the droplets are ejected in a dripping manner, avoiding any pre-acceleration before falling.
Appropriate calibration of the high-speed camera must be ensured to obtain very accurate results. The aperture size must also be carefully chosen considering parameters like the depth of field, exposure time and overall image brightness. For the camera triggering during video recording, users are also required to estimate how many frames should be recorded before triggering. This may vary with individuals, depending on the drop impact time, hence, several trial tests for practicing are recommended before actual measurements. Similarly, the light source must be properly arranged and well-diffused to minimize shadows in the image.
It is important to note and remember that the main focus of the study is the contributions of waves to the impact dynamics of the falling drops, hence the formation of regular wave structures is essential to an accurate study of the underlying physics. In scenarios where the wave structures are observed to quickly transition to three-dimensional structures, it is advised that the substrate inclination angle be reduced 14,19 to facilitate a slower transition of the wave structures.
One limitation of the technique is observed in the absence of a measuring device specifying the actual instantaneous film thickness on each region of impact. This would have provided additional details on the overall observed phenomena.
In summary, the procedure outlined in this report can also be used to study simple wave evolution dynamics, while the high-speed imaging system described can be applied to many research fields with fast dynamics such as liquid drop break-up 21,22 /coalescence 23 , granular jets 24 , etc. where important phenomena are observed at a micro timescale.

Disclosures
The authors have nothing to declare.