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Air assisted impact of drops: The effect of surface wettability
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Air assisted impact of drops: The effect of surface wettability Ramesh Kumar Singh , Deepak Kumar Mandal PII: DOI: Reference:
S0301-9322(19)30928-0 https://doi.org/10.1016/j.ijmultiphaseflow.2020.103241 IJMF 103241
To appear in:
International Journal of Multiphase Flow
Received date: Revised date: Accepted date:
26 November 2019 29 January 2020 3 February 2020
Please cite this article as: Ramesh Kumar Singh , Deepak Kumar Mandal , Air assisted impact of drops: The effect of surface wettability, International Journal of Multiphase Flow (2020), doi: https://doi.org/10.1016/j.ijmultiphaseflow.2020.103241
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Highlights
Airflow assisted impact of drops on surfaces with various wettabilities is studied.
The spreading reduces at higher airflow and Weber number for less wetting surfaces.
Combined action of airflow shear and normal imposed pressure is responsible.
The comparison of measured spread with a model improves when wettability reduces.
----------------------------*Corresponding author email: [email protected] (D. K. Mandal), Phone: +91 326 223 5058.
Air assisted impact of drops: The effect of surface wettability
Ramesh Kumar Singh, Deepak Kumar Mandal*
Department of Mechanical Engineering, Indian Institute of Technology (ISM), Dhanbad, PIN 826004, India.
Abstract
The airflow assisted impact of water drops on surfaces with various wettabilities is studied to understand the effect of shear force and normal imposed pressure provided by the incident airflow. Results show that the maximum spreading factor increases with the impact Weber number for the drops impacting a hydrophilic surface. For other surfaces, the spreading increases with the airflow for a given Weber number. However, the factor is observed to suddenly decrease at higher Weber numbers and airflow and for the less wetting surfaces. For the hydrophilic surface, as the drop continues spread, higher adhesive force between the drop and surface causes the rim to become thinner. The shear force pushes the rim outward. As the wettability decreases, a deceleration is observed at the early stage of spreading, forming fingers and thicker rim at the last stage. The thicker rim causes the viscous dissipation to decrease suddenly for the highest Weber number and airflow tested. The thicker rim resists the drop to spread. Detachment of the satellite drops on a hydrophilic surface increases with the airflow since the shear force increases. The normal imposed pressure applies pressure on the fingers and gives momentum to the lamella in the radial direction. A comparison of the measured maximum spreading with the one obtained from an existing model, is provided. Better comparison is found for the cases of less wetting surfaces, whereas good comparison is observed only at higher impact Weber numbers for the hydrophilic surface.
Keywords: Drop impact; Airflow assisted impact; Wettability; Maximum spreading factor; Normal imposed pressure.
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1. Introduction
The impact of drops on a dry solid surface is being analyzed since decades due to its vast applications in many fields, like in fire suppression (Pasandideh-Fard et al. 1996; Chandra and Avedisian 1991; Ukiwe and Kwok 2005), in automotive engines (Chandra and Avedisian 1991; Ukiwe and Kwok 2005), in spray cooling of hot surfaces (Ukiwe and Kwok 2005), in spray forming (Rioboo et al. 2001). The drop shows various outcomes after the impact depending on the wettability of the surface (Kim and Chun 2001), viscosity of the liquid (Kumar and Mandal 2019), inertia (Antonini et al. 2012), and surface tension force (Rioboo et al. 2001). The drop may spread, recede, deposit, rebound, stick, depending on the impact Weber number and the wettability of the surface (Rioboo et al. 2001). The initial stage of spreading is determined by the Weber number, whereas the last stage depends on the wettability of the surface (Schiaffino and Sonin 1997). When the Weber number is within 30 to 200, the wettability dominates and affects the deformation of the drop, whereas, the inertia force dominates when the Weber number is higher than 200 (Antonini et al. 2012). The effect of the wettability is dominant in the recoiling stage, as well, even at higher Weber numbers (Kim and Chun 2001). The drop may rebound after the impact depending on the wettability (Harley and Brunskill 1958). There are two different regimes of rebound. In the first, inertia force dominates over the surface tension force (i.e., for high impact Weber numbers) and results in less elastic rebound. However, larger energy is lost during spreading and oscillation. Second regime, known as Quasi-static regime, occurs at lower impact Weber numbers. In this regime, energy loss is minimal (Biance et al. 2006). Three different types of rebound are observed for drops impacting on heated surfaces whose temperature is near the Leidenfrost point. They are, reflection rebound, explosive rebound, and explosive detachment. These rebound occurred sequentially with decreasing wall temperature (Liang et al. 2016).
The surface temperature and Weber number both affect the detachment time of the drop, as well (Liang et al. 2016). The Weber number and the surface temperature have negligible effect on the time required to reach the maximum spreading. Different regimes
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of heat transfer (i.e., film evaporation, nucleate boiling, transition boiling, and film boiling) are observed, as well, for which suitable models exist in the open literature to describe the process (Liang and Mudawar 2017).
The rebound depends on the recoiling, and the recoiling depends on the contact angle and the impact Weber numbers (Borisov et al. 2003). The temperature of the surface is reported to affect the receding, as well (Kumar and Mandal 2020).
So, the wettability is of major focus due to its various applications, like, in water proof textiles (Kim and Chun 2001; Fukai et al. 1995), self-cleaning windows (Fukai et al. 1995; Antonini et al. 2013), and is described by the contact angle between the drop and the surface during spreading (De Gennes 1985). The spreading diameter is inversely proportional to the contact angle (Fukai et al. 1995). The advancing contact angle can be used to develop a model to obtain the maximum spreading of a drop (Pasandideh-Fard et al. 1996). Few models compare well with the measured results, but over predict the diameter during recoiling due to the change in the shape (Pasandideh-Fard et al. 1996). Better models are reported, as well, to find the maximum spread by using Young contact angle instead of the advancing contact angle with an assumption that the shape is cylindrical at the maximum spread (Ukiwe and Kwok 2005), and by incorporating the effect of surface tension (Tsurutani et al. 1990). Lee et al. (2016) provided the expression for the total time (tmax) taken by a drop to reach the maximum spreading diameter in terms of surface tension of the liquid, and modified the model of Mao et al. (1997) by incorporating tmax along with the dynamic contact angle at the maximum spread instead of the equilibrium contact angle. The time required to reach the maximum spreading mainly depends on the surface tension of the drop and has negligible effect of the surface wettability and viscosity of the drop. The surface tension plays a major role in the spreading, compared to the viscosity of the drop (Liang et al. 2019). A detailed model incorporating all the effects has been demonstrated by Laan et al. (2014). Wildeman et al. (2016) estimated viscous dissipation energy during the spreading of drop for the free and no slip boundary conditions at different impact Weber numbers.
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The wettability is responsible for the formation of fingers on the rim during spreading, as well as, during recoiling (Fukai et al. 1995). The formation of fingers is observed when a drop impacts at ambient condition (Thoroddsen and Sakakibara 1998; Allen 1975; Bang et al. 2011), and is due to the instability occurred for the rapid deceleration of rim during spreading (Allen 1975). The Ohnesorge number and surface roughness also decelerate the spreading (Tang et al. 2017). The formation of fingers during spreading provides the origin of splashing (Mundo et al. 1995; Xu et al. 2007) for which the ambient condition is reported to be important (Xu et al. 2005; Bang et al. 2011). With the decrease in the ambient pressure, the splashing decreases even at higher Weber numbers since no fingers form during spreading (Xu et al. 2005). However, increase in the air pressure, decreases the spreading diameter (Bang et al. 2011).
The underlying physics of finger formation due to the influence of the surrounding gas is well described in the recent review papers (Josserand and Thoroddsen 2016). The fingers give rise to the splashing. The effect of the surface wettability on the finger formation is reported. The surface having lower wettability, results more fingers on the outer rim of the drop during spreading, and promotes splashing (Josserand and Thoroddsen 2016). The physics behind the occurrence of splashing while impacting on a thin sheet of liquid at higher Weber numbers is addressed by Liang and Mudawar (2016). The thin film at the surface promotes splashing. Formation of crown and ejecta sheet, while impacting on a thin film is addressed, as well.
The onset of splashing and receding are delayed at higher Weber numbers for the case of gas assisted impact, i.e., when a propellant gas is incident on the surface along with the drop (Ebrahim and Ortega 2017). Aerodynamic forces like normal and shear forces provided by the gas, become important for the gas assisted impact. The deformation of the drop gets enhanced when the work done by the normal force of air is greater than 10% of the kinetic energy of the impacting drop, especially for lower Weber numbers (Diaz and Ortega 2016). Different regimes, in which the effect of normal and shear forces becomes
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negligible, exist. The effect of the propellant gas is significant on the deformation when the critical Capillary number is greater than or equal to 0.35 (Ebrahim and Ortega 2017). The maximum spreading factor increases with the airflow for a given Weber number, compared to the normal free-falling drop impact case (Ebrahim et al. 2017; Diaz and Ortega 2016; Ebrahim and Ortega 2017; Singh and Mandal 2019). However, for higher impact heights and airflow, the maximum spreading reduces, since the rim starts becoming thicker (Singh and Mandal 2019). The viscous force, surface tension, and inertia force affect the growth of rim thickness during spreading (Zang et al. 2019). When the ratio of the imposed pressure to the internal pressure of spreading becomes higher at higher Weber numbers, the imposed pressure restricts the radial flow, and limits the maximum spread (Singh and Mandal 2019). Few researches are only reported regarding the gas / air assisted impact, and therefore, a huge knowledge gap exists in the area. For example, works on the effect of the wettability when a drop is subjected to the incident air / gas flow, are limited.
Taken together, it can be concluded that although there exist many researches regarding the impact of freely falling drops, but few researches are only reported regarding the gas / air assisted impact, specifically, works on the effect of the wettability are limited. The outcome may be different for the case of air / gas assisted impact on surfaces with various wettabilities. If the maximum spreading increases for a given surface compared to case of the impact of free-falling drops, then the process becomes helpful for many industries. Hence, to fulfill this research gap, present study is executed. The effect of the normal imposed pressure and the shear force of a drop impacting surfaces with various wettabilities are studied at different Weber numbers. The setup used for the study is explained below.
2. Experimental setup and methodology
The experimental setup is explained in Singh and Mandal (2019), and is illustrated here in short. A schematic is provided in Figure 1. A vertical wind tunnel (acrylic made) was fabricated and a fan (12V, DC, 340 m3/h, outer dimension of 0.1 m × 0.1 m × 0.02 m,
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with speed regulator) was fitted at the top to generate the desired airflow towards the surface. The surface of dimension of 75 mm × 12 mm × 1 mm was placed about 10 mm above the bottom of the wind tunnel, so that the airflow becomes incident on the surface and can escape through the sideways. The bottom, with the dimension of 0.1 m × 0.1 m × 0.23 m, was considered as the test section. The cross section was rectangular, and a vertical slit of about 5 mm thickness was made on one side so that the height of the needle tip (outer diameter of the needle = 0.81 mm) from which a drop impacts, can be varied by moving the needle syringe arrangement up and down using a stand (see Figure 1). The needle syringe arrangement was kept horizontal and was firmly fixed with the stand. The tip of the needle was centrally placed, just above the surface, so that whenever one pushes the plunger of the needle, a drop (initial diameter of 3.47 ± 0.18 mm) can fall on the surface. A high-speed camera (0.0108 pixel / micron), fitted with a telecentric lens, was used to capture the entire process with the help of suitable backlights. The process was captured at three thousand frames per second.
It is important to mention that, the shape of the drop was not spherical before the impact. The spherical shape was assumed because the difference in the horizontal (Dh = 3.506 ± 0.18 mm, see the supporting material for the figure) and vertical length (Dv = 3.247 ± 0.18 mm, see the supporting material for the figure) was less than 8%. To simplify the calculation in the model, and to determine the dimensionless numbers, an equivalent 1
spherical diameter D0 was obtained (D0 = (𝐷ℎ2 𝐷𝑣 )3), as demonstrated in Mao et al. (1997), Sikalo et al. (2002), and Liang et al. (2019).
The air velocity was measured using an anemometer (Waco digital Anemometer, AVM-04A, resolution 0.01 m/s, Make: India), and was measured at five different locations in the test section to ensure uniformity in the magnitude of the airflow. The velocity was measured about 10 mm above the surface. The incident air velocities were set at 2.4 ± 0.03 m/s, 4.7 ± 0.03 m/s, and 6.8 ± 0.04 m/s, for the experiments. The Reynolds numbers, 𝑅𝑒𝑎𝑖𝑟 (
𝜌𝑎𝑖𝑟𝑉𝑎𝑖𝑟 𝐷0 𝜇𝑎𝑖𝑟
, where 𝜌𝑎𝑖𝑟 is the density of air, 𝑉𝑎𝑖𝑟 is the velocity of the airflow, 𝐷0 is the pre
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impact diameter of the drop, and 𝜇𝑎𝑖𝑟 is the dynamic viscosity of air), which indicates the role of the shear stress due to the airflow in the deformation of the drop, were 552 ± 10, 1081 ± 13, and 1564 ± 18, respectively. The numbers are mentioned as 552, 1081, and 1564, respectively, hereafter in the text.
Figure 1: Details of the experimental setup.
The heights, from which the drop was made to impact, were 4, 8, 12, 16, and 20 cm. 𝜌𝑤𝑎𝑡𝑒𝑟 𝑉02 𝐷0
The Weber numbers, 𝑊𝑒𝑤𝑎𝑡𝑒𝑟 (
𝜎
, where 𝜌𝑤𝑎𝑡𝑒𝑟 is the density of the drop, 𝑉0 is the
velocity at which the drop impacts, and σ is the surface tension between the liquid and air), were 48 ± 10, 91 ± 17, 148 ± 21, 181 ± 28, and 244 ± 41, respectively. The 𝑊𝑒𝑤𝑎𝑡𝑒𝑟 are mentioned as 48, 91, 148, 181, and 244, respectively, hereafter in the text. The impact
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velocities, obtained after tracking the pixels of consecutive images, were 0.99 ± 0.11, 1.38 ± 0.12, 1.76 ± 0.12, 1.95 ± 0.14, 2.26 ± 0.18 m/s (see the supporting material for a detailed table). It is important to mention here that, there is a negligible difference when the impact velocity of the drop is obtained using the pixel analysis and when the formula √2𝑔𝛹 (where 𝛹 is the impact height) is used. This is because the gravitational force of drop during the impact dominates over the aerodynamics forces. The maximum difference between the values of the impact velocity calculated using formula √2𝑔𝛹 (= 1.98 m/s) and using pixel analysis (2.26 ± 0.18 m/s) was 15% for the case when a drop impacts from 20 cm height and with the highest airflow. De-ionized (DI) water drop was used for the experiments.
Three surfaces with different wettabilites were used, glass (hydrophilic), Polystyrene (hydrophobic), and superhydrophobic surface (SHS). A smooth microscopic glass slide was used as the glass surface. For preparing other surfaces, a microscopic slide was first washed with methanol and water. For preparing the Polystrene surface, a solution of 99% of Toulene and 1% of Polystrene (w/w) was appiled over a clean and dry slide using a spin coater. The surface was then dried for an hour inside a closed chamber.
Table 1: Static contact angles for drops sitting on various surfaces tested. Surface
Advancing angle (°)
Receding angle (°)
Glass
53° ± 2°
4° ± 2°
Polystyrene
97° ± 3°
92° ± 3°
SHS
161° ± 3°
155° ± 3°
To prepare a superhydrophobic surface (SHS), Rust-Oleum Neverwet spray was applied on a smooth microscopic glass slide. It was a two-step coating process. First, the base coating was applied on the dry and clean slide uniformly, and then the slide was kept in a closed chamber for about 30 min for drying. Then the second, i.e., final coating was applied above the base coating from a height of about 15 cm. The surface was then left for
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at least one hour in a closed chamber for drying. The advancing and receding contact angles were measured for drops on all surfaces (see Table 1).
For conducting experiments, the syringe was first loaded with water and was fixed at a desired height. The airflow was then started and was allowed to reach the desired velocity. The surface was properly positioned and the camera was started. After adjusting the camera and the backlight for capturing the images, the camera was switched on, and the plunger of the syringe was pushed till the drop falls on the surface. Entire process was captured.
Minimum five reading were taken for a given experiment to ensure repeatability. The error reported in the paper is the maximum of the standard deviation divided by the mean of a given parameter.
3. Theoretical background
The model to determine the theoretical maximum spreading diameter for the air assisted impact, has already been demonstrated by the present authors in Singh and Mandal (2019). The energy balance equation in the model of Ukiwe and Kwok (2005) is modified by incorporating the continuous shear and normal force of air between the state when the drop touches the surface and when the drop achieves its maximum spread. The final expression for maximum spreading factor is given as, 𝑡ℎ {Wewater + 12}𝛽𝑚𝑎𝑥 ~8 𝑡ℎ )3 [ ( + (𝛽𝑚𝑎𝑥 { 3 1 − cosθa )] + (
−(
12αWeair √𝑅𝑒𝑤𝑎𝑡𝑒𝑟
4Wewater √Rewater
)−(
3Weair √𝑅𝑒𝑤𝑎𝑡𝑒𝑟 Reair
)
)} (i)
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𝑡ℎ Where, 𝛽𝑚𝑎𝑥 is the maximum spread obtained from the model, 𝜃𝑎 is the advancing
contact angle, 𝑊𝑒𝑎𝑖𝑟 is the Weber number of air (=
2 𝜌𝑎𝑖𝑟 𝑉𝑎𝑖𝑟 𝐷0
𝜎
, where, 𝜌𝑎𝑖𝑟 is the density of
air, 𝑉𝑎𝑖𝑟 is the air velocity), 𝑅𝑒𝑤𝑎𝑡𝑒𝑟 is the Reynolds number of water drop (=
𝜌𝑤𝑎𝑡𝑒𝑟 𝑉0 𝐷0 𝜇𝑤𝑎𝑡𝑒𝑟
,
where, 𝜇𝑤𝑎𝑡𝑒𝑟 the dynamic viscosity of water, 𝑉0 is the velocity at which the drop impacts), and 𝛼 is the ratio of the imposed pressure to the kinetic energy of spreading drop (= 𝑃𝑖𝑚𝑝𝑜𝑠𝑒𝑑 1 𝜌 𝑉2 2 𝑤𝑎𝑡𝑒𝑟 0
, where 𝑃𝑖𝑚𝑝𝑜𝑠𝑒𝑑 is the imposed pressure).
𝛿𝑎𝑖𝑟 is the boundary layer thickness (
2𝐷0 √𝑅𝑒𝑎𝑖𝑟
), and 𝑉𝑑 is the volume of the drop.
Since, the resultant air shear acts on the volume, so the volume should be equal to 𝜋 4
2 𝑑𝑚𝑎𝑥 𝛿𝑑𝑟𝑜𝑝 , where 𝛿𝑑𝑟𝑜𝑝 is the boundary layer thickness of water (
2𝐷0 √𝑅𝑒𝑊𝑎𝑡𝑒𝑟
) and h is the
2𝐷3
rim thickness at the maximum spread (3𝑑2 0 ). 𝑚𝑎𝑥
4. Results and discussion
The spreading at the initial phase (i.e., up to ~1 ms) is observed to become almost equal for a given drop on all the surfaces at a given Weber number (see Figure 2), i.e., the effect of wettability becomes negligible initially. As the wettability decreases, the surface tension resists a given drop to spread and forces to recoil. For higher Weber numbers, finger formation during spreading increases with the wettability, because the drop decelerates more during spreading, when the surface wettability is higher (see Figure 3 – 4). A movie demonstrating the process is provided in the supporting material. It is important to mention here that, according to the order of wettability; glass surface comes first, i.e., has the highest wettability, then Polystyrene (hydrophobic), and SHS (see Table 1). The shear force of air is found to be more effective for the hydrophilic surface (i.e., glass), because the force acts against the surface tension between drop and the surface during spreading by pushing the rim further during the last stage of spreading (see Figure 3). Maximum spreading is observed to increase with Wewater for the drops impacting on a glass surface
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(see Figure 5). The spreading increases with the airflow (Reair), as well, for a given Weber number (see Figure 5). However, the factor is observed to suddenly decrease at higher Weber numbers for the less wetting surfaces (Polystyrene, see Figure 5). This is a crucial observation and need a thorough discussion. Similar decrease for PMMA surface has already been demonstrated in Singh and Mandal (2019). PMMA is a hydrophilic surface, but the contact angles are higher than that on the glass, i.e., comparatively less wetting than the glass surface. The process couldn’t be captured for SHS, since splashing was observed at higher Weber numbers. In the present paper, the dependence of the sudden decrease on the wettability is discussed.
Figure 2: Temporal variation of spreading diameter of an impacting drop.
For the glass surface, the rim becomes thinner without forming fingers, as the drop continues to move in the radial direction (see Figure 4). The formation of the thinner rim is due to higher adhesive force between the drop and surface (since the contact angles are smaller, see Table 1). The shear force pushes the rim outward. However, as the wettability decreases (i.e., for Polystyrene and SHS), a deceleration is observed at the early stage of spreading. So, fingers and thicker rim form at the last stage (see Figure 4). At lower Weber numbers, the normal imposed pressure of air is observed to dominate for the super hydrophobic surfaces than on a hydrophilic surface. Due to the higher imposed pressure,
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the drop is forced to interact with the surface more, causing higher spread (see Figure 2 and 3). Detachment of satellite drops increase with the airflow since the shear force increases (see Figure 4). When the surface tension force dominates, the drop stops spreading and starts recoiling (height decreases, see Figure 6). The spreading at higher Weber number (i.e., 244) shows a negligible retraction for the glass surface, whereas, the retraction is observed to increase with the Weber number for the comparatively less wetting surfaces, Polystyrene, and SHS (see Figure 2, 4, and 6). At higher Weber numbers, the spreading is controlled by inertia and viscous force for the glass surface, whereas the inertia force gets dissipated into viscous force and surface energy for the comparatively lower wetting surfaces. The rise during recoiling depends on the morphology of the surface at all Weber numbers.
Figure 3: Sequential image of temporal spreading diameter on different surface at Wewater = 244.
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Figure 4: Sequential images of a water drop impacting a superhydrophobic surface at a Wewater of 148.
Figure 5: Variation of the maximum spreading diameter (non-dimensionalized with the initial drop diameter) with the Wewater for (a) Glass, (b) Polystyrene, and (c) Superhydrophic surface.
Taken together, there are few issues to address. One is to explain the effect of the normal imposed pressure, which is responsible for the formation of satellite drops, 𝛽 max to increase, and the retraction velocity to vary, compared to the free – falling cases. The other is the continuous increase of the maximum spreading factor (𝛽 max) with We for the drops impacting a glass surface, and the sudden decrease for the cases of less wetting surfaces. A
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comparison of the measured 𝛽 max with the one obtained from an existing model (Singh and Mandal 2019), which was demonstrated earlier by the present authors, is provided as well, to check the applicability of the model and to bring out important physics, if any. The comparison is however better for the cases of less wetting surfaces (i.e., Polystyrene and SHS) at almost all Wewater, whereas, good comparison is observed only at higher Wewater for the case of the glass surface.
Figure 6: Temporal variation of rise in height for various air velocities.
4.1. The effect of the normal imposed pressure due to the incident airflow
For the airflow assisted impacts, the airflow imposes a continuous normal and shear force of air on the drop, mainly during spreading (see Figure 7). The impact Weber number is the same when a given drop impacts with the air, compared to the still air, because the aerodynamic force given to a small drop does not dominate over the gravitational force.
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Figure 7: Possible effect of the incident airflow on spreading.
The normal imposed pressure of air affects the impact while spreading and receding both. The imposed pressure is observed to increase with the contact angle, i.e., the spreading is more on the superhydrophobic surface than hydrophobic surface (i.e., Polystyrene) at a given Weber and Reynolds number of air (see Figure 3, 4, and 5). The normal imposed pressure suppresses the maximum rise of the drop on a given surface while receding as the Reynolds number of air increases (see Figure 6). At moderate Weber numbers (i.e., Wewater > 91), the fingers on the rim are generated and splashing occurs on the super hydrophobic surface. When the generated fingers are exposed to the shear force of air, the number of satellite drops increase (see Figure 4). Hence, the satellite drops in splashing increase with the Reynolds number of air and results in lesser availability of mass for the recoiling.
For the hydrophobic surface, i.e., Polystyrene, the normal pressure dominates when the impact Weber number is lower. At lower Weber numbers (i.e., Wewater ≤ 91), the effect of the normal imposed pressure is observed to be high, so the drop interacts with the surface more. Hence, the increment in the spreading occurs for a given drop (see Figure 2 and 3). So, the spreading generally increases with the imposed pressure.
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Figure 8: Sequential image (at an angle of 45° with the horizontal plane) of water drop impacting a glass and a Polystyrene surface along with various airflow.
As explained earlier, as the drop continues to move in the radial direction after impacting on the glass surface, higher adhesive force between the drop and surface makes the rim thinner without forming fingers (see Figure 3, 5). The air shear force pushes the
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thinner rim outward, making the maximum spread comparatively higher than the freefalling case (see Figure 5). However, finger and thicker rim formation are observed at the last stage of spreading (see Figure 5) for the lower wetting surfaces, since the deceleration rate increases at the early stage of spreading. The shear force is not effective and the spreading factor decreases at the higher impact Weber numbers (Wewater of 244 and Reair of 1564, see Figure 5). The drops after impacting the glass surface at higher Weber number (Wewater = 244) just spreads and show negligible retraction, whereas the retraction increases with the decrease in the wettability of the surface (i.e., the increase in the contact angles), i.e., for Polystyrene, and SHS (see Figure 2,3 and 6).
The retraction depends on the surface properties for all Weber numbers, unlike in the spreading where the surface properties play a vital role for the lower Weber numbers only. The surface energy at the maximum spread gets converted to the inertia force during recoiling, i.e., the drop moves toward the centre. For glass, the equlibrium contact angle is smaller, so the recoiling on the surface is slower (since surface tension is lower) at lower impact Weber numbers (Wewater < 91). Whereas, at higher Weber numbers (Wewater ≥ 91), large amount of energy is lost in viscous dissipation due to larger wetting area. The energy gets converted to the surface energy to retract after the maximum spread. The energy does not dominate over the adhesive force and results in negligible recoiling for the glass surface at higher Weber numbers (see Figure 6). As the contact angle increases, the energy available at the maximum spread to retract the drop dominates over the adhesive force and results in faster retraction. Therefore, the rebound time decreases with the increase in the contact angle (see Figure 2 and 6). The available energy also increases with the impact Weber number due to larger wetting area. The available energy gets converted into the reverse kinetic energy of drop during recoiling which further gets converted to potential energy of the drop, and causes the drop to rise on the surface. If the kinetic energy during recoiling is large, then the drop may bounce on the non wettable surface, whereas the drop rises and oscillates before coming to equlibrium state on the partially wetting surfaces (see Figure 3).
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As the Reynolds number of air increases, the normal imposed pressure of air reduces the maximum rise of the drop on a given surface (while rebounding, see Figure 6). The maximum rise and oscillation of the drop over the surface increase with the contact angle (see Figure 6), as well. Drop oscillates faster on the hydrophobic surface than the hydrophilic surface (i.e., glass, see Figure 6 and 8), so a drop comes to rest faster on the hydrophilic surface. The shear force and the normal imposed pressure given by air, resist the drop to recoil and hence, the retraction time increases. The retraction increases with the velocity of air. The detachment of the drop during recoiling is observed on the highly wettable surface, i.e., glass, at moderate Weber numbers (i.e., 144) due to the restriction by shear force of air during recoiling (see Figure 8).
4.2. What happens to the maximum spread due to the incident airflow?
As demonstrated earlier, maximum spread is observed to increase with Wewater for the drops impacting on a glass surface (see Figure 5), and suddenly decrease at higher Weber numbers (Wewater = 244) and at highest airflow (Reair = 1564) for the less wetting surfaces (Polystyrene, see Figure 5). The shear force and the normal imposed pressure significantly affect the present impact process. The two cause both of the surface energy and the viscous dissipation to decrease suddenly at the highest Weber number and Reynolds number tested (see Figure 9). The sharp decrease in the surface energy is due to the finger formation on the rim by the imposed pressure. As the thicker rim move slowly in the radial direction, the viscous dissipation reduces. That sudden change in rim thickness restricts the spreading of the drop. The effects of these forces are elaborated below.
The enhancement of the spreading diameter due to air shear force mainly depends on the rim thickness, which depends on the decellaration rate. The lower the wettability of a surface, the higher the decellation rate during spreading of a drop. Since glass has higher wettability at a given impact Weber number and airflow, the decellaration rate in the spreading is lowest among all tested. So, thinner rim forms at the last stage of spreading. The shear force given by the air pushes the rim in the radial direction and causes the drop to
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spread further (see Figure 3 and 7). The thickness of the rim depends on the deceleration rate (see Figure 3, 4, and 8). The deceleration occurs due to the combined action of viscous force, surface tension force, and imposed air pressure. When the velocity of air is low, the inertia force dominates over the imposed pressure. Hence, the deceleration becomes negligible. The deceleration is observed to be higher when the drop impacts with higher Weber number (Wewater of 244) and with a higher airflow (Reair of 1564, thicker rim, see Figure 3 and 5). The deceleration rate further increases with the contact angle, due to higher surface tension, which resist the drop to spread (see Figure 2 – 4). The shear force of air resists the surface tension force, because the drop stops spreading and starts recoiling (see Figure 3 – 4). Hence, to assist the spreading for the air assisted impact, air shear force must dominate over the surface tension force at the last stage of spreading. As the contact angle increases (see Table 1), the effect of shear force decreases. The shear force dominates on the hydrophilic surface compared to that on SHS (see Figure 3 and 5).
The normal imposed pressure and the shear force affects only during the last stage of the spreading, when the impact energy given to the drop is dissipated into viscous and surface energy. It is reported in the previous literature that the pressure is significant when the work done by the pressure is greater than 10% of the kinetic energy of the spreading drop (Ebrahim and Ortega 2017). The pressure is significant on less wetting surfaces at lower Reynolds numbers for the present case, since the viscous dissipation is negligible on the entire surface because of less spreading velocity. Therefore, the surface energy plays a major role at lower Weber numbers to counter the kinetic energy, whereas at higher Weber numbers, inertia force dominates over viscous and surface force. The competing surface energy and kinetic energy generates fingers at the rim periphery while spreading even at lower Weber numbers. Higher the surface energy (i.e., higher contact angle), higher the fingers formed at a given Weber number (for Polystyrene and SHS). At lower Weber numbers, the surface energy retards the flow in the radial direction, since the kinetic energy does not dominate. Hence, more mass is accumulated at the periphery of the rim. The generated finger gets momentum in the transverse direction due to lower surface energy.
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The normal imposed pressure of air applies pressure on the generated fingers and gives momentum to the lamella in the radial direction to spread more (see Figure 5).
Figure 9: Amount of (i) surface energy utilized, and (ii) viscous dissipation, from the initial energy at maximum spread for different surfaces.
The ratio of surface energy and initial energy increases with the decrease in the wettability (see Figure 9i) and results in the faster deceleration (i.e., kinetic energy loss during spreading) on less wetting surfaces at a given impact Weber number. Therefore, the normal imposed pressure increases at lower Reynolds numbers. At a given Weber number, the surface energy increases with the contact angle although the wetting area decreases (see Figure 9i). The loss due to the viscous dissipation decreases with the increase in the contact angle (see Figure 9ii). The kinetic energy is converted into surface energy, as well. The higher the contact angle, the higher the surface energy at the maximum spread due to the growth of the rim thickness. Further increase in the surface energy due to airflow is because of the enhancement in spreading diameter (see Figure 5 and 9i). The slopes of the variations in Figure 9i at lower impact Weber numbers are steeper than that at higher Weber numbers. The steepness decreases with the decreasing wettability at lower impact Weber numbers. For SHS, since the viscous dissipation is negligible, the kinetic energy is only countered by the surface energy. Therefore, the dominance of the impact Weber number decreases with
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the increase in the surface energy or contact angle. The dominance of surface energy at low Weber number makes the drop to rebound completely on SHS, whereas, partial rebound or splashing occurs with the increase in Weber numbers. There is a decrease in the slope with the increase in contact angle (see Figure 9i). With the increment in the contact angle, the surface repellency increases and results in less viscous dissipation between the surface and drop (see Figure 9ii). Therefore, higher kinetic energy is converted into the surface energy. Thicker rim also forms at this stage for less wetting surfaces. The rim moves slower and resists the drop to spread, causing a sudden decrease in the maximum spread (see Figure 5).
The data points are missing on the SHS surface for the Wewater > 91 due to the occurrence of splashing. Splashing disintegrate the outer rim of drop into tiny droplet, and that process is difficult to process.
4.3. Comparison of the measured maximum spreading diameter with the one obtained from the model
The maximum spreading diameter is compared with the one obtained from the model (see Figure 10) provided by Singh and Mandal (2019). The model takes the effect of the imposed pressure and shear force of the incident airflow into account. The comparison is better at higher impact Weber numbers (see Figure 10), because the assumption to calculate the boundary layer thickness (𝛿𝑑𝑟𝑜𝑝 =
2𝐷0 √ 𝑅𝑒𝑊𝑎𝑡𝑒𝑟
) works better at higher impact
velocity (i.e., higher Weber numbers and Reynolds number), due to lower boundary layer thickness. The loss between surface and drop decreases with the decrease in the boundary layer thickness during spreading. The comparison of the measured maximum spreading factors (as x axis) with the theoretically obtained ones (as y axis) is provided in the supporting material for the interested readers.
The comparison is observed to improve as one moves from higher to lower wettabilities (see Figure 10). At low Wewater, the adhesive force between the drop and the
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surface decreases with the decrease in wettability. The model incorporates the rim thickness at the maximum spread, and is greatly influenced by the wettability of the surface. As discussed earlier, lower wetting surfaces give thicker rim at the maximum spread.
Figure 10: The comparison of the experimentally measured maximum spreading factor with the same obtained from the model provided by Singh and Mandal (2019). 'Re_air' and 'Re_air_th' in the legend represent measured βmax, and βmax from the model, respectively, at a given Reynolds number.
The comparison shows that, the incident airflow dominates when the wettability of the surface and Wewater are lower, whereas,the airflow dominates at higher Wewater due to lower rim thickness for the surfaces with higher wettabilities.
Since the predictive capability of the present model is poor at lower Weber numbers (see Figure 10), it is important to provide justifications. Most of the models present in the literature are derived using the fitting of experimental data with the Weber number and Reynolds number. The effect of inertia, viscous and surface tension force are considered to predict the maximum spreading factor. At high impact Weber and Reynolds numbers, these models show good agreement on all wetting surfaces, but the discrepancy increases with the decrease in the surface wettability at low impact Weber numbers. The effect of the wettability plays a vital role in predicting the maximum spreading factor at lower impact Weber numbers. Few models (Chandra et al. 1991; Pasendideh-Fard et al. 1995; Mao et al. 1997; Ukiwe and Kwok 2005) incorparated the effect of the wettability in terms of contact angle using the energy balance method. The energy conservation are applied based on many assumptions during the radial flow of drop on the dry surface. These assumptions are made based on different impact conditions. So, it is difficult to say which assumptions are
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more accurate in different impact conditions. For example, the assumption of cylindrical disk at the maximum spread of a drop is not valid for higher impact Weber numbers, and high viscosity of liquids, whereas the the development of boundary layer thickness is not valid at lower Weber numbers for non-viscous liquids. Various definition of contact angle in different models derived using energy based method are confusing and complicated too. As the Young’s contact angle defined in the model of Ukiwe and Kwok (2005) is applicable for smooth and heterogeneous surface, whereas advancing contact angle defined in Pasendideh-Fard (1996) model depends upon the impact condition like the viscosity of drop. The calculation of the time required to reach maximum spreading diameter on surface in the model of Pasendideh-Fard (1996) is applicable for non-viscous liquids. Therefore, these model are not consistent to give a good prediction of spreading factor in all impact conditions.
However, many assumptions are valid for non viscous liquids (i.e., water) even though the models are not consistent, like incorporating the rim thickness and wettability of the surface in terms of contact angle, fulfills our motive. So, the predictive capability few times becomes poor.
The comparison of the measured maximum spread for free falling drops with other existing models is provided in the supporting material.
5. Conclusions
The air assisted impact of water drops on surfaces with various wettabilities is studied to bring out the effect of the shear force and normal imposed pressure of air. Three surfaces, glass, Polystyrene, and SHS, are chosen. The study is conducted for five impact heights, and three velocities of the incident airflow.
Results show that the effect of surface wettability is negligible initially (i.e., up to ~1 ms), since the spreading at the initial phase is almost equal for a given drop on all the
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surfaces at a given Weber number. Maximum spreading is observed to increase with Wewater for the drops impacting on a glass surface. The spreading increases with the airflow (Reair) as well, for a given Weber number. However, the factor is observed to suddenly decrease at higher Weber numbers (244) and Reair (1564) for the less wetting surfaces. So, there is a dependence of the maximum spreading on the wettability.
For the glass surface, higher adhesive force between the drop and surface causes the rim to become thinner without forming fingers, as the drop continues to move in the radial direction. The shear force pushes the rim outward. However, as the wettability decreases (i.e., for Polystyrene and SHS), a deceleration is observed at the early stage of spreading, forming fingers and thicker rim at the last stage. The thicker rim moves slowly in radial direction, causing the viscous dissipation to decrease suddenly for the highest Weber numbers (244) and Reair (1564) tested. The deceleration of thicker rim resists the drop to spread. Hence, the maximum spread reduces suddenly.
The deceleration is observed to be higher when the drop impacts with high Weber number and with a high airflow. The deceleration rate is dependent on the contact angle, as well. The rate increases with the contact angle, due to higher surface tension, and resist the drop to spread.
At lower Weber numbers, the normal imposed pressure of air is observed to dominate for the super hydrophobic surfaces than on a hydrophilic surface. Due to the higher imposed pressure, the drop is forced to interact with the surface more, causing higher spread. Detachment of satellite drops on hydrophilic surface (i.e., glass) increases with the airflow since the shear force increases. At lower Weber numbers, the surface energy retards the flow in the radial direction, since the kinetic energy does not dominate. Hence, more mass is accumulated at the periphery of the rim. The generated finger gets momentum in the transverse direction due to lower surface energy. The normal imposed pressure of air applies pressure on the generated fingers and gives momentum to the lamella in the radial direction to spread more.
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A comparison of the measured 𝛽 max with the one obtained from an existing model (Singh and Mandal 2019), is provided. Better comparison is found for the cases of less wetting surfaces (i.e., Polystyrene and SHS) at almost all We, whereas, good comparison is observed only at higher We for the case of the glass surface.
Future work can be focused on the airflow assisted impact of various drops on heated surfaces. The surface with elevated temperatures, can change the liquid viscosity, which may have a direct influence on the shear force given by the incident airflow on the drop. Novel physics will possibly come out.
Acknowledgements
The work is supported by the Science and Engineering Research Board (SERB) of the Department of Science and Technology (DST), Government of India (Project number: ECR / 2016 / 000026). Author statement Re: “Air assisted impact of drops: The effect of surface wettability" by Ramesh Kumar Singh and Deepak Kumar Mandal. To our knowledge, it is the first systematic study which explores the influence of the incident airflow when a drop impacts on surfaces with various wettabilities. The authors acknowledge that none of the material in the manuscript has been published or is under consideration for publication elsewhere. If you need any further information, please do not hesitate to contact me. Ramesh Kumar Singh: Conceptualization, Methodology, Validation. Deepak Kumar Mandal: Supervision, Analysis, Software, Resources.
To whom it may concern The authors wish to confirm that there are no known conflicts of interest associated with this publication. The authors confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship
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but are not listed. The authors further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. References
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Air assisted impact of drops: The effect of surface wettability Ramesh Kumar Singh , Deepak Kumar Mandal PII: DOI: Reference:
S0301-9322(19)30928-0 https://doi.org/10.1016/j.ijmultiphaseflow.2020.103241 IJMF 103241
To appear in:
International Journal of Multiphase Flow
Received date: Revised date: Accepted date:
26 November 2019 29 January 2020 3 February 2020
Please cite this article as: Ramesh Kumar Singh , Deepak Kumar Mandal , Air assisted impact of drops: The effect of surface wettability, International Journal of Multiphase Flow (2020), doi: https://doi.org/10.1016/j.ijmultiphaseflow.2020.103241
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Highlights
Airflow assisted impact of drops on surfaces with various wettabilities is studied.
The spreading reduces at higher airflow and Weber number for less wetting surfaces.
Combined action of airflow shear and normal imposed pressure is responsible.
The comparison of measured spread with a model improves when wettability reduces.
----------------------------*Corresponding author email: [email protected] (D. K. Mandal), Phone: +91 326 223 5058.
Air assisted impact of drops: The effect of surface wettability
Ramesh Kumar Singh, Deepak Kumar Mandal*
Department of Mechanical Engineering, Indian Institute of Technology (ISM), Dhanbad, PIN 826004, India.
Abstract
The airflow assisted impact of water drops on surfaces with various wettabilities is studied to understand the effect of shear force and normal imposed pressure provided by the incident airflow. Results show that the maximum spreading factor increases with the impact Weber number for the drops impacting a hydrophilic surface. For other surfaces, the spreading increases with the airflow for a given Weber number. However, the factor is observed to suddenly decrease at higher Weber numbers and airflow and for the less wetting surfaces. For the hydrophilic surface, as the drop continues spread, higher adhesive force between the drop and surface causes the rim to become thinner. The shear force pushes the rim outward. As the wettability decreases, a deceleration is observed at the early stage of spreading, forming fingers and thicker rim at the last stage. The thicker rim causes the viscous dissipation to decrease suddenly for the highest Weber number and airflow tested. The thicker rim resists the drop to spread. Detachment of the satellite drops on a hydrophilic surface increases with the airflow since the shear force increases. The normal imposed pressure applies pressure on the fingers and gives momentum to the lamella in the radial direction. A comparison of the measured maximum spreading with the one obtained from an existing model, is provided. Better comparison is found for the cases of less wetting surfaces, whereas good comparison is observed only at higher impact Weber numbers for the hydrophilic surface.
Keywords: Drop impact; Airflow assisted impact; Wettability; Maximum spreading factor; Normal imposed pressure.
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1. Introduction
The impact of drops on a dry solid surface is being analyzed since decades due to its vast applications in many fields, like in fire suppression (Pasandideh-Fard et al. 1996; Chandra and Avedisian 1991; Ukiwe and Kwok 2005), in automotive engines (Chandra and Avedisian 1991; Ukiwe and Kwok 2005), in spray cooling of hot surfaces (Ukiwe and Kwok 2005), in spray forming (Rioboo et al. 2001). The drop shows various outcomes after the impact depending on the wettability of the surface (Kim and Chun 2001), viscosity of the liquid (Kumar and Mandal 2019), inertia (Antonini et al. 2012), and surface tension force (Rioboo et al. 2001). The drop may spread, recede, deposit, rebound, stick, depending on the impact Weber number and the wettability of the surface (Rioboo et al. 2001). The initial stage of spreading is determined by the Weber number, whereas the last stage depends on the wettability of the surface (Schiaffino and Sonin 1997). When the Weber number is within 30 to 200, the wettability dominates and affects the deformation of the drop, whereas, the inertia force dominates when the Weber number is higher than 200 (Antonini et al. 2012). The effect of the wettability is dominant in the recoiling stage, as well, even at higher Weber numbers (Kim and Chun 2001). The drop may rebound after the impact depending on the wettability (Harley and Brunskill 1958). There are two different regimes of rebound. In the first, inertia force dominates over the surface tension force (i.e., for high impact Weber numbers) and results in less elastic rebound. However, larger energy is lost during spreading and oscillation. Second regime, known as Quasi-static regime, occurs at lower impact Weber numbers. In this regime, energy loss is minimal (Biance et al. 2006). Three different types of rebound are observed for drops impacting on heated surfaces whose temperature is near the Leidenfrost point. They are, reflection rebound, explosive rebound, and explosive detachment. These rebound occurred sequentially with decreasing wall temperature (Liang et al. 2016).
The surface temperature and Weber number both affect the detachment time of the drop, as well (Liang et al. 2016). The Weber number and the surface temperature have negligible effect on the time required to reach the maximum spreading. Different regimes
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of heat transfer (i.e., film evaporation, nucleate boiling, transition boiling, and film boiling) are observed, as well, for which suitable models exist in the open literature to describe the process (Liang and Mudawar 2017).
The rebound depends on the recoiling, and the recoiling depends on the contact angle and the impact Weber numbers (Borisov et al. 2003). The temperature of the surface is reported to affect the receding, as well (Kumar and Mandal 2020).
So, the wettability is of major focus due to its various applications, like, in water proof textiles (Kim and Chun 2001; Fukai et al. 1995), self-cleaning windows (Fukai et al. 1995; Antonini et al. 2013), and is described by the contact angle between the drop and the surface during spreading (De Gennes 1985). The spreading diameter is inversely proportional to the contact angle (Fukai et al. 1995). The advancing contact angle can be used to develop a model to obtain the maximum spreading of a drop (Pasandideh-Fard et al. 1996). Few models compare well with the measured results, but over predict the diameter during recoiling due to the change in the shape (Pasandideh-Fard et al. 1996). Better models are reported, as well, to find the maximum spread by using Young contact angle instead of the advancing contact angle with an assumption that the shape is cylindrical at the maximum spread (Ukiwe and Kwok 2005), and by incorporating the effect of surface tension (Tsurutani et al. 1990). Lee et al. (2016) provided the expression for the total time (tmax) taken by a drop to reach the maximum spreading diameter in terms of surface tension of the liquid, and modified the model of Mao et al. (1997) by incorporating tmax along with the dynamic contact angle at the maximum spread instead of the equilibrium contact angle. The time required to reach the maximum spreading mainly depends on the surface tension of the drop and has negligible effect of the surface wettability and viscosity of the drop. The surface tension plays a major role in the spreading, compared to the viscosity of the drop (Liang et al. 2019). A detailed model incorporating all the effects has been demonstrated by Laan et al. (2014). Wildeman et al. (2016) estimated viscous dissipation energy during the spreading of drop for the free and no slip boundary conditions at different impact Weber numbers.
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The wettability is responsible for the formation of fingers on the rim during spreading, as well as, during recoiling (Fukai et al. 1995). The formation of fingers is observed when a drop impacts at ambient condition (Thoroddsen and Sakakibara 1998; Allen 1975; Bang et al. 2011), and is due to the instability occurred for the rapid deceleration of rim during spreading (Allen 1975). The Ohnesorge number and surface roughness also decelerate the spreading (Tang et al. 2017). The formation of fingers during spreading provides the origin of splashing (Mundo et al. 1995; Xu et al. 2007) for which the ambient condition is reported to be important (Xu et al. 2005; Bang et al. 2011). With the decrease in the ambient pressure, the splashing decreases even at higher Weber numbers since no fingers form during spreading (Xu et al. 2005). However, increase in the air pressure, decreases the spreading diameter (Bang et al. 2011).
The underlying physics of finger formation due to the influence of the surrounding gas is well described in the recent review papers (Josserand and Thoroddsen 2016). The fingers give rise to the splashing. The effect of the surface wettability on the finger formation is reported. The surface having lower wettability, results more fingers on the outer rim of the drop during spreading, and promotes splashing (Josserand and Thoroddsen 2016). The physics behind the occurrence of splashing while impacting on a thin sheet of liquid at higher Weber numbers is addressed by Liang and Mudawar (2016). The thin film at the surface promotes splashing. Formation of crown and ejecta sheet, while impacting on a thin film is addressed, as well.
The onset of splashing and receding are delayed at higher Weber numbers for the case of gas assisted impact, i.e., when a propellant gas is incident on the surface along with the drop (Ebrahim and Ortega 2017). Aerodynamic forces like normal and shear forces provided by the gas, become important for the gas assisted impact. The deformation of the drop gets enhanced when the work done by the normal force of air is greater than 10% of the kinetic energy of the impacting drop, especially for lower Weber numbers (Diaz and Ortega 2016). Different regimes, in which the effect of normal and shear forces becomes
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negligible, exist. The effect of the propellant gas is significant on the deformation when the critical Capillary number is greater than or equal to 0.35 (Ebrahim and Ortega 2017). The maximum spreading factor increases with the airflow for a given Weber number, compared to the normal free-falling drop impact case (Ebrahim et al. 2017; Diaz and Ortega 2016; Ebrahim and Ortega 2017; Singh and Mandal 2019). However, for higher impact heights and airflow, the maximum spreading reduces, since the rim starts becoming thicker (Singh and Mandal 2019). The viscous force, surface tension, and inertia force affect the growth of rim thickness during spreading (Zang et al. 2019). When the ratio of the imposed pressure to the internal pressure of spreading becomes higher at higher Weber numbers, the imposed pressure restricts the radial flow, and limits the maximum spread (Singh and Mandal 2019). Few researches are only reported regarding the gas / air assisted impact, and therefore, a huge knowledge gap exists in the area. For example, works on the effect of the wettability when a drop is subjected to the incident air / gas flow, are limited.
Taken together, it can be concluded that although there exist many researches regarding the impact of freely falling drops, but few researches are only reported regarding the gas / air assisted impact, specifically, works on the effect of the wettability are limited. The outcome may be different for the case of air / gas assisted impact on surfaces with various wettabilities. If the maximum spreading increases for a given surface compared to case of the impact of free-falling drops, then the process becomes helpful for many industries. Hence, to fulfill this research gap, present study is executed. The effect of the normal imposed pressure and the shear force of a drop impacting surfaces with various wettabilities are studied at different Weber numbers. The setup used for the study is explained below.
2. Experimental setup and methodology
The experimental setup is explained in Singh and Mandal (2019), and is illustrated here in short. A schematic is provided in Figure 1. A vertical wind tunnel (acrylic made) was fabricated and a fan (12V, DC, 340 m3/h, outer dimension of 0.1 m × 0.1 m × 0.02 m,
6
with speed regulator) was fitted at the top to generate the desired airflow towards the surface. The surface of dimension of 75 mm × 12 mm × 1 mm was placed about 10 mm above the bottom of the wind tunnel, so that the airflow becomes incident on the surface and can escape through the sideways. The bottom, with the dimension of 0.1 m × 0.1 m × 0.23 m, was considered as the test section. The cross section was rectangular, and a vertical slit of about 5 mm thickness was made on one side so that the height of the needle tip (outer diameter of the needle = 0.81 mm) from which a drop impacts, can be varied by moving the needle syringe arrangement up and down using a stand (see Figure 1). The needle syringe arrangement was kept horizontal and was firmly fixed with the stand. The tip of the needle was centrally placed, just above the surface, so that whenever one pushes the plunger of the needle, a drop (initial diameter of 3.47 ± 0.18 mm) can fall on the surface. A high-speed camera (0.0108 pixel / micron), fitted with a telecentric lens, was used to capture the entire process with the help of suitable backlights. The process was captured at three thousand frames per second.
It is important to mention that, the shape of the drop was not spherical before the impact. The spherical shape was assumed because the difference in the horizontal (Dh = 3.506 ± 0.18 mm, see the supporting material for the figure) and vertical length (Dv = 3.247 ± 0.18 mm, see the supporting material for the figure) was less than 8%. To simplify the calculation in the model, and to determine the dimensionless numbers, an equivalent 1
spherical diameter D0 was obtained (D0 = (𝐷ℎ2 𝐷𝑣 )3), as demonstrated in Mao et al. (1997), Sikalo et al. (2002), and Liang et al. (2019).
The air velocity was measured using an anemometer (Waco digital Anemometer, AVM-04A, resolution 0.01 m/s, Make: India), and was measured at five different locations in the test section to ensure uniformity in the magnitude of the airflow. The velocity was measured about 10 mm above the surface. The incident air velocities were set at 2.4 ± 0.03 m/s, 4.7 ± 0.03 m/s, and 6.8 ± 0.04 m/s, for the experiments. The Reynolds numbers, 𝑅𝑒𝑎𝑖𝑟 (
𝜌𝑎𝑖𝑟𝑉𝑎𝑖𝑟 𝐷0 𝜇𝑎𝑖𝑟
, where 𝜌𝑎𝑖𝑟 is the density of air, 𝑉𝑎𝑖𝑟 is the velocity of the airflow, 𝐷0 is the pre
7
impact diameter of the drop, and 𝜇𝑎𝑖𝑟 is the dynamic viscosity of air), which indicates the role of the shear stress due to the airflow in the deformation of the drop, were 552 ± 10, 1081 ± 13, and 1564 ± 18, respectively. The numbers are mentioned as 552, 1081, and 1564, respectively, hereafter in the text.
Figure 1: Details of the experimental setup.
The heights, from which the drop was made to impact, were 4, 8, 12, 16, and 20 cm. 𝜌𝑤𝑎𝑡𝑒𝑟 𝑉02 𝐷0
The Weber numbers, 𝑊𝑒𝑤𝑎𝑡𝑒𝑟 (
𝜎
, where 𝜌𝑤𝑎𝑡𝑒𝑟 is the density of the drop, 𝑉0 is the
velocity at which the drop impacts, and σ is the surface tension between the liquid and air), were 48 ± 10, 91 ± 17, 148 ± 21, 181 ± 28, and 244 ± 41, respectively. The 𝑊𝑒𝑤𝑎𝑡𝑒𝑟 are mentioned as 48, 91, 148, 181, and 244, respectively, hereafter in the text. The impact
8
velocities, obtained after tracking the pixels of consecutive images, were 0.99 ± 0.11, 1.38 ± 0.12, 1.76 ± 0.12, 1.95 ± 0.14, 2.26 ± 0.18 m/s (see the supporting material for a detailed table). It is important to mention here that, there is a negligible difference when the impact velocity of the drop is obtained using the pixel analysis and when the formula √2𝑔𝛹 (where 𝛹 is the impact height) is used. This is because the gravitational force of drop during the impact dominates over the aerodynamics forces. The maximum difference between the values of the impact velocity calculated using formula √2𝑔𝛹 (= 1.98 m/s) and using pixel analysis (2.26 ± 0.18 m/s) was 15% for the case when a drop impacts from 20 cm height and with the highest airflow. De-ionized (DI) water drop was used for the experiments.
Three surfaces with different wettabilites were used, glass (hydrophilic), Polystyrene (hydrophobic), and superhydrophobic surface (SHS). A smooth microscopic glass slide was used as the glass surface. For preparing other surfaces, a microscopic slide was first washed with methanol and water. For preparing the Polystrene surface, a solution of 99% of Toulene and 1% of Polystrene (w/w) was appiled over a clean and dry slide using a spin coater. The surface was then dried for an hour inside a closed chamber.
Table 1: Static contact angles for drops sitting on various surfaces tested. Surface
Advancing angle (°)
Receding angle (°)
Glass
53° ± 2°
4° ± 2°
Polystyrene
97° ± 3°
92° ± 3°
SHS
161° ± 3°
155° ± 3°
To prepare a superhydrophobic surface (SHS), Rust-Oleum Neverwet spray was applied on a smooth microscopic glass slide. It was a two-step coating process. First, the base coating was applied on the dry and clean slide uniformly, and then the slide was kept in a closed chamber for about 30 min for drying. Then the second, i.e., final coating was applied above the base coating from a height of about 15 cm. The surface was then left for
9
at least one hour in a closed chamber for drying. The advancing and receding contact angles were measured for drops on all surfaces (see Table 1).
For conducting experiments, the syringe was first loaded with water and was fixed at a desired height. The airflow was then started and was allowed to reach the desired velocity. The surface was properly positioned and the camera was started. After adjusting the camera and the backlight for capturing the images, the camera was switched on, and the plunger of the syringe was pushed till the drop falls on the surface. Entire process was captured.
Minimum five reading were taken for a given experiment to ensure repeatability. The error reported in the paper is the maximum of the standard deviation divided by the mean of a given parameter.
3. Theoretical background
The model to determine the theoretical maximum spreading diameter for the air assisted impact, has already been demonstrated by the present authors in Singh and Mandal (2019). The energy balance equation in the model of Ukiwe and Kwok (2005) is modified by incorporating the continuous shear and normal force of air between the state when the drop touches the surface and when the drop achieves its maximum spread. The final expression for maximum spreading factor is given as, 𝑡ℎ {Wewater + 12}𝛽𝑚𝑎𝑥 ~8 𝑡ℎ )3 [ ( + (𝛽𝑚𝑎𝑥 { 3 1 − cosθa )] + (
−(
12αWeair √𝑅𝑒𝑤𝑎𝑡𝑒𝑟
4Wewater √Rewater
)−(
3Weair √𝑅𝑒𝑤𝑎𝑡𝑒𝑟 Reair
)
)} (i)
10
𝑡ℎ Where, 𝛽𝑚𝑎𝑥 is the maximum spread obtained from the model, 𝜃𝑎 is the advancing
contact angle, 𝑊𝑒𝑎𝑖𝑟 is the Weber number of air (=
2 𝜌𝑎𝑖𝑟 𝑉𝑎𝑖𝑟 𝐷0
𝜎
, where, 𝜌𝑎𝑖𝑟 is the density of
air, 𝑉𝑎𝑖𝑟 is the air velocity), 𝑅𝑒𝑤𝑎𝑡𝑒𝑟 is the Reynolds number of water drop (=
𝜌𝑤𝑎𝑡𝑒𝑟 𝑉0 𝐷0 𝜇𝑤𝑎𝑡𝑒𝑟
,
where, 𝜇𝑤𝑎𝑡𝑒𝑟 the dynamic viscosity of water, 𝑉0 is the velocity at which the drop impacts), and 𝛼 is the ratio of the imposed pressure to the kinetic energy of spreading drop (= 𝑃𝑖𝑚𝑝𝑜𝑠𝑒𝑑 1 𝜌 𝑉2 2 𝑤𝑎𝑡𝑒𝑟 0
, where 𝑃𝑖𝑚𝑝𝑜𝑠𝑒𝑑 is the imposed pressure).
𝛿𝑎𝑖𝑟 is the boundary layer thickness (
2𝐷0 √𝑅𝑒𝑎𝑖𝑟
), and 𝑉𝑑 is the volume of the drop.
Since, the resultant air shear acts on the volume, so the volume should be equal to 𝜋 4
2 𝑑𝑚𝑎𝑥 𝛿𝑑𝑟𝑜𝑝 , where 𝛿𝑑𝑟𝑜𝑝 is the boundary layer thickness of water (
2𝐷0 √𝑅𝑒𝑊𝑎𝑡𝑒𝑟
) and h is the
2𝐷3
rim thickness at the maximum spread (3𝑑2 0 ). 𝑚𝑎𝑥
4. Results and discussion
The spreading at the initial phase (i.e., up to ~1 ms) is observed to become almost equal for a given drop on all the surfaces at a given Weber number (see Figure 2), i.e., the effect of wettability becomes negligible initially. As the wettability decreases, the surface tension resists a given drop to spread and forces to recoil. For higher Weber numbers, finger formation during spreading increases with the wettability, because the drop decelerates more during spreading, when the surface wettability is higher (see Figure 3 – 4). A movie demonstrating the process is provided in the supporting material. It is important to mention here that, according to the order of wettability; glass surface comes first, i.e., has the highest wettability, then Polystyrene (hydrophobic), and SHS (see Table 1). The shear force of air is found to be more effective for the hydrophilic surface (i.e., glass), because the force acts against the surface tension between drop and the surface during spreading by pushing the rim further during the last stage of spreading (see Figure 3). Maximum spreading is observed to increase with Wewater for the drops impacting on a glass surface
11
(see Figure 5). The spreading increases with the airflow (Reair), as well, for a given Weber number (see Figure 5). However, the factor is observed to suddenly decrease at higher Weber numbers for the less wetting surfaces (Polystyrene, see Figure 5). This is a crucial observation and need a thorough discussion. Similar decrease for PMMA surface has already been demonstrated in Singh and Mandal (2019). PMMA is a hydrophilic surface, but the contact angles are higher than that on the glass, i.e., comparatively less wetting than the glass surface. The process couldn’t be captured for SHS, since splashing was observed at higher Weber numbers. In the present paper, the dependence of the sudden decrease on the wettability is discussed.
Figure 2: Temporal variation of spreading diameter of an impacting drop.
For the glass surface, the rim becomes thinner without forming fingers, as the drop continues to move in the radial direction (see Figure 4). The formation of the thinner rim is due to higher adhesive force between the drop and surface (since the contact angles are smaller, see Table 1). The shear force pushes the rim outward. However, as the wettability decreases (i.e., for Polystyrene and SHS), a deceleration is observed at the early stage of spreading. So, fingers and thicker rim form at the last stage (see Figure 4). At lower Weber numbers, the normal imposed pressure of air is observed to dominate for the super hydrophobic surfaces than on a hydrophilic surface. Due to the higher imposed pressure,
12
the drop is forced to interact with the surface more, causing higher spread (see Figure 2 and 3). Detachment of satellite drops increase with the airflow since the shear force increases (see Figure 4). When the surface tension force dominates, the drop stops spreading and starts recoiling (height decreases, see Figure 6). The spreading at higher Weber number (i.e., 244) shows a negligible retraction for the glass surface, whereas, the retraction is observed to increase with the Weber number for the comparatively less wetting surfaces, Polystyrene, and SHS (see Figure 2, 4, and 6). At higher Weber numbers, the spreading is controlled by inertia and viscous force for the glass surface, whereas the inertia force gets dissipated into viscous force and surface energy for the comparatively lower wetting surfaces. The rise during recoiling depends on the morphology of the surface at all Weber numbers.
Figure 3: Sequential image of temporal spreading diameter on different surface at Wewater = 244.
13
Figure 4: Sequential images of a water drop impacting a superhydrophobic surface at a Wewater of 148.
Figure 5: Variation of the maximum spreading diameter (non-dimensionalized with the initial drop diameter) with the Wewater for (a) Glass, (b) Polystyrene, and (c) Superhydrophic surface.
Taken together, there are few issues to address. One is to explain the effect of the normal imposed pressure, which is responsible for the formation of satellite drops, 𝛽 max to increase, and the retraction velocity to vary, compared to the free – falling cases. The other is the continuous increase of the maximum spreading factor (𝛽 max) with We for the drops impacting a glass surface, and the sudden decrease for the cases of less wetting surfaces. A
14
comparison of the measured 𝛽 max with the one obtained from an existing model (Singh and Mandal 2019), which was demonstrated earlier by the present authors, is provided as well, to check the applicability of the model and to bring out important physics, if any. The comparison is however better for the cases of less wetting surfaces (i.e., Polystyrene and SHS) at almost all Wewater, whereas, good comparison is observed only at higher Wewater for the case of the glass surface.
Figure 6: Temporal variation of rise in height for various air velocities.
4.1. The effect of the normal imposed pressure due to the incident airflow
For the airflow assisted impacts, the airflow imposes a continuous normal and shear force of air on the drop, mainly during spreading (see Figure 7). The impact Weber number is the same when a given drop impacts with the air, compared to the still air, because the aerodynamic force given to a small drop does not dominate over the gravitational force.
15
Figure 7: Possible effect of the incident airflow on spreading.
The normal imposed pressure of air affects the impact while spreading and receding both. The imposed pressure is observed to increase with the contact angle, i.e., the spreading is more on the superhydrophobic surface than hydrophobic surface (i.e., Polystyrene) at a given Weber and Reynolds number of air (see Figure 3, 4, and 5). The normal imposed pressure suppresses the maximum rise of the drop on a given surface while receding as the Reynolds number of air increases (see Figure 6). At moderate Weber numbers (i.e., Wewater > 91), the fingers on the rim are generated and splashing occurs on the super hydrophobic surface. When the generated fingers are exposed to the shear force of air, the number of satellite drops increase (see Figure 4). Hence, the satellite drops in splashing increase with the Reynolds number of air and results in lesser availability of mass for the recoiling.
For the hydrophobic surface, i.e., Polystyrene, the normal pressure dominates when the impact Weber number is lower. At lower Weber numbers (i.e., Wewater ≤ 91), the effect of the normal imposed pressure is observed to be high, so the drop interacts with the surface more. Hence, the increment in the spreading occurs for a given drop (see Figure 2 and 3). So, the spreading generally increases with the imposed pressure.
16
Figure 8: Sequential image (at an angle of 45° with the horizontal plane) of water drop impacting a glass and a Polystyrene surface along with various airflow.
As explained earlier, as the drop continues to move in the radial direction after impacting on the glass surface, higher adhesive force between the drop and surface makes the rim thinner without forming fingers (see Figure 3, 5). The air shear force pushes the
17
thinner rim outward, making the maximum spread comparatively higher than the freefalling case (see Figure 5). However, finger and thicker rim formation are observed at the last stage of spreading (see Figure 5) for the lower wetting surfaces, since the deceleration rate increases at the early stage of spreading. The shear force is not effective and the spreading factor decreases at the higher impact Weber numbers (Wewater of 244 and Reair of 1564, see Figure 5). The drops after impacting the glass surface at higher Weber number (Wewater = 244) just spreads and show negligible retraction, whereas the retraction increases with the decrease in the wettability of the surface (i.e., the increase in the contact angles), i.e., for Polystyrene, and SHS (see Figure 2,3 and 6).
The retraction depends on the surface properties for all Weber numbers, unlike in the spreading where the surface properties play a vital role for the lower Weber numbers only. The surface energy at the maximum spread gets converted to the inertia force during recoiling, i.e., the drop moves toward the centre. For glass, the equlibrium contact angle is smaller, so the recoiling on the surface is slower (since surface tension is lower) at lower impact Weber numbers (Wewater < 91). Whereas, at higher Weber numbers (Wewater ≥ 91), large amount of energy is lost in viscous dissipation due to larger wetting area. The energy gets converted to the surface energy to retract after the maximum spread. The energy does not dominate over the adhesive force and results in negligible recoiling for the glass surface at higher Weber numbers (see Figure 6). As the contact angle increases, the energy available at the maximum spread to retract the drop dominates over the adhesive force and results in faster retraction. Therefore, the rebound time decreases with the increase in the contact angle (see Figure 2 and 6). The available energy also increases with the impact Weber number due to larger wetting area. The available energy gets converted into the reverse kinetic energy of drop during recoiling which further gets converted to potential energy of the drop, and causes the drop to rise on the surface. If the kinetic energy during recoiling is large, then the drop may bounce on the non wettable surface, whereas the drop rises and oscillates before coming to equlibrium state on the partially wetting surfaces (see Figure 3).
18
As the Reynolds number of air increases, the normal imposed pressure of air reduces the maximum rise of the drop on a given surface (while rebounding, see Figure 6). The maximum rise and oscillation of the drop over the surface increase with the contact angle (see Figure 6), as well. Drop oscillates faster on the hydrophobic surface than the hydrophilic surface (i.e., glass, see Figure 6 and 8), so a drop comes to rest faster on the hydrophilic surface. The shear force and the normal imposed pressure given by air, resist the drop to recoil and hence, the retraction time increases. The retraction increases with the velocity of air. The detachment of the drop during recoiling is observed on the highly wettable surface, i.e., glass, at moderate Weber numbers (i.e., 144) due to the restriction by shear force of air during recoiling (see Figure 8).
4.2. What happens to the maximum spread due to the incident airflow?
As demonstrated earlier, maximum spread is observed to increase with Wewater for the drops impacting on a glass surface (see Figure 5), and suddenly decrease at higher Weber numbers (Wewater = 244) and at highest airflow (Reair = 1564) for the less wetting surfaces (Polystyrene, see Figure 5). The shear force and the normal imposed pressure significantly affect the present impact process. The two cause both of the surface energy and the viscous dissipation to decrease suddenly at the highest Weber number and Reynolds number tested (see Figure 9). The sharp decrease in the surface energy is due to the finger formation on the rim by the imposed pressure. As the thicker rim move slowly in the radial direction, the viscous dissipation reduces. That sudden change in rim thickness restricts the spreading of the drop. The effects of these forces are elaborated below.
The enhancement of the spreading diameter due to air shear force mainly depends on the rim thickness, which depends on the decellaration rate. The lower the wettability of a surface, the higher the decellation rate during spreading of a drop. Since glass has higher wettability at a given impact Weber number and airflow, the decellaration rate in the spreading is lowest among all tested. So, thinner rim forms at the last stage of spreading. The shear force given by the air pushes the rim in the radial direction and causes the drop to
19
spread further (see Figure 3 and 7). The thickness of the rim depends on the deceleration rate (see Figure 3, 4, and 8). The deceleration occurs due to the combined action of viscous force, surface tension force, and imposed air pressure. When the velocity of air is low, the inertia force dominates over the imposed pressure. Hence, the deceleration becomes negligible. The deceleration is observed to be higher when the drop impacts with higher Weber number (Wewater of 244) and with a higher airflow (Reair of 1564, thicker rim, see Figure 3 and 5). The deceleration rate further increases with the contact angle, due to higher surface tension, which resist the drop to spread (see Figure 2 – 4). The shear force of air resists the surface tension force, because the drop stops spreading and starts recoiling (see Figure 3 – 4). Hence, to assist the spreading for the air assisted impact, air shear force must dominate over the surface tension force at the last stage of spreading. As the contact angle increases (see Table 1), the effect of shear force decreases. The shear force dominates on the hydrophilic surface compared to that on SHS (see Figure 3 and 5).
The normal imposed pressure and the shear force affects only during the last stage of the spreading, when the impact energy given to the drop is dissipated into viscous and surface energy. It is reported in the previous literature that the pressure is significant when the work done by the pressure is greater than 10% of the kinetic energy of the spreading drop (Ebrahim and Ortega 2017). The pressure is significant on less wetting surfaces at lower Reynolds numbers for the present case, since the viscous dissipation is negligible on the entire surface because of less spreading velocity. Therefore, the surface energy plays a major role at lower Weber numbers to counter the kinetic energy, whereas at higher Weber numbers, inertia force dominates over viscous and surface force. The competing surface energy and kinetic energy generates fingers at the rim periphery while spreading even at lower Weber numbers. Higher the surface energy (i.e., higher contact angle), higher the fingers formed at a given Weber number (for Polystyrene and SHS). At lower Weber numbers, the surface energy retards the flow in the radial direction, since the kinetic energy does not dominate. Hence, more mass is accumulated at the periphery of the rim. The generated finger gets momentum in the transverse direction due to lower surface energy.
20
The normal imposed pressure of air applies pressure on the generated fingers and gives momentum to the lamella in the radial direction to spread more (see Figure 5).
Figure 9: Amount of (i) surface energy utilized, and (ii) viscous dissipation, from the initial energy at maximum spread for different surfaces.
The ratio of surface energy and initial energy increases with the decrease in the wettability (see Figure 9i) and results in the faster deceleration (i.e., kinetic energy loss during spreading) on less wetting surfaces at a given impact Weber number. Therefore, the normal imposed pressure increases at lower Reynolds numbers. At a given Weber number, the surface energy increases with the contact angle although the wetting area decreases (see Figure 9i). The loss due to the viscous dissipation decreases with the increase in the contact angle (see Figure 9ii). The kinetic energy is converted into surface energy, as well. The higher the contact angle, the higher the surface energy at the maximum spread due to the growth of the rim thickness. Further increase in the surface energy due to airflow is because of the enhancement in spreading diameter (see Figure 5 and 9i). The slopes of the variations in Figure 9i at lower impact Weber numbers are steeper than that at higher Weber numbers. The steepness decreases with the decreasing wettability at lower impact Weber numbers. For SHS, since the viscous dissipation is negligible, the kinetic energy is only countered by the surface energy. Therefore, the dominance of the impact Weber number decreases with
21
the increase in the surface energy or contact angle. The dominance of surface energy at low Weber number makes the drop to rebound completely on SHS, whereas, partial rebound or splashing occurs with the increase in Weber numbers. There is a decrease in the slope with the increase in contact angle (see Figure 9i). With the increment in the contact angle, the surface repellency increases and results in less viscous dissipation between the surface and drop (see Figure 9ii). Therefore, higher kinetic energy is converted into the surface energy. Thicker rim also forms at this stage for less wetting surfaces. The rim moves slower and resists the drop to spread, causing a sudden decrease in the maximum spread (see Figure 5).
The data points are missing on the SHS surface for the Wewater > 91 due to the occurrence of splashing. Splashing disintegrate the outer rim of drop into tiny droplet, and that process is difficult to process.
4.3. Comparison of the measured maximum spreading diameter with the one obtained from the model
The maximum spreading diameter is compared with the one obtained from the model (see Figure 10) provided by Singh and Mandal (2019). The model takes the effect of the imposed pressure and shear force of the incident airflow into account. The comparison is better at higher impact Weber numbers (see Figure 10), because the assumption to calculate the boundary layer thickness (𝛿𝑑𝑟𝑜𝑝 =
2𝐷0 √ 𝑅𝑒𝑊𝑎𝑡𝑒𝑟
) works better at higher impact
velocity (i.e., higher Weber numbers and Reynolds number), due to lower boundary layer thickness. The loss between surface and drop decreases with the decrease in the boundary layer thickness during spreading. The comparison of the measured maximum spreading factors (as x axis) with the theoretically obtained ones (as y axis) is provided in the supporting material for the interested readers.
The comparison is observed to improve as one moves from higher to lower wettabilities (see Figure 10). At low Wewater, the adhesive force between the drop and the
22
surface decreases with the decrease in wettability. The model incorporates the rim thickness at the maximum spread, and is greatly influenced by the wettability of the surface. As discussed earlier, lower wetting surfaces give thicker rim at the maximum spread.
Figure 10: The comparison of the experimentally measured maximum spreading factor with the same obtained from the model provided by Singh and Mandal (2019). 'Re_air' and 'Re_air_th' in the legend represent measured βmax, and βmax from the model, respectively, at a given Reynolds number.
The comparison shows that, the incident airflow dominates when the wettability of the surface and Wewater are lower, whereas,the airflow dominates at higher Wewater due to lower rim thickness for the surfaces with higher wettabilities.
Since the predictive capability of the present model is poor at lower Weber numbers (see Figure 10), it is important to provide justifications. Most of the models present in the literature are derived using the fitting of experimental data with the Weber number and Reynolds number. The effect of inertia, viscous and surface tension force are considered to predict the maximum spreading factor. At high impact Weber and Reynolds numbers, these models show good agreement on all wetting surfaces, but the discrepancy increases with the decrease in the surface wettability at low impact Weber numbers. The effect of the wettability plays a vital role in predicting the maximum spreading factor at lower impact Weber numbers. Few models (Chandra et al. 1991; Pasendideh-Fard et al. 1995; Mao et al. 1997; Ukiwe and Kwok 2005) incorparated the effect of the wettability in terms of contact angle using the energy balance method. The energy conservation are applied based on many assumptions during the radial flow of drop on the dry surface. These assumptions are made based on different impact conditions. So, it is difficult to say which assumptions are
23
more accurate in different impact conditions. For example, the assumption of cylindrical disk at the maximum spread of a drop is not valid for higher impact Weber numbers, and high viscosity of liquids, whereas the the development of boundary layer thickness is not valid at lower Weber numbers for non-viscous liquids. Various definition of contact angle in different models derived using energy based method are confusing and complicated too. As the Young’s contact angle defined in the model of Ukiwe and Kwok (2005) is applicable for smooth and heterogeneous surface, whereas advancing contact angle defined in Pasendideh-Fard (1996) model depends upon the impact condition like the viscosity of drop. The calculation of the time required to reach maximum spreading diameter on surface in the model of Pasendideh-Fard (1996) is applicable for non-viscous liquids. Therefore, these model are not consistent to give a good prediction of spreading factor in all impact conditions.
However, many assumptions are valid for non viscous liquids (i.e., water) even though the models are not consistent, like incorporating the rim thickness and wettability of the surface in terms of contact angle, fulfills our motive. So, the predictive capability few times becomes poor.
The comparison of the measured maximum spread for free falling drops with other existing models is provided in the supporting material.
5. Conclusions
The air assisted impact of water drops on surfaces with various wettabilities is studied to bring out the effect of the shear force and normal imposed pressure of air. Three surfaces, glass, Polystyrene, and SHS, are chosen. The study is conducted for five impact heights, and three velocities of the incident airflow.
Results show that the effect of surface wettability is negligible initially (i.e., up to ~1 ms), since the spreading at the initial phase is almost equal for a given drop on all the
24
surfaces at a given Weber number. Maximum spreading is observed to increase with Wewater for the drops impacting on a glass surface. The spreading increases with the airflow (Reair) as well, for a given Weber number. However, the factor is observed to suddenly decrease at higher Weber numbers (244) and Reair (1564) for the less wetting surfaces. So, there is a dependence of the maximum spreading on the wettability.
For the glass surface, higher adhesive force between the drop and surface causes the rim to become thinner without forming fingers, as the drop continues to move in the radial direction. The shear force pushes the rim outward. However, as the wettability decreases (i.e., for Polystyrene and SHS), a deceleration is observed at the early stage of spreading, forming fingers and thicker rim at the last stage. The thicker rim moves slowly in radial direction, causing the viscous dissipation to decrease suddenly for the highest Weber numbers (244) and Reair (1564) tested. The deceleration of thicker rim resists the drop to spread. Hence, the maximum spread reduces suddenly.
The deceleration is observed to be higher when the drop impacts with high Weber number and with a high airflow. The deceleration rate is dependent on the contact angle, as well. The rate increases with the contact angle, due to higher surface tension, and resist the drop to spread.
At lower Weber numbers, the normal imposed pressure of air is observed to dominate for the super hydrophobic surfaces than on a hydrophilic surface. Due to the higher imposed pressure, the drop is forced to interact with the surface more, causing higher spread. Detachment of satellite drops on hydrophilic surface (i.e., glass) increases with the airflow since the shear force increases. At lower Weber numbers, the surface energy retards the flow in the radial direction, since the kinetic energy does not dominate. Hence, more mass is accumulated at the periphery of the rim. The generated finger gets momentum in the transverse direction due to lower surface energy. The normal imposed pressure of air applies pressure on the generated fingers and gives momentum to the lamella in the radial direction to spread more.
25
A comparison of the measured 𝛽 max with the one obtained from an existing model (Singh and Mandal 2019), is provided. Better comparison is found for the cases of less wetting surfaces (i.e., Polystyrene and SHS) at almost all We, whereas, good comparison is observed only at higher We for the case of the glass surface.
Future work can be focused on the airflow assisted impact of various drops on heated surfaces. The surface with elevated temperatures, can change the liquid viscosity, which may have a direct influence on the shear force given by the incident airflow on the drop. Novel physics will possibly come out.
Acknowledgements
The work is supported by the Science and Engineering Research Board (SERB) of the Department of Science and Technology (DST), Government of India (Project number: ECR / 2016 / 000026). Author statement Re: “Air assisted impact of drops: The effect of surface wettability" by Ramesh Kumar Singh and Deepak Kumar Mandal. To our knowledge, it is the first systematic study which explores the influence of the incident airflow when a drop impacts on surfaces with various wettabilities. The authors acknowledge that none of the material in the manuscript has been published or is under consideration for publication elsewhere. If you need any further information, please do not hesitate to contact me. Ramesh Kumar Singh: Conceptualization, Methodology, Validation. Deepak Kumar Mandal: Supervision, Analysis, Software, Resources.
To whom it may concern The authors wish to confirm that there are no known conflicts of interest associated with this publication. The authors confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship
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but are not listed. The authors further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. References
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