Experimental study of the onset of downstream motion of adhering droplets in turbulent shear flows

Experimental Thermal and Fluid Science 109 (2019) 109843

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Experimental study of the onset of downstream motion of adhering droplets in turbulent shear flows

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Beawer Barwari , Sebastian Burgmann, Artur Bechtold, Martin Rohde, Uwe Janoske Chair of Fluid Mechanics, University of Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: Air flow Adhering droplet Hydrodynamic instabilities Liquid-surface interaction Critical air velocity

In this study, the dynamics of adhering liquid droplets on various solid surfaces in shear flow, which is driven by a controlled air flow, are investigated experimentally. A series of experiments are carried out to understand the effects of fluid properties, wetting characteristics, droplet sizes and flow velocities. A rectangular Plexiglaschannel is used for the experiments. Droplets are placed on the bottom wall of that channel. The droplet shape and contour position are measured by the transmitted light technique and are further analyzed by using image processing. The shear flow leads to a deformation and oscillation of the droplet and eventually to a movement downstream. Three regimes are identified: regime I corresponds to a deformation and oscillation, whereas in regime II the deformed and still oscillating droplets moves downstream. Regime III characterizes a continuous downstream movement in a gliding manner. The start of regime II, i.e. the onset of downstream motion is defined by a critical air velocity. Results show that the contact angle hysteresis and surface energy have a strong influence on the critical air velocity for the onset of droplet motion, as well as the wetting characteristics and droplet volume. To find a global empirical law for the critical velocity, a dimensionless approach is derived based on the droplet Reynolds number and a modified Laplace number. The empirical law is in good agreement with experimental data and may serve as an estimator of the critical velocity for the onset of droplet motion.

1. Introduction In many technical applications droplets occur due to various conditions (rain, spray, condensation, etc.) and adhere to the surfaces of the machine, housing etc. [1,2]. Often these droplets experience a gas flow passing the surface where the droplet is located. The additional force due to an air flow may be selectively used to remove droplets on surfaces, e.g. for water management in fuel cells [3] or in cleaning processes [4]. Generally spoken, the droplet is eventually moved when the magnitude of the external force overcomes the magnitude of the adhesive force. The influence of different external effects, e.g. surface vibration [5,6], gravitation [7,8], magnetic and electric fields [9,10], and shear flow [11,12] have been investigated in the past. For the latter case Dussan [11] and Durbin [12] already noticed that a droplet begins to move over the surface, if some critical velocity of the air flow is exceeded. Before the droplet detaches, the droplet undergoes a contour deformation. Droplets with a higher volume show a visible contour deformation compared to the droplets of significantly smaller volume [15–17]. Additionally, an oscillation of the droplet has been observed that exhibits characteristic frequencies [13,14]. It has to be stated that the mechanism and condition for the onset of droplet motion are not

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completely understood. In the following a short overview of the characteristics of droplets on surfaces and the detachment of droplets is given. For a liquid droplet on a horizontal substrate, three interfaces are formed due to adhesion and cohesion forces as shown in Fig. 1: (1) the interface between the fluid and the surrounding gas phase, (2) the interface between the fluid and the substrate and (3) the interface between the gas phase and the substrate. The angle between the substrate surface and the tangent along the droplet surface is defined as the contact angle θS. Three wetting behaviors can be defined depending on the contact angle: complete wetting (θS = 0°), partial wetting (0 < θS < 90°), and non-wetting (90 < θS < 180°). These particular behaviors can be influenced by the surface tension of the liquid σlg, the surface energy of the substrate σsg and the interfacial tension between the fluid and the substrate σsl. Milne and Amirfazli [18] and Moghtadernejad et al. [19,20] used surfaces with different wettabilities to analyze the influence of droplet contact angle on the detachment mechanism. They found that adhesion is higher for a droplet on a hydrophilic surface than on a hydrophobic surface under same conditions. Therefore, a lower force is needed to detach the droplets on hydrophobic surfaces as compared to

Corresponding author. E-mail address: [email protected] (B. Barwari).

https://doi.org/10.1016/j.expthermflusci.2019.109843 Received 21 December 2018; Received in revised form 8 April 2019; Accepted 6 June 2019 Available online 07 June 2019 0894-1777/ © 2019 Elsevier Inc. All rights reserved.

Experimental Thermal and Fluid Science 109 (2019) 109843

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Nomenclature

Subscripts

dh dw H R2 t v V xc

A adv crit ch d e g lg rec S sg sl

height of the droplet (m) width of the droplet (m) height of the channel (m) coefficient of determination time (s) velocity (m/s) volume (m3) position of droplet of center of gravity (m)

Greek letters θ μ ρ σ Ф

contact angle (°) dynamic viscosity (Pa⋅s) density (kg/m3) surface tension (N/m) mass fraction (%)

air flow advancing critical channel droplet ethanol glycerine liquid-gas receding static solid-gas solid-liquid

Superscripts d p

dispersive polar

Dimensionless groups Abbreviation La Re θS

Laplace number Reynolds number Young contact angle

cSW PMMA

coated silicon wafer acrylic glass

history of gas-diffusion-layers based on its aging behavior. They have concluded that aged materials make the removal of the water droplets more difficult compared to unused surfaces. Several investigations have shown that the movement of droplet is dependent on the fluid properties. Thoreau et al. [32] used oil droplets in a laminar water flow to investigate the influence of surfactants on the droplet detachment. They concluded that adding surfactants to the shearing fluid reduces the critical velocity due to the generation of surface tension gradients along the surface of the droplet. Gupta and Basu [33] also investigated a disperse system of water and oil and found that oil droplets with less interfacial tension exhibit a visible deformation compared to the droplets with a high interfacial tension. Seevaratnam et al. [34] and Mandani and Amirfazli [35] studied the instability of oil droplets of different viscosity in a laminar water flow. They have observed three different regimes “sliding, crawling, and detachment by lift-off from the surface” depending on the droplet size, flow rate and viscosity ratio. The authors pointed out that the droplet detachment by lift-off from the surface becomes more pronounced as the droplet viscosity increases. Fan et al. [36] investigated droplets of pure water, pure glycerine and a water-glycerine solution in a turbulent channel gas flow. According to their study, the critical velocity strongly depends on the contact angle hysteresis and droplet volume, but weakly on the viscosity of the liquid droplet. Similarly, Fu et al. [37] investigated droplets of a water-glycerine solution on a flat plate boundary layer flow configuration. They confirmed that the critical velocity strongly depends on the contact angle hysteresis and droplet size. Despite numerous investigations in this field, the phenomena of dynamic wetting and the onset of droplet movement are not completely described and understood. The present work tries to fill the gap with a systematic investigation of the droplet dynamics considering the combination of the influence of fluid properties, wetting behavior and droplet sizes under the influence of a turbulent air flow. A method to identify the critical velocity is developed. The results of the current study may serve to predict the onset of the droplet movement.

hydrophilic surfaces. The results from Bixler and Bhushan [21] demonstrated that the lotus effect induced through nanostructured coating reduces drag, increases contact angle, and improves the selfcleaning efficiency. According to the experimental investigations by Shastry et al. [22] and Li et al. [23], the transport of droplet can be controlled by introducing a wetting gradient on the surface. Adhered droplets on these surfaces are easily transported in the direction of higher wettability due to an external force. Lv et al. [24] performed a material analysis on an inclined plate by using rough hydrophobic substrates. The results show that the beginning of water droplet sliding under gravity always starts with the detachment of the rear contact line of the droplet. Hao et al. [25] confirmed this dynamic mechanism in their study by investigating the droplet behavior in wall-parallel air flow on superhydrophobic horizontal surfaces. Hu et al. [26] experimentally analyzed the surface topography using micro-grooved-surfaces with different direction and dimensions. The authors determined that the critical velocity for the onset of droplet movement is higher for a vertically micro-grooved surface (related to the direction of air flow) compared to a parallel orientation. In many experimental investigations, porous materials or gas-diffusion-layers are analyzed to gain a better understanding of water management on fuel cells. According to the investigations by Kumbur et al. [27], Cai et al. [28] and Theodorakakos et al. [29] the channel flow rate and the surface properties of gas-diffusion-layer affect the droplet detachment: more hydrophobic surfaces lead to a more pronounced droplet removal. Ha et al. [30] analyzed the gas-diffusion surfaces under dry and wet conditions and have observed that droplets can be removed with lower air flow on the dry surfaces compared to wet surfaces. Das et al. [31] examined the

Fig. 1. Schematic representation of an adhering droplet on a substrate. 2

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2. Experiments

reduction of the viscosity. With the addition of ethanol, the density decreases only slightly, but the decrease of the surface tension is more pronounced (see Table 1). The rheological measurements show that all investigated liquids have a Newtonian behavior, since they exhibit a constant viscosity over a very large range of shear rate. Additionally, material properties are taken into account, i.e. wetting behavior is studied for all mentioned fluid mixtures. In this study acrylic glass (abbr. PMMA) and coated silicon wafers (abbr. cSW) are used for the investigation of the wetting behavior. The silicon wafers are coated with the anti-sticking agent Fluorooctatrichlorosilane. For further information on the manufacturing process, readers are referred to [38]. By studying these two materials, a wider range of moderate static contact angles can be investigated: as Fig. 4 shows the static contact angles for the investigated liquid droplets are between 45° to 105°, i.e. hydrophilic to hydrophobic behavior is investigated. As can be seen from the example of water-glycerine solutions, the static contact angles are higher for cSW than PMMA, but remain almost constant for the particular material for different mixtures. On the other hand, for waterethanol droplets the static contact angles on cSW decrease drastically when the ethanol portion is increased. Water-ethanol droplets are not investigated on PMMA due to uncontrolled risk of corrosion. Finally, the surface energy of PMMA and cSW is calculated according to [39,40]. Therefore, contact angle measurements are performed for four different liquids (water, glycerine, ethanol, and diiodomethane) and their dispersive and polar part of the corresponding surface tension (σ = σd + σp) are taken from [41]. Based on these values the surface energy is calculated and these are the results: for waterglycerine on PMMA: σsg = (33.4 + 13.7) mN/m, for water-glycerine on cSW: σsg = (11.6 + 3.5) mN/m; and for water-ethanol on cSW: σsg = (10.4 + 8.6) mN/m. Thus, PMMA has a water-attracting effect and cSW has a water-repellent effect.

2.1. Method Fig. 2 shows a sketch of the test rig. A Plexiglas-channel of rectangular cross-section is used. The lower channel-wall at the measurement section can be changed such that different substrates may be examined. It is also possible to modify the upper channel wall to obtain different channel heights. In this investigation, the height and width of the channel are 22 mm and 22 mm, respectively. The channel length is 700 mm, which is sufficient to provide fully developed laminar or turbulent flow at the measurement section. In addition, a honeycomb structure is integrated at the entrance to support the development of controlled velocity profiles. The droplets are placed manually on the bottom wall of the channel in the measurement section using a microliter syringe. The air-flow-rate that passes through the Plexiglas-channel is adjusted by a mass flow controller. In each experiment the flow velocity is linearly increased by 0.258 m/s per second which corresponds to 7.5 l/ min. The ramp time and velocity step are kept constant for all experiments. The experiment itself takes 30 s in the maximum. The experiments are stopped when the droplet reaches the gliding phase. This happens at Reynolds numbers between 2.500 and 22.200 (with respect to bulk velocity and hydraulic diameter of the channel), depending on the fluid and material properties. The velocity profile of the air flow in the channel is measured with a constant-temperature anemometer (TSI, Type 1750 CTA) by using a hot-film sensor with a standard boundary layer probe (TSI, Model 1218) at the same location where the droplet is placed (Fig. 2). The hot-film sensor is calibrated in a tube with an undisturbed and well-defined flow field previously to the experiments. The mean velocity profiles in the symmetry plane of the measurement section are exemplarily shown in Fig. 3 for three different channel’s Reynolds numbers Rech (2.500, 12.400 and 22.200), which are calculated based on the hydraulic diameter and bulk mean velocity. These correspond to a mean velocity range of 1.7 to 15.5 m/s. The droplet behavior is measured with high temporal resolution by a high-speed camera (MotionBLITZ EoSens Cube7) with a built-in system lens. The frame rate for all measurements is set to 120 fps. The shadowgraph technique is used, i.e. the measurement section and the droplet within is located between the camera and an LED. The images of the droplet provide a strong contrast. To assess the instantaneous contour of the droplet, the images of the droplet shadow are automatically analyzed by an in-house tool which is based on an edge detection algorithm. The first image of the droplet is always recorded before the air flow starts in order to determine the static contact angle. A number of 115 different test series are carried out. Each test series is repeated 6 times under the same conditions. This results in a total number of 690 experiments. In this analysis, various droplet volumes ranging from 7.8 to 39.9 µl are investigated. The height of the droplet varies between 0.9 and 2.7 mm. This maximum height of the droplet is less than 13% of the channel height. Accordingly, the droplet is within the strongest velocity gradient of the channel flow (compare Fig. 3).

3. Experimental results and discussion 3.1. Motion pattern In a first step, the contours of the non-moving (“static”) and moving droplets are analyzed to categorize the behavior of the droplet into different phenomena. With the help of these visual observations, the process of the droplet detachment can be defined in a second step. Fig. 5 exemplarily illustrates the contour of adhering and moving droplets on cSW for a 39.9 µl water-glycerine droplet. The time t = 0 corresponds to the start of the experiment, i.e. when the flow is started.

2.2. Fluid and material properties Various water-glycerine and water-ethanol solutions are used for the liquid droplets. The specific mixtures and the corresponding fluid properties at 25 °C are shown in Table 1. The fluid properties of the liquid solutions are measured as follows: the density ρ with an analytical balance, the surface tension σ with a tensiometer using Du Noüy’s ring method, and the dynamic viscosity μ with a rotational viscometer. Adding glycerine into water leads to a significant rise of viscosity, but only a slight change in density and surface tension (see Table 1). Water and ethanol form an azeotropic mixture. Increasing the ethanol mass fraction up to 40–50% leads to an increase in the viscosity of the mixture. A further increase in the ethanol amount results in the

Fig. 2. Schematic of experimental setup for the investigation of the droplet behavior. 3

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Fig. 3. Mean streamwise velocity profiles for the three investigated Reynolds numbers in the measurement section and schematic representation of the contour shape of the droplets.

droplet contour is only observed for a glycerine mass fraction of 70% and higher (see Fig. 5). For pure water droplets and droplets of low glycerine-content, it can be observed that the moving droplet almost retains its “static” shape during the movement. The wetted area slightly decreases or remains the same as for the non-moving case. Due to the air flow the mass distribution within the droplet is shifted downstream and the droplet slightly grows in height. The droplet contour becomes steeper at the downstream side, whereas at the upstream side the droplet becomes flatter. This becomes more prominent for higher glycerin mass fraction where a small tail at the rear of the droplet is formed. For glycerine mass fraction of 90% and higher, the tail increases as the droplet travels further in the downstream direction, i.e. in these cases the wetted surface increases. For comparison, Fig. 6 shows the 39.9 µl-water-glycerine droplets on PMMA. Again, all “static” droplets have a similar shape. Compared to cSW, the droplets are flatter and wider due to the smaller static contact angles (≈100° for cSW and ≈70° for PMMA). The droplets are deformed due to the air flow but they are deformed stronger than for cSW. As for PMMA, the deformation becomes more pronounced with increasing glycerine content. Based on these observations, it can be concluded that three phenomena can occur for continuously moving droplets on surfaces at contact angles around 90°: (1) the “compact” shape for low droplet viscosities (compact shape is defined as a deformed droplet with smaller wetted area than for the non-moving case, whereas the height of the droplet increases); (2) the tail formation in connection with a possible droplet separation at the rear of the droplet; (3) the formation of rivulets for high droplet viscosities. The onset of these three phenomena also depends on the surface energy. The tail formation on PMMA occurs at lower viscosities compared to cSW due to the waterattracting effect of PMMA. No droplet separation is observed for cSW, since this might be an effect of surface roughness and cSW is smoother than PMMA. Observations show that smaller droplets typically remain in the “compact” shape during their movement. Independent of the material, it is observed that droplet movement downstream takes longer as the glycerine mass fraction increases and the droplet volume decreases. Fig. 7 illustrates the contour of the adhering and moving droplet for different water-ethanol solutions on cSW. In contrast to the previous cases the static contact angles decrease as the ethanol mass fraction increases due to the reduction in surface tension. As a result, the droplets are flatter in the “static” state for higher ethanol mass fractions. As in the previous case at contact angles around 90° the deformation and movement of the droplet shows the compact shape-phenomenon. Due to the low viscosity of the water-ethanol mixtures, no tail- and rivulet formation is present in this case. But for ethanol mass fractions of 80% and higher, a small tail forms on the rear side of the droplet although

Table 1 Composition and thermophysical properties values of selected liquids used for the droplet at 25 °C. Water-glycerine solutions Фg (%) 0 10 20 30 40 50 60 70 80 90 100

ρ

σ 3

(kg/m ) 997.05 1006.26 1031.38 1055.58 1083.63 1123.55 1124.79 1182.74 1208.20 1244.94 1258.20

(mN/m) 71.96 70.38 69.82 67.52 66.77 65.77 66.42 65.76 64.89 64.23 63.50

Water-ethanol solutions μ (mPa s) 0.89 1.31 1.76 2.41 4.25 5.14 9.27 21.75 60.90 185.45 1132.90

Фe (%) 0 10 20 30 40 50 60 70 80 90 100

ρ 3

(kg/m ) 997.05 985.18 985.92 961.44 950.01 941.17 906.59 868.04 848.72 813.87 784.31

σ

μ

(mN/m) 71.96 51.91 41.30 35.68 32.62 30.81 29.48 28.39 27.16 25.91 24.32

(mPa s) 0.89 1.41 1.96 2.34 2.53 2.51 2.38 2.14 1.86 1.56 1.21

Fig. 4. Static contact angle for PMMA and cSW are plotted against the glycerine mass fraction and the ethanol mass fraction for droplets with 7.8 µl volume.

In the “static” state (dashed contour), i.e. without the effect of air flow, the droplet contour is identical for all glycerine mass fractions. With increasing air flow which corresponds to increase of time the droplet contour is deformed. Measurements show that additionally an oscillation of the droplet occurs before the droplet moves. The contours shown here and in the following images are the temporal mean over a small time span, i.e. the contour oscillation is not visible in this depiction. At a specific time, the droplet moves downstream and deformation becomes stronger. As can be seen a very prominent deformation of the 4

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Fig. 5. Typical patterns of static and moving droplets (39.9 µl) in shear flow on cSW for glycerine mass fraction from 0 to 100% (contours correspond to time step: — 12 s 18 s — 21 s — 24 s). vt is the air velocity to the last time step for each investigated case. 0s

the droplet is much “flatter” than in the previous cases. Tail formation means that the receding contact angle flattens, whereas the advancing contact angle remains constant. Again, it can be observed that deformation increases with droplet size, i.e. the shape of the moving droplet differs to a higher extent from the “static” shape. Summing up, for all investigated cases the adhering droplets are deformed and start to oscillate when the air flow is increased. Eventually they start to move over the surface. Note the phenomenon of adhering droplets in increasing shear flow can be divided into three regimes: in the first one, the droplets do not move but oscillate, in the second one the droplets move a little while still oscillating (which occurs in a stick-slip-manner) and in the third regime, the droplets move downstream in a stable gliding manner (which become apparent in the figures before). Only in some investigated cases, i.e. for high droplet viscosity on PMMA-substrate, the distinction between these regimes can hardly be made. The droplet deforms in a way that it exhibits a long tail, i.e. the droplet spreads along the surface (see Fig. 6). This dropletspreading resembles the phenomenon of rivulets. In these cases, an onset of droplet movement cannot be clearly identified, since their droplet-shape is lost. In the following we focus on those cases, where droplet detachment and movement is clearly detectable.

3.2. Definition of the onset of droplet movement As shown in the previous section, as air velocity increases the droplets start to move downstream. Based on the plurality of experiments and the repetition of each single case it can be concluded that droplet movement happens when a specific critical velocity of air flow is exceeded. This critical velocity cannot be determined solely by the visualizations shown above. Therefore, in the following section, a method for the description of the transition of droplet adhesion to droplet movement is described. As mentioned above for rising air velocity, before the droplet moves downstream in a stable gliding phase (regime III) it starts to move in a stick-slip-manner and stronger oscillation (regime II). This movement happens even when the air velocity is kept constant over time in the following. Therefore it is important to find the corresponding critical velocity which initializes that motion. Some attempts for the definition of droplet detachment can already be found in literature. In [25], the critical velocity for onset of the droplet movement is identified by using the point at which the advancing and receding contact lines are moved for the first time for superhydrophobic surfaces. In [42], droplet detachment is defined as the time when advancing and receding contact line move with the same velocity. In Fig. 8 the positions of the advancing and receding contact line are

Fig. 6. Typical patterns of static and moving droplets (39.9 µl) in shear flow on PMMA for glycerine mass fraction from 0 to 100% (contours correspond to time step: 12 s 18 s — 21 s — 24 s). vt is the air velocity to the last time step for each investigated case. — 0s 5

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Fig. 7. Typical patterns of static and moving droplets (left 7.8 µl and right 39.9 µl) in shear flow on cSW for various ethanol mass fraction (contours correspond to 12 s 18 s — 21 s — 24 s). vt is the air velocity to the last time step for each investigated case. time step: — 0 s

Fig. 9 shows two typical examples: water droplets on PMMA (top) and on cSW (bottom). The x-axis represents the distance travelled by the center of gravity xc of the droplet and its value is normalized by the actual width of the droplet dw. The y-axis represents the ratio of the droplet velocity of the center of gravity vd to the air flow velocity vA. As can be seen, the graphs are very similar. In both cases, the graph can be segmented into three different regimes. The images on the right side show four successive images for these three different regimes. In regime I, the droplet does not move but oscillates around the half-axis. In regime II, the droplet accelerates in the downstream direction and performs a stick-slip motion. At this state, the center of gravity is located to a greater extent on the right semi-axis. In regime III, the droplet begins to move continuously over the surface without a further change of the contour shape and almost no oscillation. The beginning of the droplet movement is determined by the change of the curvature and the spreading of the points, as marked with the dotted box in Fig. 9. Each point belongs to a single measurement time and the time step between the points is the same. To make it even more clear that there is a significant change of the droplet behavior, in Fig. 10 the derivation of the curve from the Fig. 9 is plotted. It can be clearly seen that there is a significant change of the droplet acceleration. That change of the droplet acceleration is defined as the transition from regime I to II. The onset of movement that is initializes by this acceleration corresponds to a distinct air velocity which is called critical

exemplarily plotted over the time for water droplets (39.9 µl) on PMMA and cSW. It can be seen that both curves are congruent for cSW. Consequently, the two contact lines move with the same velocity over the surface. In comparison, the contact lines on PMMA move at different velocities. However, these definitions and methods mentioned above are not suitable in this case. In this study, onset of movement is defined as that moment in which the droplet moves for the first time over the surface and enters the regime II. It may happen at lower velocities that the droplet slightly stutters over the surface. As can be seen in Fig. 8, for very low velocities the droplet in total does not move (0 to approx. 9 s). With increasing air flow there is a slight movement of the contact lines still within regime I. However, this droplet movement is not a stick-slip motion (which belongs to regime II). The air flow is linearly increased but for technical reasons the increase is in small time steps, i.e. for a small period of time the air flow is kept constant. In regime I the movement of the droplet occurs from time to time (not in every case) when the velocity increases again but the droplet remains on that position for the complete time step. In regime II the droplet exhibits a stick-slip motion even when the air flow velocity is kept constant. This effect has been verified in additional experiments where the air flow rate is kept constant on different levels. In order to clearly identify the onset of droplet motion a dimensionless description of droplet position versus velocity is used.

Fig. 8. The typical movement behavior of water droplets (39.9 µl) on PMMA (left) and cSW (right). The position of the advancing and receding contact line is plotted against the time. 6

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Fig. 9. Definition of the onset of droplet movement and method for determining the critical velocity for PMMA (top) and cSW (bottom) using the example of water droplets (39.9 µl).

investigated cases the critical velocity is determined which is the local air velocity at the level of the droplet height. Fig. 11 shows the individual critical velocities as a function of the droplet volume and the mass fraction for water-glycerine droplets on PMMA (top), for waterglycerine droplets on cSW (middle), and for water-ethanol droplets on

velocity. 3.3. Critical droplet detachment velocity Based on the definition and analysis presented above, for all

Fig. 10. Derivation of vd/vA (droplet velocity/air flow velocity) as a function of xc/dw for PMMA (left) and for cSW (right) using the example of water droplets (39.9 µl). 7

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Fig. 11. Critical velocity required to initiate droplet movement as a function of the droplet volume (left) and mass fraction (right) for water-glycerine mixtures on PMMA (top), for water-glycerine mixtures on cSW (middle), and for water-ethanol mixtures on cSW (bottom).

highest droplet contour surface is further away from substrate compared to the smaller droplet. Additionally, it can be deduced that the critical velocity increases for all droplet volumes independent of the material properties, as the glycerine mass fraction increases. This behavior is related to an increase of the contact angle hysteresis and only to a small extent to an increase of viscosity as becomes apparent in Fig. 12. The higher the difference between cosθrec and cosθadv, the greater the surface forces. Apart from viscous damping, the adhesive force increases due to the larger wetted area between the droplet and the solid surface.

cSW (bottom). The error bars represent the spreading of the results as a consequence of the repetition of each experiment. As can be seen, the statistical deviation is low. There are some findings that can be directly deduced from these graphs. In all cases, the critical air velocity required to initiate a droplet movement decreases as the droplet volume increases and this result agrees with literature [18]. All investigated fluid-material systems show a similar trend. The 39.9 μl-droplet needs a 15–25% lower air flow velocity to start moving as compared to the 7.8 μl-droplet. This is because a larger droplet is subjected to a higher local velocity, as the 8

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Fig. 12. Critical velocity required to initiate droplet movement as a function of (a) droplet volume, (b) droplet viscosity, (c) static contact angle, (d) contact angle hysteresis and (e) surface energy for water-glycerine mixtures on PMMA and cSW, and for water-ethanol mixtures on cSW.

The effect of the ethanol mass fraction does not show a clear trend as shown in Fig. 11 (bottom). Several effects counteract in this case, e.g. viscosity, contact angle hysteresis and surface tension. In Fig. 12 the results of all experiments are shown together in several diagrams. Based on that representation the effect of droplet volume, viscosity, static contact angle, contact angle hysteresis and surface energy might be distinguished. As already shown, if the droplet size increases the critical velocity decreases slightly. Increasing droplet viscosity leads to a slight increase of the critical velocity. As can be deduced from Fig. 12(c) there is no direct link between the static

The critical velocities for cSW are more than 50% lower compared to those of PMMA. As already mentioned earlier, the contact angles for cSW are higher than the contact angles for PMMA. The distance between the substrate and the highest droplet contour surface increases. Accordingly, the droplet is subjected to a higher local velocity with respect to the velocity profile (compare Fig. 3). Furthermore, the drag force counteracts a lower surface force due to the reduction of the contact angle hysteresis. As observed in experiments, the droplet on cSW exhibits a more compact shape. This results in a small wetted area between the droplet and the surface, resulting in lower capillary force. 9

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contact angle and the critical velocity. Although the static contact angles for water-glycerine droplets on PMMA are larger than the static contact angles for ethanol droplets on cSW, higher critical velocities are required. On the other hand, there is a clear dependence between the critical velocity and the contact angle hysteresis. With increasing the contact angle hysteresis, the critical velocity increases. Contact angle hysteresis might influenced by the surface roughness. The coated silicon wafers are supposed to be less rough than the PMMA substrate. But that does not seem to be the dominant effect here. As can be clearly seen in Fig. 12(e) the surface energy seems to be the main influencing parameter for the critical velocity. Since the contact angle hysteresis is directly linked to the surface energy the two diagrams show a similar trend. However, there is a spreading of the points in each diagram since several effects superpose. Therefore, it is suitable to find a non-dimensional description of this phenomenon. In the following an empirical, non-dimensional law is developed.

tan

⎛ 16Vd dw = ⎜ ⎜ π tan θS 1 + 1 tan2 θS 2 3 2 ⎝

(

⎛ 2tan2 θ2S V d dh = ⎜ ⎜ π 1 + 1 tan2 θS 3 2 ⎝

(

La =

π ⎡ ⎛ dw ⎞2 + d2h⎤ dh 3 ⎥ 6 ⎢ ⎣ ⎝ 2 ⎠ ⎦

1 3

(4)

1 3

(5)

ρd σsl dw μd2

(6)

where ρd and μd are the liquid droplet properties and the droplet width dw is taken as the characteristic length. As discussed earlier, the critical velocity strongly depends on the material properties (surface energy of the substrate). For this reason, the influence of wetting is included in the Laplace number in terms of the interfacial tension σsl between the liquid droplet and the substrate surface, instead of only the surface tension σlg of the liquid droplet, which is commonly used (compare Fig. 1). The interfacial tension is calculated according to Young's approach as follows:

σsl = σsg − cosθS σlg

(1)

(7)

where σsg is the surface energy of the material. Using the interfacial tension, it is possible to take into account the effects wetting characteristics additionally to fluid properties and the droplet geometry. Fig. 14 shows the plot of the critical droplet Reynolds number as a function of the reciprocal value of the Laplace number. The experimental results in this work can be accurately well reproduced with the derived relationship. There is a strong exponential relation between Recrit and La. Thus, the curve can be characterized by an exponential relation which is fitted as:

In this context, ρd is the density and μd is the dynamic viscosity of the liquid droplet and dh represents the droplet height. When the droplet starts to move, its velocity is unknown a priori. The maximum possible velocity of the droplet would be the gas velocity (not considering any pinning effects of the contact line). Based on this conclusion the critical velocity required for the onset of droplet movement is taken as the characteristic velocity for the droplet Reynolds number. As mentioned above, the droplet geometry changes during the transition from regime I to II. Hence, and also due to the oscillation, there is no fixed geometry. To be able to asses all investigated cases in the same way some assumptions are made. In most technical applications, the droplet volume is known and not its shape on a distinct surface. Furthermore, we assume that the static contact angle is also known. Therefore, all geometry values are deduced from the droplet volume and the static contact angle in this empirical model. The droplet on the surface is assumed as a spherical cap. The volume can be expressed by

Vd =

)

⎞ ⎟ ⎟ ⎠

)

⎞ ⎟ ⎟ ⎠

The second dimensionless parameter, the Laplace number, is defined as the product of surface forces and inertial forces of a fluid divided by the square of the frictional force. In this analysis, we use a modified approach of the Laplace number. It is defined as follows:

It would be useful to develop a predictive tool based on simple parameters that could be easily obtained from simple experiments and data sets. For that reason, we derive a simple empirical law for the description of the critical velocity based on our experimental results. The approach takes into account the Young equation and the two important dimensionless numbers, which are identified using Buckingham’s π theorem: the critical droplet Reynolds number, Recrit, and a modified Laplace number, La. The Reynolds number describes the ratio of inertial forces to viscous forces. It is calculated as follows:

ρd dh vcrit μd

(3)

According to the assumption that the droplet is a spherical cap, the droplet width and droplet height can be obtained combining the Eqs. (2) and (3):

3.4. Empirical model of critical droplet detachment velocity

Recrit =

θS 2d = h 2 dw

1 −0.483 Recrit = 62.713 ⎛ ⎞ ⎝ La ⎠

(8) −0.483

2 θS ρd vcrit ⎛ 2tan 2 V d ⎜ μd ⎜ π 1 + 1 tan2 θS 3 2 ⎝

(

⎛ ⎞ ⎜ ⎟ ⎞ ⎜ ⎟ 1 62.713 = ⎟ 1 ⎟ ⎜ ⎟ 3 ⎞ ⎟ ⎜ ρd σsl ⎛ ⎠ 16Vd ⎜ μd2 ⎜ π tan θS ⎛1 + 1 tan2 θS ⎞ ⎟ ⎟ 2 ⎝ 3 2 ⎠⎠ ⎝ ⎝ ⎠ 1 3

)

(2)

(9) The coefficient of determination, R2, is found to be 0.83. However, some limitations still exist in the model, as for higher Laplace numbers

where the width is considered to be equal to the depth. Moreover, it is possible to define the droplet width dw and droplet height dh by the static contact angle θS. In Fig. 13, the half-angle method is shown, which is commonly used in many technical applications for characterization of the wetting behavior [43,44]. The half-angle method allows the contact angle to be determined directly from the “static” droplet geometry. This method is applied by drawing a line from the threephase point to the vertex of the circle. This line has an angle which is half of the static contact angle as can be seen in Fig. 13. Comparing the results of this approach with the measurement data of this work, we found a reasonable agreement and it seems to be a valid simple approach. The contact angle is determined assuming the droplet shape as part of a circle as follows:

Fig. 13. Schematic representation of the half-angle method. 10

Experimental Thermal and Fluid Science 109 (2019) 109843

B. Barwari, et al.

is velocity versus position of the droplet and acceleration versus position of the droplet the onset of movement and therefore the critical velocity can be clearly extracted. Based on aforementioned definition, the effects of viscosity, wetting behavior and droplet volume are analyzed. For increasing droplet volume, the droplet experiences higher velocities at the droplet top as it covers a larger area of the velocity gradient at the channel wall. Hence, larger droplets lead to lower critical velocities. It is found in agreement with literature that hydrophobicity leads to a decrease of the critical velocity. The onset of droplet motion is reached faster (i.e. at lower velocities) by decreasing the surface tension of the liquid. Additionally, it is found that the critical velocity increases slightly with increasing viscosity. However, the increase of viscosity in this case is directly related to an increase of contact angle hysteresis. Contact angle hysteresis and surface energy have the most significant effect on the increase or decrease of the critical velocity. Contact angle hysteresis is influenced by the surface roughness. The coated silicon wafers are supposed to be less rough than the PMMA substrate. But this effect seems to be negligible in this case. In literature several analytical approaches can be found to predict the critical velocity. But these models are based on the assumption of a rigid shape of the droplet and corresponding approaches are used for the drag coefficient. The process of droplet detachment corresponds to an oscillation of the droplet, i.e. the droplet surface is deformed and oscillates at characteristic frequencies [13,14]. It can be assumed that the dynamics of the droplet plays an important role concerning the balance of forces, but neither the deformation nor the oscillation have been taken into account in the previous studies [32,36,37]. The drag coefficient is the most difficult parameter to be assessed for an analytical model, because it changes with the droplet shape and the Reynolds number. There is no validated formula for flexible, cap-like structures. Hence, an empirical approach based on the broad parameter range investigated in this work seems to be suitable. Applying π theorem, two characteristic dimensionless numbers are found that describe the problem: Reynolds number and a modified Laplace number. In this case, the modified approach of the Laplace number takes into account the Young equation to consider the wetting properties. The empirical law covers the experimental data with an R2 of 0.83. As we demonstrate in this work based on a “static” state, the critical velocity for the onset of droplet motion may be predicted with good accuracy using the empirical law. Correspondingly, the critical velocity can be calculated based on simple parameters (droplet volume, static contact angle and fluid properties) they could easily obtain from simple experiments and data sets.

Fig. 14. The critical droplet Reynolds number for the droplet detachment as a function of the reciprocal value of the Laplace number for all investigated droplets.

(higher surface tension, lower viscosity, and larger droplets) there is a stronger deviation from the fitting-curve. Deviations can be attributed to the assumptions which are made for this model concerning the geometry of the droplet, since larger droplets deform stronger from the assumed spherical cap shape. In this study, the droplet Reynolds number Recrit is between 16 and 19.000, and the Laplace number La is in the range of 0.06 to 115.000. As can be seen this empirical approach may be used in this wide range to determine the critical velocity for the droplet detachment only as a function of the static contact angle, the droplet volume and fluid properties (see Eq. (10)).

vcrit 2 θS μ ⎛ 2tan 2 V d = d⎜ ρd ⎜ π 1 + 1 tan2 θS 3 2 ⎝

(

−1 3

)

⎞ ⎟ ⎟ ⎠

⎛ ρ σsl ⎛ 16Vd 62.713 ⎜ d 2 ⎜ ⎜ μd ⎜ π tan θS 1 + 1 tan2 θS 2 3 2 ⎝ ⎝

(

)

⎞ ⎟ ⎟ ⎠

1 3

0.483

⎞ ⎟ ⎟ ⎠ (10)

4. Conclusions In this work the behavior of liquid droplets adhering to the wall of a rectangular channel under the influence of air flow are investigated. The critical velocity to detach a droplet from a surface has been analyzed over a broad range of droplet volumes, fluid properties and wetting behavior. In the first step, the global behavior and the onset of droplet movement is thoroughly investigated because there is no common definition in literature. It is found that the droplets exhibit the same behavior before movement starts, i.e. a deformation of the droplet with a shift of the center of gravity downstream occurs while the whole droplet is oscillating (regime I). As long as viscosity is not too large for slightly hydrophilic surfaces (e.g. glycerine-content of 70% in a water droplet on a PMMA-surface) the droplet starts to move downstream in a “compact” shape when a critical velocity is reached. In case of larger viscosity, droplets deform in a way that they exhibit a long tail and an onset of movement cannot be clearly identified, as the droplets start to span over a wide distance. At a distinct velocity the droplets start to move over the surface. This movement phase can be divided into two regimes: in the first one, the droplets move but still oscillate (regime II) and in the second, the droplets move downstream in a stable gliding manner (regime III). In this work the velocity that leads to the onset of movement at regime II is defined as the critical velocity. Using a dimensionless description which

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