Characterization of Low Weber Number Post-Impact Drop-Spread Dynamics by a Damped Harmonic System Model A Thesis Submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in the Department of Mechanical Engineering of the College of Engineering 2011 By Sandeep K. Gande BE.(Mechanical), National Institute of Technology, Warangal, AP, India, 2008 Committee Co-Chairs: Dr. Milind A. Jog and Dr. Raj M. Manglik
ABSTRACT The post-impact spread, recoil, and shape oscillations of a droplet impinging on a dry horizontal substrate at low Weber numbers (We < ~ 30) are modeled as the behavior of a secondorder damped harmonic system. Droplets of six different Newtonian liquids (acetic anhydride, 4:3 aqueous glycerin, ethylene glycol, glycerin, propylene glycol, and water) impinging on hydrophobic (Teflon) and hydrophilic (glass) substrates are considered. These liquids are selected so as to cover a wide range of viscosities and surface tension coefficients. Photographic images of the post-impact spread-recoil process obtained using a high-speed digital video camera (2000 frames per second) at different Weber numbers are analyzed. A MATLAB based numerical tool was developed to obtain the temporal variations of droplet height and spread from the high-speed images. The results are presented in terms of the flatness factor (the ratio of liquid height to the droplet diameter prior to impact) and the spread factor (the ratio of liquid spread to the droplet diameter prior to impact). It is observed that the transient flatness and spread factor variations on a hydrophobic substrate at low Weber number resemble the damped harmonic response of a mass-springdamper system. During the spread, recoil, and shape oscillations, the surface tension force acts as a spring and liquid viscosity provides the damping. Due to contact angle hysteresis, the frequency of oscillations for the transient flatness factor variation is slightly different from that for the spread factor variation. Semi-empirical correlations are developed for the oscillation frequency and the damping factor as a function of drop Weber and Reynolds numbers. The predictions of temporal variations of the flatness factor and the spread factor from these ii
equations agree very well with experimental measurements on a hydrophobic substrate (Teflon). The drop spread behavior on a glass substrate, because of its hydrophilic nature, does not follow the response of a mass-spring-damper system with a linear spring; perhaps a non-linear spring stiffness might capture its spread dynamics. iii
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ACKNOWLEDGEMENTS The satisfaction and euphoria that accompany the successful completion of any task would be incomplete without the mention of people who made it possible, whose constant guidance and encouragement crowns all efforts with success. I sincerely thank Dr. Milind Jog and Dr. Raj Manglik, who are generous enough to spare their precious time in giving me the guidance, encouragement and valuable suggestions. I would like to express my gratitude and love towards my parents for having made me what I am today. Last, but not the least, I would like to thank my friends who were a constant source of support and help throughout my Research. v
Table of Contents Page Abstract Acknowledgements Table of Contents List of Tables List of Figures Nomenclature i v vi viii ix xii Chapters 1. Introduction 1 1.1 Motivation 1 1.2 Scope of the present work 3 2. Literature Review 4 3. Model Development 7 3.1 Introduction 7 3.2 Damped Harmonic System Model 11 3.3 Analysis of Experimental Data 13 3.4 Spread factor variation 15 3.5 Flatness factor variation 17 4. Results and Discussion 20 4.1 Predictions of transient flatness variations on Teflon surface 21 4.2 Predictions of transient spread variations on Teflon surface 35 vi
4.3 Transient flatness factor variations on a Glass substrate 48 5. Conclusions and Recommendations for Future work 55 5.1 Conclusions 55 5.2 Recommendations for future work 57 References 58 Appendix A 61 Appendix B 71 vii
Table 3.1 4.1 4.2 4.3 4.4 4.5 List of Tables Physical properties of the pure liquids. Experimental data for droplet impact on a Teflon substrate (Ravi, 2011). Harmonic oscillation variables for predicting the flatness factor on a Teflon substrate Harmonic oscillation variables for predicting spread factor on a Teflon substrate Experimental data for droplet impact on a glass substrate (Ravi, 2011) Harmonic oscillation variables for predicting flatness factor on a glass substrate Page 13 20 23 36 48 49 viii
List of Figures Figure 1.1 1.2 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 Various stages of droplet post-impact spread-recoil dynamics: a) At the time of impact, b) Maximum spread, c) Maximum recoil, d) Equilibrium Instantaneous spread and height during droplet spread-recoil process A schematic showing similarities in a simple harmonic (mass-spring-damper) system and the droplet-substrate system Comparison of damped harmonic motion responses with droplet substrate dynamics. Schematic diagram of the experimental setup (Ravi, 2011). Droplet image showing indent and formation of a thick rim during drop spreading (Ravi, 2011) Figure showing the equilibrium shape of a large droplet (Ravi, 2011). Experimental and predicted transient flattening factor variation, Substrate: Teflon, Water droplet, d 0 = 3.087 mm, We = 21.16, Re = 2179.36 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Water droplet, d 0 = 3.095 mm, We = 10.61, Re = 1545.28 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Water droplet, d 0 = 2.108 mm, We = 9.03, Re = 1176.59 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Acetic Anhydride droplet, d 0 = 2.295 mm, We = 24.07, Re = 1711.37 Page 1 2 7 10 14 15 19 24 25 26 27 ix
4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Aqueous-Glycerin droplet, d 0 = 2.883 mm, We = 19.00, Re = 90.53 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Propylene Glycol droplet, d 0 = 2.306 mm, We = 16.49, Re = 22.87 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Propylene Glycol droplet, d 0 = 1.669 mm, We = 11.93, Re = 16.55 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Ethylene Glycol droplet, d 0 = 2.659 mm, We = 19.24, Re = 103.43 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Ethylene Glycol droplet, d 0 = 1.624 mm, We = 9.40, Re = 56.51 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Glycerin droplet, d 0 = 2.781 mm, We = 23.39, Re = 3.08 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Water droplet, d 0 = 3.087 mm, We = 21.16, Re = 2179.36 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Water droplet, d 0 = 3.095 mm, We = 10.61, Re = 1545.289 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Water droplet, d 0 = 2.108 mm, We = 9.03, Re = 1176.59 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Acetic Anhydride droplet, d 0 = 2.295 mm, We = 24.07, Re = 1711.37 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Aqueous-Glycerin droplet, d 0 = 2.883 mm, We = 19.00, Re = 90.53 29 30 31 32 33 34 38 39 40 41 42 x
4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Ethylene Glycol droplet, d 0 = 2.659 mm, We = 19.24, Re = 103.43 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Ethylene Glycol droplet, d 0 = 1.624 mm, We = 9.40, Re = 56.51 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Propylene Glycol droplet, d 0 = 2.306 mm, We = 16.49, Re = 22.87 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Propylene Glycol droplet, d 0 = 1.669 mm, We = 11.93, Re = 16.55 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Glycerin droplet, d 0 = 2.781 mm, We = 23.39, Re = 3.08 Experimental and predicted transient spreading factor variation, Substrate: Glass, Ethylene Glycol droplet, d 0 = 2.697 mm, We = 19.52, Re = 104.95 Experimental and predicted transient spreading factor variation, Substrate: Glass, Glycerin droplet, d 0 = 2.809 mm, We = 23.62, Re = 3.11 Experimental and predicted transient spreading factor variation, Substrate: Glass, Water droplet, d 0 = 3.053 mm, We = 20.93, Re = 2155.94 Experimental and predicted transient spreading factor variation, Substrate: Glass, Propylene Glycol droplet, d 0 = 2.446 mm, We = 17.49, Re = 24.26 Experimental and predicted transient spreading factor variation, Substrate: Glass, Water droplet, d 0 = 2.424 mm, We = 10.38, Re = 1353.19 43 44 45 46 47 50 51 52 53 54 xi
Nomenclature Symbol Description Units d 0 Droplet diameter before impact mm V 0 Droplet impact velocity m/s D Instantaneous droplet spread mm D max Maximum droplet spread after the impact mm D eq Equilibrium droplet diameter after impact mm h Instantaneous droplet height mm h min Droplet height at maximum spread mm H eq Equilibrium droplet height after the impact mm V cap Volume of the droplet calculated from d 0 mm 3 Re Reynolds number = -- We Weber number = -- Oh k c m Ohnesorge number -- Spring constant Damping coefficient Mass y(0) Initial displacement m Initial velocity m/s xii
Greek Letters Symbol Description Units α Viscous Damping factor 1/s β Spreading factor = / -- δ Flattening factor = / -- θ eq Equilibrium Contact Angle deg λ Capillary length mm µ Liquid viscosity Pa.s Liquid density kg/m 3 σ Liquid surface tension N/m τ Non-Dimensional time = / -- Oscillation frequency of an un-damped system 1/s xiii
Chapter 1 Introduction 1.1 Motivation The post-impact spread, recoil and shape oscillations of a spray droplet on a solid surface forms a critical part of many engineering processes such as spray cooling, near net shape manufacturing, spray cooling, ink-jet printing, spray painting and crop spraying (Yarin, 2006). In these applications the droplet Weber number is kept low to avoid liquid splashing and droplet rebound. When a liquid droplet impinges on a horizontal dry surface at low Weber number (We < ~ 30), the liquid spreads in an axisymmetric fashion. The spread reaches a maximum when part of the initial kinetic energy is consumed in viscous dissipation and the remainder is converted to surface energy. The contact angle then changes from the advancing angle to the receding contact angle and the liquid recoils to form a liquid column. These spread-recoil oscillations continue until the viscous forces completely damp the motion. The various stages in this process are shown in Fig. 1.1 a b c d Fig. 1.1 Various stages of droplet post-impact spread-recoil dynamics: a) At the time of impact, b) Maximum spread, c) Maximum recoil, d) Equilibrium. 1
The maximum spread and the strength of the recoil is governed by the impact velocity, drop size, liquid properties (density, surface tension, and viscosity), and surface wettability. A simple model to predict the post-impact spread-recoil dynamics of a droplet is highly desirable to design, control, and improve a variety of applications where the drop post-impact spreading is encountered. There are a large number of experimental and computational investigations of the drop impact dynamics available in the literature (Yarin, 2006) beginning with Worthington (1876). In most applications, the process takes only a few milliseconds to complete. In last two decades, using high-speed cameras, detailed images of the process have been captured for a variety of liquid-substrate combinations. The drop behavior has been reported in terms of the temporal variations of instantaneous drop height (h) and spread (D) (see Fig. 1.2) or the spread factor (the ratio of instantaneous spread to drop diameter (d 0 ) prior to impact) and flatness factor (the ratio of height to drop diameter). However a simple model to predict the spread-recoil dynamics is not available in the literature. Fig. 1.2 Instantaneous spread and height during droplet spread-recoil process. 2
1.2 Scope of the present work In the present study, the post-impact spread, recoil, and shape oscillations for a droplet impacting on a dry horizontal surface at low Weber number (< ~ 30) are modeled as a second order damped harmonic system. Droplets of six different Newtonian liquids (acetic anhydride, 4:3-aqueous glycerin (a solution with 4 parts water and 3 parts glycerin by volume), ethylene glycol, glycerin, propylene glycol, and water) impacting on hydrophobic (Teflon) and hydrophilic (glass) substrates are considered. Using experimental data for the six Newtonian liquids that encompass a large range of viscosities and surface tension coefficients, semiempirical correlations are developed to characterize the drop post-impact behavior using a damped harmonic system model. The correlations for the spring constant and the damper constant as a function of drop Weber and Reynolds numbers are presented for a hydrophobic substrate that can predict the transient spread and flatness variations. Also, a detailed analysis of spread and flatness factor variations on a hydrophilic substrate (glass) is carried out. In this case, the behavior did not conform to a simple damped mass-spring-damper system with a linear spring. 3
Chapter 2 Literature Review The post-impact spread, recoil, and shape oscillations of a spray droplet on a solid surface are encountered in a variety of engineering applications. These include spray coating, near net-shape manufacturing, spray cooling, ink-jet printing, spray painting, and crop spraying, among others. Not surprisingly, a large number of experimental, analytical and computational investigations of the drop impact dynamics available in the literature (Yarin, 2006; Gatne et al. 2009). The first experimental study was carried out by Worthington (1876). In spite of the rudimentary imaging techniques of the time, he was able to record drop shapes during post-impact spreading of milk and mercury drops on glass. In the last three decades, using high-speed cameras, detailed images of the process have been captured for a variety of liquid-substrate combinations (Asai 1993, Chandra & Avedisian 1991, Mao et al. 1997, Sikalo et al. 2005, Gatne et al. 2009, among others). The drop behavior has been reported in terms of the temporal variations of instantaneous drop spread (D) and height (h); or in dimensionless quantities, the spread factor (the ratio of instantaneous spread to drop diameter prior to impact, D/d 0 ) and the flatness factor (the ratio of instantaneous height to drop diameter, h/d 0 ). Six different outcomes of drop impact process (deposition, prompt or corona splash, receding breakup, partial or complete rebound) have been cataloged by Rioboo et al. (2001). The qualitative effects of the impact velocity, drop size, liquid properties (densityρ, surface tensionσ, and viscosityμ), and surface wettability on the spread outcomes have been well documented (Rioboo et al. 2001; Yarin, 2006). In terms of dimensionless quantities, the outcome of drop impact process have been characterized by Weber 2 number ( We = ρv d 0 σ ), Reynolds number ( Re = ρvd μ ), Ohnesorge number ( 4
Oh = We Re = μ ρσ d 0 ), and the contact angle as a measure of the surface wettability. Deposition (i.e. spread recoil oscillations without any splash, breakup, or rebound) is the most desired drop behavior in many spray applications and it is the focus of the present investigation. During deposition, a liquid droplet impinging on a horizontal dry surface at low Weber number, depending on the liquid viscosity, spreads as a thin circular disk with a thick outer rim. The spread reaches a maximum when part of the initial kinetic energy is consumed in viscous dissipation and the remainder is converted to surface energy. Then the contact angle undergoes hysteresis as it changes from the advancing angle to the receding contact angle. During contact angle hysteresis, the drop shape changes, however there is no motion of the contact line and the spread remains essentially the same (Dechelette et al. 2010). Then the liquid recoils to form a liquid column. These spread-recoil oscillations continue until the viscous forces completely damp the motion. Mao et al. (1997), Gatne (2006), Ravi (2011), among others, provide measurements of transient droplet spread and height on hydrophilic (glass) and hydrophobic (Teflon and paraffin wax) substrate with a number of liquids that cover a wide range of viscosity and surface tension. Their measurements show that at Weber number below about 30, the droplet height tends to oscillate about their equilibrium value and resemble the response of a massspring-damper system. However, at We > ~ 30, the drop spread is large and drop height no longer oscillates about the equilibrium position. The behavior at higher Weber numbers does not resemble a second-order damped harmonic motion. By estimating initial kinetic energy, viscous dissipation and surface tension, and employing the principle of energy conservation at the point of maximum spread, several correlations have 5
been proposed to predict the maximum spread of the droplet (Jones, 1971; Chandra and Avedisian, 1991; Asai et al., 1993; Pandideh-Fard et al., 1995; Mao et al., 1997; Ukiwe & Kwok, 2005; Ravi 2011). However, there is no unanimity in their predictions over a large range of Weber numbers and for different liquid-substrate combinations. Roisman et al. (2002) developed a model for drop spread and receding by considering the drop as a thin circular disk with a think outer rim. The model predictions agreed with their experimental measurements during spreading. However the model is unable to predict drop shape oscillations. Computational modeling of the process is challenging as it requires accurate tracking and prediction of the continuously deforming gas-liquid interface. Although, successful simulations for a single drop impact have been carried out using the Arbitrary-Lagrangian-Eulerian (Fukai et al. 1995; Ganesan & Tobiska 2006), Volume-of-Fluid (Gunjal et al. 2005; Sikalo et al. 2005; Sanjeev et al. 2008; Sanjeev 2008) and Level-Set (Ding & Spelt, 2007) methods, these simulations tend to be computationally very intensive. Furthermore, the process of spray atomization in practical applications gives rise to non-uniform droplet diameters and velocities which makes it very difficult to use the single droplet simulations for process control. As such a simple model that predicts the dynamics of the drop surface interactions would be desirable. Such a model is developed in this thesis by considering the process of drop spread and recoil oscillations as analogous to the behavior of a simple damped harmonic system. The model is based on detailed temporal variations of the droplet spread and height that were obtained using experimental data for six different liquids that cover a large range of liquid properties (surface tension and viscosity). 6
Chapter 3 Model Development 3.1 Introduction A simple 2 nd order damped harmonic system consists of a mass, a linear spring, and a damper (see Fig. 3.1) and inertial, restoring and damping forces govern its motion. When the mass is given initial velocity, its kinetic energy is used to do work against the linear spring and the viscous damper. The displacement of the mass follows a damped sinusoidal response and the system attains its equilibrium position once the initial kinetic energy is completely dissipated via viscous damping. As shown in Figure 3.1, similar forces are responsible for spread-recoil oscillations of an impinging droplet. The drop inertia is analogous to mass, the surface tension force acts as the spring, and viscous force provides the damping. Fig. 3.1 A schematic showing similarities in a simple harmonic (mass-spring-damper) system and the droplet-substrate system 7
It is hypothesized that the spread-recoil dynamics of an impinging droplet, characterized in terms of temporal variations of drop spread and centerline height, can be modeled as a damped harmonic system. The typical response of a second-order damped harmonic motion and possible behaviors of the droplet-substrate system are presented in the Figure 3.2 below. Figure 3.2 (1.A) shows typical behavior of an under-damped system where its response is sinusoidal and the amplitude decays exponentially with time. A similar behavior is often observed in a droplet-substrate system in its variation of the centerline height (Gatne, 2006; Ravi, 2011). Depending on the physical properties of the liquid, oscillations tend to fall in two types as shown in Figures 3.2 (1.B) and (1.C). If the viscosity of the drop is high, drop height variation tends to decay exponentially and the droplet attains its equilibrium shape when droplet spread reaches its equilibrium value. This behavior is depicted in part 1.B. However, if the viscosity of the liquid is low, then the drop continues to undergo shape oscillations after the drop spread has reached its equilibrium value. The viscous dissipation during shape change without any change in the liquidsolid contact is very low. In this case, the liquid shape tends to oscillates for a long time with only a small change in amplitude in each oscillation. This behavior is shown as region 2 of Figure 3.2 (1.C). Low viscosity liquids, for example, acetic anhydride and water, tend to exhibit this type of behavior. The damping factor in region 2 is much smaller than that in region 1. We have modeled region 1 where the drop height variation is restricted to the time period that corresponds to a change in the spread factor. 8
In Figure 3.2 (2.A), a typical over-damped system behavior is depicted. Figure 3.2 (2.B) and (2.C) show two possible types of droplet-substrate spread variations. Low viscosity liquids often exhibit temporal spread variations of the type shown in Fig. 3.2 (2.B) on hydrophilic substrates where the droplet attains its equilibrium spread with very little retraction. Figure 3.2(2.C) represents a behavior which is often observed on hydrophilic substrates with very viscous liquids (Gatne, 2006; Ravi 2011). In this case the droplet reaches the maximum spread and then undergoes a very slow retraction. The variation in region 2 does not resemble a simple damped harmonic system behavior. As such we have modeled only region 1 as a mass-spring -damper system in this study. 9
Typical damped harmonic system response Possible behaviors of a droplet-substrate system (1B) (1A) (1C) (2B) (2A) Fig. 3.2 Comparison of damped harmonic motion responses with droplet substrate dynamics. (2C) 10
3.2 Damped Harmonic System Model The differential equation of the motion of the mass in a mass-spring-damper system which is disturbed with an initial displacement and initial velocity is given by (Pain, 1993): my && + cy& + ky = 0 (3.1) Here m is mass, c is damping coefficient, and k is the spring constant, y is the extension of the spring and the dots indicate time derivatives. Two initial conditions are required to solve this 2 nd order equation. These are initial displacement [ y (0) ] and the initial velocity [ y& (0) ]. The general solution of Eqn. (1) is: αt 2 ( ) ( cosγ sinγ ) yt = e A t+ B t (3.2) Where α = cm, γ = ω 2 ( α 2) 2, km 1 γ α 2 ω =, A= y( 0), and B = y & ( 0) + y ( 0) The four quantities viscous damping factor α, frequency of an un-damped system ω, initial displacement [ y (0) ] and the initial velocity [ y& (0) ]) uniquely define the transient variation of the displacement of the mass in a mass-spring-damper system. The damping coefficient determines the decay in the amplitude whereas both spring constant and the damping coefficient govern the frequency of oscillations. Now the task is to relate α, ω, y (0), and y& (0) to liquid properties and initial conditions of the droplet substrate system. As shown in Fig. 3.1, viscous and interfacial properties govern the evolution of the drop shape. Work done by the inertia force against viscous force is responsible for energy dissipation 11
in the system and it results in damping the drop shape oscillations. The ratio of the inertia force to viscous force is given by the Reynolds number. Therefore the viscous damping coefficient α is correlated as a function of the Reynolds number. α = f 1 ( Re ) (3.3) When the droplet spreads on the substrate, the liquid-air interface is stretched and part of the initial kinetic energy is used to do work against the action of the surface tension force. Not surprisingly, if we compare two liquids with nearly the same viscosity, for example, water and acetic anhydride, the liquid with low surface tension coefficient (Acetic Anhydride) produces larger spread compared to high surface tension fluid (water). The surface tension force acts as the spring force in the damped harmonic system by arresting the spread and by initiating recoil of the liquid to its original shape. Once again, similar to the mass-spring-damper system, the shape change goes beyond the equilibrium position by forming a liquid column which increases the potential energy of the liquid. The shape oscillations continue until the motion is completely damped by viscosity. Therefore, the surface tension coefficient affects the frequency of the shape oscillations. The fluid viscosity acts to slow the fluid motion and it affects the frequency of shape oscillations. The ratio of inertia to surface tension force and the ratio of inertia force to viscous force are expressed in dimensionless form in terms of the Weber number and the Reynolds number, respectively. Therefore it is expected that the frequency of oscillation will be a function of both the Weber number and the Reynolds number. ω = f 2 ( We, Re) (3.4) 12
The initial velocity can be expressed in terms of the impact velocity. 3.3 Analysis of Experimental Data To develop the damped harmonic system model, experimental data for six different Newtonian liquids are considered. The liquids are chosen to cover a large range of viscosity and surface tension values as shown in the table below. Equilibrium contact angle values are obtained from Ravi (2011). He measured the equilibrium contact angle values on glass surface using capillary rise technique and on Teflon surface from the images of sessile drop. Table 3.1 Physical properties of the pure liquids. Properties θ eq [deg] θ eq [deg] ρ [kg/m³] σ [m N/m] µ [m Pa. s] λ [mm] Liquid on Teflon on Glass Water 1000.0 72.70 1.000 2.722 121 23.7 Acetic Anhydride 1077.0 31.98 0.806 1.740 74 18.3 Ethylene Glycol 1115.3 48.00 16.000 2.095 90 22.7 Aqueous-Glycerin (4:3) 1180.0 66.90 22.970 2.404 107 47 Propylene Glycol 1033.0 36.00 52.000 1.885 97 20.5 Glycerin 1257.0 65.16 749.000 2.299 120 120 The experimental data for droplet spread-recoil dynamics from Ravi (2011) are used. A schematic of his experimental setup is shown in the Fig. 3.3. The droplet impact dynamics was captured using high speed camera Hi Dcam-II (NAC image technology) at 2000 frames per second. Droplets were generated from a precision syringe that had provision for interchangeable needles to alter the drop size. The impact velocity of the droplet was changed by moving the syringe vertically on a stand. About 4000 images were obtained for one droplet from impact to reaching final equilibrium shape. He repeated each experiment three times to insure repeatability. 13
To analyze the large number of images obtained from Ravi (2011), a Matlab based computer code was developed. This computer code for image analysis is given in Appendix A and it provides the instantaneous drop spread and height from each frame. Fig. 3.3 Schematic diagram of the experimental setup (Ravi, 2011). The experimental data reported in Ravi (2011) was obtained with the high speed camera aligned horizontally with the substrate. As such each frame showed only the frontal view with zero degrees angle of elevation. A typical image obtained with the high speed camera during drop spreading is shown in Fig. 3.4. For a short duration during the spreading process, low viscosity liquid drop attains a shape of thin disc surrounded by a thick rim. The height of the liquid at the centerline becomes less than the thickness of the rim. As such only the rim thickness can be obtained. During this process, the experimental data for liquid thickness are not expected to match the predictions of the developed model. This limitation is not pertinent to glycerin and 14
propylene glycol droplets. The high viscosity liquids do not tend to form a thick rim while they spread on a horizontal substrate at We < ~ 30. Fig. 3.4 Droplet image showing indent and formation of a thick rim during drop spreading (Ravi, 2011) 3.4 Spread Factor Variation In dimensionless form, the drop spread can be expressed as the spread factor β = D/d 0. For mass-spring-damper system, as t, the amplitude goes to zero when the system attains the final equilibrium position. However, in case of a liquid droplet, as t, the liquid attains the shape of a sessile drop. This is the final equilibrium position. The spread factor is equal to 0 at t = 0 and reaches the spread factor of a sessile drop as t. Therefore general form of Eqn. (1) needs to be modified as follows: eq ( 1 1 1 1 ) α1t 2 β β = e A cosγ t+ B sinγ t (3.5) where A 1 1 α1 = β eq, and B1 = & β ( 0) + A1 γ 2 1 15
Note that when droplet reaches maximum spread during each spread-recoil oscillation, contact angle hysteresis is observed. The advancing contact angle changes to a lower receding contact angle. The shape of the drop and consequently the drop flatness factor changes in this process however the spread factor remains the same during contact angle hysteresis. As such the frequency of oscillations for the spread factor is expected to be slightly different from that for the flatness factor. When the droplet spreads in an axi-symmetric manner, the rate of increase in spread is twice the contact line velocity. Therefore, the initial rate of change of spread is specified as twice the impact velocity. In the model developed here, experimental values of equilibrium spread were used. However, based on the shape of the sessile droplet it is possible to evaluate the equilibrium spread factor as follows: V cap π d = 3 0 6 If equilibrium contact angle, θ < 90 o then 3V cap Deq = 2sinθ 2 π (1 cos θ )(1+ sin θ 2 cos θ ) 13 If θ = 90 o then D eq 12V cap = π 13 If θ > 90 o then D eq 3V cap = 2sinθ 2 π cos θ(2 + sin θ) 13 D and β eq = d eq 0 16
3.5 Flatness Factor Variation In dimensionless form the drop centerline height can be expressed as the flatness factor δ = h/d 0. The droplet centerline height varies from the initial drop diameter to the final height of a sessile drop. Therefore the dimensionless flatness factor starts at 1 and goes to the value of sessile drop at equilibrium. To accommodate the non-zero value of flatness factor as t the general form of the equation is changed to: eq ( 2 2 2 2 ) α2t 2 δ δ = e A cosγ t+ B sinγ t (3.6) 2 where A = δ ( ) δ, and B = & δ ( 0) + 2 0 eq 1 α A 2 2 γ 2 2 Furthermore, in case of water droplet, the height after the first recoil can be greater than the initial drop diameter or flatness factor greater than 1. It was found that this behavior can be correlated in terms of harmonic motion if we consider the height variation from the point of the maximum spread. Therefore the model for simple harmonic motion is considered to apply from the point of maximum spread or minimum drop centerline height. Because analytical models of the initial spread are available in the literature (Roisman et al. 2002, as well as many correlations of the maximum spread factor), it is possible to calculate the height of the drop at maximum spread apriori. The time in Eqn. 3.5 indicates time from the maximum spread or from minimum drop height. As the droplet reaches its maximum spread, the liquid velocity goes to zero. The shape recoil begins with a low velocity. The rate of change of flatness factor at the beginning of recoil is 17
considered as 5% of the initial impact velocity. This small value of initial rate of change provided improved agreement with the experimental data compared to a value of zero. As the shape of the drop begins to change at the onset of recoil the contact angle decreases. Only after the contact angle becomes the receding contact angle, the contact line begins to move. There is variation in the liquid velocity across the liquid cross-section. The small but non-zero initial velocity may be a result of this velocity variation. We have used available experimental data for the droplet equilibrium height and minimum height to obtain our correlations. These values can also be calculated from considering sessile droplet shape as a truncated sphere and obtain the equilibrium droplet height using the static contact angle as follows: The volume of the droplet prior to impact, based on droplet diameter of d 0 V cap π d = 3 0 6 If equilibrium contact angle, θ < 90 o then h eq 2 3 Vcap (1 cos θ ) = 2 π(1+ sin θ cos θ) 13 If θ = 90 o then h eq 3V cap = 2π 13 If θ > 90 o then h eq 2 3 Vcap (cosθ 1) = 2 π cos θ(4 + 2sin θ) 13 18
h and δ eq = d eq 0 When the droplet diameter is much larger than the capillary length, the above method of evaluating h eq may introduce some error because the droplet equilibrium shape may deviate from section of a sphere due to its weight. This is illustrated in Fig. 3.5 below. Fig. 3.5 Figure showing the equilibrium shape of a large droplet (Ravi, 2011). Height at maximum spread of the droplet is calculated by considering it as circular disc with maximum spread D max and height h min. The maximum spread can be obtained from correlations available in the literature. h 2d 3 0 min = 2 3Dmax 19
Chapter 4 Results and Discussion Experimental data for transient spread and height variations for six different liquids from Ravi (2011) was used for developing the simple damped harmonic system model. The physical properties of the liquids are given in Table 3.1. By changing droplet velocity and size, ten cases were considered for the six liquids as shown in Table 4.1. Table 4.1 Experimental data for droplet impact on a Teflon substrate (Ravi, 2011) Liquid d 0 [mm] D max [mm] V 0 [m/s] We Re Oh*10 3 Based on d 0 Oh*10 3 Based on λ Water (I) 3.087 6.256 0.706 21.16 2179.37 2.11 2.25 Water (II) 3.095 4.799 0.499 10.61 1545.29 2.11 2.25 Water (III) 2.108 3.359 0.558 9.03 1176.60 2.55 2.25 Acetic Anhydride 2.295 5.814 0.558 24.07 1711.37 2.87 3.29 Ethylene Glycol (I) 2.659 4.624 0.558 19.24 103.44 42.41 47.78 Ethylene Glycol (II) 1.624 2.929 0.499 9.40 56.51 54.26 47.78 Aqueous-Glycerin (4:3) 2.883 4.726 0.611 19.00 90.54 48.15 52.72 Propylene Glycol (I) 2.306 4.284 0.499 16.49 22.87 177.55 196.41 Propylene Glycol (II) 1.669 2.660 0.499 11.93 16.55 208.72 196.41 Glycerin 2.781 3.570 0.660 23.40 3.08 1569.22 1726.15 20
4.1 Predictions of transient flatness factor variations on a Teflon substrate When a water droplet impacts on a Teflon substrate at Weber number below about 20, it experiences strong recoil that leads to the formation of a liquid column. At higher Weber numbers, the column breaks to form one or more smaller droplets that fall back to the substrate (Gatne et al., 2007). For We < ~20, the water column height exceeds the diameter of the droplet or the flatness factor becomes greater than 1. The experimental data for transient flatness factor variation for water shows that the behavior after the initial spread follows a damped harmonic motion. Therefore we have considered the variation of the flatness factor from its minimum value or at the beginning of the first recoil. There is considerable work on analytical models of the initial spread in the literature (see for example, Roisman et al. 2002) and many correlations are available to predict the maximum spread. As such it is possible to calculate the height of the drop at maximum spread apriori. A Matlab based code (Appendix B) was developed to calculate a damped harmonic system response to obtain values of frequency of oscillation and the viscous damping factor for each case by minimizing the absolute error between experimental data and predicted response for first three cycles. The frequency can be related to the time difference between successive peaks and the damping factor is related to the ratio of the amplitude between successive peaks. Regression analysis was carried out to correlate the damping factor with Re and the oscillation frequency with Re and We. The correlations are: 21
For flatness factor: ( ) ( ) ω2 = exp 1.44 + 1.173 Re We + 5.78 Re We 0.25 0.5 2 13 23 ( ) α = 1 6.66 + 5.29 Re 0.24 Re δ (0) h d = min 0 (4.1) & δ (0) 0.05V d = 0 0 The values of minimum height (h min ), initial droplet diameter (d 0 ), and the initial velocity along with values of frequency and damping factor are given in Table 4.2. It can be seen that the damping factor increases as Re 1/3 decreases whereas the frequency increases with increasing (Re We) -1/4. Using these parameters the correlations given in Eqn. 4.1 were obtained by regression analysis. As described earlier, the variation of flatness factor can be obtained using ( 2 2 2 2 ) α2t 2 eq = e A cos t+ B sin t where ( ) δ δ γ γ A δ δ 2 =, and B = δ ( 0) + 2 0 eq 1 & α A 2 2 γ 2 2 The transient variations of flatness factor obtained from the above correlations are plotted to compare with the experimental observations for three sizes of water droplets in Figs. 4.1, 4.2, and 4.3. It is seen that the predicted flatness factor variation agrees well with experimental data for first three oscillations (dimensionless time of about 20). Beyond this the predictions underestimate the maximum amplitude of oscillations. However the frequency of oscillations is predicted well for the entire time. Of the liquids considered here, water has the highest surface tension and low viscosity. This combination of properties results in a strong recoil of a water drop after the initial spread. After about three oscillations, the contact line no longer moves and 22
the further oscillations indicate periodic deformation of the liquid-air interface without a change in the liquid-solid contact area. The viscous dissipation and damping is low during this phase. A mass-spring-damper system has a fixed damping coefficient and a prediction based on a fixed value of damping coefficient tends to underestimate the oscillation amplitude. The behavior shown in Fig. 4.3 corresponds to a small droplet where the drop diameter is smaller than the capillary length. The first recoil is under predicted in this case. Table 4.2 Harmonic oscillation variables for predicting the flatness factor on a Teflon substrate (Re We) -1/4 Re 1/3 ω 2 [1/s] α 2 [1/s] h(0) [mm] h eq [mm] Water (I) 0.06824 12.9648 0.26367 0.046333 0.496 2.102 Water (II) 0.08838 11.5603 0.25552 0.044502 0.860 2.210 Water (III) 0.09849 10.5566 0.28128 0.044566 0.640 1.560 Acetic Anhydride 0.07177 11.7853 0.26552 0.044743 0.397 0.708 Ethylene Glycol (I) 0.14971 4.69424 0.32147 0.077617 0.8584 1.425 Ethylene Glycol (II) 0.24777 3.5009 0.45176 0.112120 0.570 1.020 Aqueous-Glycerin (4:3) 0.16624 4.3560 0.33782 0.084538 1.060 1.770 Propylene Glycol (I) 0.21769 2.9181 0.40220 0.148499 0.538 1.126 Propylene Glycol (II) 0.26675 2.5484 0.48877 0.190017 0.480 0.890 Glycerin 0.34317 1.4552 0.69985 1.885135 1.400 1.800 23
Fig. 4.1 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Water droplet, d 0 = 3.087 mm, We = 21.16, Re = 2179.36 24
Fig. 4.2 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Water droplet, d 0 = 3.095 mm, We = 10.61, Re = 1545.28 25
Fig. 4.3 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Water droplet, d 0 = 2.108 mm, We = 9.03, Re = 1176.59 26
Fig. 4.4 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Acetic Anhydride droplet, d 0 = 2.295 mm, We = 24.07, Re = 1711.37 27
Figure 4.4 shows a comparison of predictions of flatness factor variation with experimental data for acetic anhydride droplet. Acetic anhydride has the lowest viscosity of all the liquids considered in this study. Once again the first two oscillations are predicted very well. The frequency appears to change beyond dimensionless time of about 10 when the drop spread reaches its equilibrium value. At about dimensionless time of 10, the spread factor attains its equilibrium value and the liquid-solid contact line does not move thereafter. Hence the change in flatness factor beyond τ ~ 10 reflects droplet height change during shape oscillations without any change in the spread factor. The viscous dissipation is low during this process compared to spread-recoil prior to τ ~ 10. Both viscous effects and surface tension force determine the frequency of oscillation of a damped harmonic system. As such the change in frequency of oscillation beyond dimensionless time of about 10 is due to the change in viscous effects. Figures 4.5 to 4.9 compare the experimental measurements of flatness factor with predicted values using Equation 4.1 for aqueous glycerin (4:3), propylene glycol, and ethylene glycol. These liquids are highly viscous compared to water. The importance of viscosity relative to surface tension can be expressed as the Ohnesorge number (Oh) and the values of Ohnesorge number for these liquids are one or two-orders of magnitude higher than that for water as seen in Table 4.1. The high viscosity tends to damp the spread-recoil oscillations and their behavior as predicted by the damped harmonic system model agrees very well with experimental measurements. The Ohnesorge number for the glycerin droplet is about 800 times that of water droplet and as seen in Figure 4.10, the glycerin droplet undergoes only one recoil after the initial spread. The high viscosity of glycerin quickly damps its motion. Once again, the damped 28
harmonic system based model is able to accurately predict the flatness factor response of this high viscosity fluid droplet. Fig. 4.5 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Aqueous-Glycerin droplet, d 0 = 2.883 mm, We = 19.00, Re = 90.53 29
Fig. 4.6 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Propylene Glycol droplet, d 0 = 2.306 mm, We = 16.49, Re = 22.87 30
Fig. 4.7 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Propylene Glycol droplet, d 0 = 1.669 mm, We = 11.93, Re = 16.55 31
Fig. 4.8 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Ethylene Glycol droplet, d 0 = 2.659 mm, We = 19.24, Re = 103.43 32
Fig. 4.9 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Ethylene Glycol droplet, d 0 = 1.624 mm, We = 9.40, Re = 56.51 33
Fig. 4.10 Experimental and predicted transient flattening factor variation, Substrate: Teflon, Glycerin droplet, d 0 = 2.781 mm, We = 23.39, Re = 3.08 34
4.2 Predictions of transient spread variation on a Teflon substrate Similar to the procedure described in section 4.1, the experimental data for the six liquids are used to develop correlations for the damping factor and oscillation frequency for predicting the spread factor variations on a Teflon substrate. When a droplet impacts on a substrate its spread factor increases from zero to a maximum value. The low Ohnesorge number droplets tend to undergo several spread and retraction oscillations with a large change in the liquid-solid contact area during each cycle. The high Ohnesorge number liquids tend to have fewer spreadretraction oscillations and the droplet reaches its equilibrium spread in a short duration. Using experimental measurements of the temporal variations of the spread factor, the values of damping coefficient and oscillation frequency were obtained by regression analysis as: 1 1 ( ) 0.625 ω = 0.19 + 9.91 Re We α = + β (0) = 0 1 2 0.18 13.85Re 29.43Re ln(re) & β (0) 2V d = 0 0 (4.2) The damping factor and the oscillation frequency for each case are listed in Table 4.3. As described earlier, the correlations listed in Eqn. 4.2 can be used to predict the spread factor response eq ( 1 1 1 1 ) α1t 2 e A cos t B sin t β β = γ + γ 35
where A 1 1 α2 = β eq, and B1 = & β ( 0) + A1 γ 2. The equilibrium or final spreads of the droplets are given in Table 4.3 and β eq = Deq d0. 1 Table 4.3 Harmonic oscillation variables for predicting spread factor on a Teflon substrate Liquid (Re We) -1/4 Re ω 1 [1/s] α 1 [1/s] D eq [mm] Water (I) 0.06823 2179.37 0.20205 0.18631 3.60 Water (II) 0.09227 1545.29 0.21560 0.18970 3.60 Water (III) 0.09849 1176.60 0.22017 0.19162 2.17 Acetic Anhydride 0.07019 1711.37 0.20294 0.18802 4.27 Ethylene Glycol (I) 0.14971 103.44 0.27595 0.30114 3.47 Ethylene Glycol (II) 0.24777 42.91 0.49283 0.44268 2.34 Aqueous-Glycerin (4:3) 0.16624 82.66 0.30167 0.32854 3.50 Propylene Glycol (I) 0.21769 24.85 0.40912 0.58422 3.78 Propylene Glycol (II) 0.26674 16.55 0.55417 0.71529 2.63 Glycerin 0.34314 3.08 0.87356 1.18634 3.46 Of the six liquids considered here, water has low viscosity (1 mpa-s) with high surface tension (~73 mn/m). Acetic anhydride has low viscosity (0.8 mpa-s) but its surface tension is 36
low as well (~32 mn/m) and other liquid have viscosity values that are much higher than water. The spread factor variation for water droplets is shown in Figs. 4.11 to 4.13 for We = 21.16, 10.61, and 9.03, respectively. The combination of low viscosity and high surface tension of water leads to a very strong recoil followed by multiple spread-recoil oscillations. For the larger droplets (d 0 ~ 3.1 mm), the prediction of maximum spread and oscillation frequency agrees well with experimental data. For the smaller droplet (d 0 ~ 2.1 mm), the maximum spread is overpredicted and the extent of recoil is under-predicted. Figures 4.14 to 4.17 depict the temporal variation of spread factors and compare them with experimental measurements for acetic anhydride and ethylene glycol droplets. For these liquids, the droplet tends to undergo strong recoil after the initial spread and the drop reaches a value of the spread factor that is close to its equilibrium value. The damped harmonic system model predicts this spread factor behavior and the strong agreement between experimental measurements and predictions is evident from the figures. The spread factor variation with time for highly viscous propylene glycol and glycerin are shown in Figures 4.18 to 4.20. The droplets are seen to spread and reach its equilibrium position and large spread-recoil oscillations are not observed. With extremely high viscosity of propylene glycol and glycerin, such behavior is not surprising. The predictions based on mass-springdamper damped harmonic system (Equation 4.2) agree well with experimental measurements. 37
Fig. 4.11 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Water droplet, d 0 = 3.087 mm, We = 21.16, Re = 2179.36 38
Fig. 4.12 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Water droplet, d 0 = 3.095 mm, We = 10.61, Re = 1545.289 39
Fig. 4.13 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Water droplet, d 0 = 2.108 mm, We = 9.03, Re = 1176.59 40
Fig. 4.14 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Acetic Anhydride droplet, d 0 = 2.295 mm, We = 24.07, Re = 1711.37 41
Fig. 4.15 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Aqueous-Glycerin droplet, d 0 = 2.883 mm, We = 19.00, Re = 90.53 42
Fig. 4.16 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Ethylene Glycol droplet, d 0 = 2.659 mm, We = 19.24, Re = 103.43 43
Fig. 4.17 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Ethylene Glycol droplet, d 0 = 1.624 mm, We = 9.40, Re = 56.51 44
Fig. 4.18 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Propylene Glycol droplet, d 0 = 2.306 mm, We = 16.49, Re = 22.87 45
Fig. 4.19 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Propylene Glycol droplet, d 0 = 1.669 mm, We = 11.93, Re = 16.55 46
Fig. 4.20 Experimental and predicted transient spreading factor variation, Substrate: Teflon, Glycerin droplet, d 0 = 2.781 mm, We = 23.39, Re = 3.08 47
4.3 Transient flatness factor variations on a glass substrate: Transient flatness factor variations for five liquids (water, aqueous glycerin (4:3), ethylene glycol, propylene glycol, and glycerin) from Ravi (2011) were used to investigate the drop spread dynamics on a hydrophilic substrate (glass). Acetic anhydride was not considered because its height became very small as it spread easily on glass and the relative uncertainty in obtaining the liquid height using the Matlab-based tool increased. Table 4.4 provides the velocity and diameter values prior to impact for all the cases considered. For each case the flatness factor variation was modeled as damped harmonic motion and the relevant damping factor and frequency are listed in Table 4.5 Table: 4.4 Experimental data for droplet impact on a glass substrate (Ravi, 2011) d 0 [mm] D max [mm] V 0 [m/s] We Re Oh*10 3 Based on d 0 Oh*10 3 Based on λ Water (I) 3.054 7.548 0.706 20.93 2155.84 2.12 2.25 Water (II) 2.425 6.112 0.558 10.39 1353.19 2.38 2.25 Ethylene Glycol 2.698 5.202 0.558 19.52 104.95 42.10 47.78 Aqueous-Glycerin (4:3) 2.767 4.585 0.611 18.24 86.90 49.15 52.73 Propylene Glycol 2.447 4.284 0.499 17.49 24.26 172.39 196.41 Glycerin 2.809 3.536 0.660 23.63 3.11 1561.50 1726.15 48
Table 4.5 Harmonic oscillation variables for predicting flatness factor variations on a glass substrate Liquid (Re We) -1/4 Re 1/3 ω 2 α 2 h(0) [mm] h eq [mm] Water (I) 0.06860 12.9183 0.24867 0.03000 0.2858 0.93171 Water (II) 0.09183 11.0608 0.26800 0.03500 0.2258 0.57171 Ethylene Glycol 0.14862 4.7170 0.30287 0.04345 0.5748 0.92071 Aqueous-Glycerin (4:3) 0.15848 4.4293 0.31867 0.04799 0.8641 1.42007 Propylene Glycol 0.22031 2.8949 0.34867 0.69950 0.6842 0.94008 Glycerin 0.34146 1.4601 0.41866 0.70499 1.3861 1.73201 Experimental measurements for the temporal variations of flatness factor during spread-recoil dynamics on a glass substrate are shown in Figs. 4.21 to 4.26. The best fit for a damped harmonic mass-spring-damper motion are shown as a solid red line in each figure. It is seen that the change in surface wettability significantly alters the spread-recoil dynamics. The minimum flatness factors on glass are lower than that on Teflon. This corresponds to the larger maximum spread of the droplet on glass compared to that on Teflon. The recoil is weaker and leads to a lower value of the maximum flatness factor compared to Teflon. Furthermore, the variation of flatness factor is not symmetric about the equilibrium value. A damped harmonic system model is only able to predict the behavior of a glycerin droplet. Due to high viscosity of glycerin the 49
droplet its motion is highly damped and it does not undergo flatness factor oscillations but attains its equilibrium height as it spreads on the glass substrate. Fig. 4.21 Experimental and predicted transient spreading factor variation, Substrate: Glass, Ethylene Glycol droplet, d 0 = 2.697 mm, We = 19.52, Re = 104.95 50
Fig. 4.22 Experimental and predicted transient spreading factor variation, Substrate: Glass, Glycerin droplet, d 0 = 2.809 mm, We = 23.62, Re = 3.11 51
Fig. 4.23 Experimental and predicted transient spreading factor variation, Substrate: Glass, Water droplet, d 0 = 3.053 mm, We = 20.93, Re = 2155.94 52
Fig. 4.24 Experimental and predicted transient spreading factor variation, Substrate: Glass, Propylene Glycol droplet, d 0 = 2.446 mm, We = 17.49, Re = 24.26 53
Fig. 4.25 Experimental and predicted transient spreading factor variation, Substrate: Glass, Water droplet, d 0 = 2.424 mm, We = 10.38, Re = 1353.19 54
Chapter 5 Conclusions and Recommendations for Future Work 5.1 Conclusions A second-order damped harmonic system model was developed to predict the temporal variations of droplet dimensionless spread (spread factor) and dimensionless centerline height (flatness factor) during post-impact spread-recoil oscillations of a liquid droplet impacting on a solid substrate. Correlations for the damping factor and oscillation frequency were obtained using regression analysis of experimental measurements of transient droplet spread and height available in Ravi (2011). A total of six different liquids were considered for this analysis which covered a large range of liquid properties (viscosity and surface tension). High speed photographic images for ten cases with different combinations of droplet sizes and impact velocities were analyzed. A computational tool was developed in Matlab to process the large number of images (4000 for each case) to determine the drop spread and maximum height in each photograph. It is seen that the droplet spread factor and flatness factor oscillate about their equilibrium position on a hydrophobic surface at We < ~ 30 and the variations resembles the behavior of a mass-linear-spring-damper system. Initial spread or height, initial velocity, damping factor, and oscillation frequency uniquely define a response of a 2 nd order damped harmonic system. Correlations for the damping factor and the frequency are provided for drop spread-recoil oscillations on Teflon. The spread and flattening factor variations predicted by these correlations agree well with experimental data for the hydrophobic substrate. For We < ~ 30, the flatness 55
factor variation on a Teflon substrate follows the damped harmonic system response as: ( 1 1 1 1 ) α1t 2 eq = e A cos t+ B sin t where A 1 β eq β β γ γ 1 α B = & β 0 + A =, and ( ) 1 1 1 γ1 2 and ω 1 = exp 1.44 + 1.173( Re We) + 5.78( Re We) 0.25 0.5, 13 23 ( ) α 1 = 1 6.66 + 5.29 Re 0.24 Re, β (0) = hmin d0, and & β (0) = 0.05V0 d0. The spread factor variation is given by ( 2 2 2 2 ) α2t 2 eq = e A cos t+ B sin t where ( ) δ δ γ γ A δ δ 2 =, B = δ ( 0) + 2 0 eq 1 α & A, 2 2 γ 2 2 ( ) 0.625 1 2 ω 2 = 0.19 + 9.91 Re We, α 2 = 0.18 + 13.85Re 29.43Re ln(re), δ (0) = 0 and & δ (0) = 2V d. 0 0 On a hydrophilic surface, droplet spread is larger and recoil is weaker compared to that on a hydrophobic surface and droplet height oscillations are not symmetrical about the equilibrium position. When the drop height variation on a glass substrate is compared with a damped harmonic behavior of a mass-spring-damper system with a linear spring, the maximum height is under predicted for each oscillation. 56
5.2 Recommendations for future work The present study provides a simple damped harmonic system based model to predict the transient droplet dynamics in terms of droplet Weber and Reynolds numbers that can be determined from physical properties of the liquid, droplet diameter and impact velocity. The model agrees well with experimental data for drop spread and height on Teflon. The recommendations to improve the model are: 1. The spread-recoil process on a hydrophilic substrate produces larger spreads and weaker recoils compared to those on hydrophobic substrates. The droplet centerline height and spread oscillations are not symmetric about the equilibrium position. A mass-spring-damper system with a non-linear spring may be suitable to model this behavior. 2. The surface wettability should be used to correlate the non-linearity of the spring constant. This would require consideration of the adhesion force due to the substrateliquid contact. 3. Experimental data for spread and height during post-impact spread-recoil dynamics should be obtained for a variety of surfaces with different surface wettability. Using these data the model presented here can be extended to incorporate surface wettability. 57
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APPENDIX A Matlab code for measuring droplet spread (D) and flatness (h) on the substrate: Droplet diameter before impact, transient spread and flatness measurements over the solid substrate are automated using matlab code. An image containing the known value of syringe diameter is used for the calibration purpose. Figure below shows the adopted measuring process. 61
The following code is used to measure the required values from the images: clearall closeall clc %Input the image used to Caliberate RGB = imread(uigetfile('*.jpg', 'Select JPG-file')); RGB = im2bw(rgb, graythresh(rgb)); %Select the image which is used to calibrate %Image is stored as binary values %Crop the image if necessary RGB = ~RGB; RGB = imcrop(rgb, [0 0 512 256]); %Image is restored by swapping binary values %Crop the image which is used to calibrate if necessary l = max(find(rgb(20,:)))-min(find(rgb(20,:))); %Finding number of pixels occupied by the syringe niddle imshow(rgb) %Show image to the viewer %%Image reading and giving required input parameters files = dir('*.jpg'); for k = 1:(numel(files)-1) RGB = imread(files(k).name); if k < 2 RGB = rgb2gray(rgb); RGB = im2bw(rgb,graythresh(rgb)); RGB = ~RGB; RGB = imcrop(rgb, [1 1 184 221]); %Stores number of JPEG files count in the variable files %For-loop to extract all images in the folder one after the other %Stores the image information in RGB %Image processing for the first image in the folder %Image is stored as black and white in the same variable RGB %Image is stored as binary values in the same variable RGB %Image is restored by swapping the binary values %Crop the image if any discrepencies are arised 62
RGB = imfill(rgb, 'holes'); RGB1 = ~RGB; imshow(rgb1) impixelregion %Close the holes formed because of the light focus %Store the swaped binary image in the new variable RGB1 %Show the new image stored in RGB1 %This tool is used to find the reference values to be taken for measurement x1 = input('enter left side reference pixel \n x1='); %Following four values are taken as input for reference values y1 = input('enter right side reference pixel \n y1='); x2 = input('enter left side reference pixel \n x2='); y2 = input('enter right side reference pixel \n y2='); %%Make the image suitable for the measurement dim = size(rgb); forww = x2:1:dim(2) %Size of the new cropped image is stored in the variable 'dim' %This loop makes the substrate surface smooth RGB1(y2+1,ww) = 0; RGB1(y2+2,ww) = 0; RGB1(y2+3,ww) = 0; RGB1(y2+4,ww) = 0; end for w = 1 : x1 %This look makes the substrate surface smooth RGB1(y1+1,w) = 0; RGB1(y1+2,w) = 0; RGB1(y1+3,w) = 0; RGB1(y1+4,w) = 0; end %%Evaluation of parameters required for measurement 63
for i = 1:dim(2) C2(1,i) = max(find(rgb1(:,i))); C3(1,i) = min(find(rgb(:,i))); end r1=round((y1+y2)/2); [C,Z]=min(C2); r2=min(c2); [c,z] = min(c2); r3= min(c3); %%Show horizontal lines to clearly visualize the height holdon x = [r3]; y = [z]; plot(y,x, 'b*'); holdon x = [1 dim(2)]; y = [c c]; plot(x,y,'color','r'); %Shows the horizontal line to clearly visualize the height holdon x = [1 dim(2)]; y = [r1 r1]; plot(x,y, 'Color', 'r'); %Shows the horizontal line to clearly visualize the height %%Height Measurement height(k,1) = {((r1-c)*1.19)/l }; %Calculated height value 64
text(dim(2)-150,0.5*(r1+c),['\leftarrow h =',height(k,1)], 'Color', 'red') %Displays text on the image viewer %%Spread Measurement q = z; m=0; a = 1; while a == 1 if C2(1,q) == y1 a=0; else q = q-1; m=m+1; a=1; end end c1 = m; holdon x=[q q]; y=[dim(1) 0]; plot(x, y, 'Color', 'r'); %Shows vertical line to clearly visualize the spread q = Z; m=0; a = 1; while a == 1 65
if C2(1,q) == y2 a=0; else q =q+1; m=m+1; a=1; end end c2 = m; holdon x=[q q]; y=[dim(1) 0]; plot(x, y, 'Color', 'r'); spread(k,1) = {(c1 + c2)*(1.19/l)}; %Shows vertical line to clearly visualize the spread %Calculated spread value text(z,dim(1)-30,['\uparrow d =',spread(k,1)], 'Color', 'red') else RGB = rgb2gray(rgb); %Image processing for other images %Same above process is adopted from here for all the other images RGB = im2bw(rgb,graythresh(rgb)); RGB = ~RGB; RGB = imcrop(rgb, [1 1 184 221]); RGB = imfill(rgb, 'holes'); RGB1 = ~RGB; %%Mekes the image suitable for measurement dim = size(rgb); 66
for w = 1 : x1 RGB1(y1+1,w) = 0; RGB1(y1+2,w) = 0; RGB1(y1+3,w) = 0; RGB1(y1+4,w) = 0; end forww = x2:1:dim(2) RGB1(y2+1,ww)=0; RGB1(y2+2,ww)=0; RGB1(y2+3,ww)=0; RGB1(y2+4,ww)=0; end %%Evaluation of parameters required for the measurement for i = 1:dim(2) C2(1,i) = max(find(rgb1(:,i))); C3(1,i) = min(find(rgb(:,i))); end imshow(rgb1) r1=round((y1+y2)/2); [C,Z]=min(C2); [c,z]=min(c3); r3=min(c3); r2=min(c2); %%Show Horizontal lines to visualize the height 67
holdon x = [r3]; y = [z]; plot(y,x, 'b*'); holdon x = [1 dim(2)]; y = [c c]; plot(x,y,'color','r'); holdon x = [1 dim(2)]; y = [r1 r1]; plot(x,y, 'Color', 'r'); height(k,1) = {((r1-c)*1.19)/l }; text(dim(2)-150,0.5*(r1+c),['\leftarrow h =',height(k,1)], 'Color', 'red') q = Z; m=0; a = 1; while a == 1 if C2(1,q) == y1 a=0; else q = q-1; m=m+1; a=1; 68
end end c1 = m; holdon x=[q q]; y=[dim(1) 0]; plot(x, y, 'Color', 'r'); q = Z; m=0; a = 1; while a == 1 if C2(1,q) == y2 a=0; else q =q+1; m=m+1; a=1; end end c2 = m; holdon x=[q q]; y=[dim(1) 0]; plot(x, y, 'Color', 'r'); 69
spread(k,1) = {(c1 + c2)*(1.19/l)}; text(z,dim(1)-30,['\uparrow d =',spread(k,1)], 'Color', 'red') pause(0.01) end end %Post processing Time(:,1) = 0.0005 : 0.0005 : (0.0005*(numel(files)-1)) ; %Store time values starting from 0.5 ms Title = { 'Time', 'Spread', 'Height'}; %Title bar to designate values being stored in excel sheet xlswrite('a.xls', Title, 'Sheet1', 'A1'); %Following commands allows to include headings in the excel sheet xlswrite('a.xls', Time, 'Sheet1', 'A2'); xlswrite('a.xls', spread, 'Sheet1', 'B2'); xlswrite('a.xls', height, 'Sheet1', 'C2'); Z = xlsread('a.xls'); figure, plot (Z(:,1),Z(:,3)) xlabel('time') %Restore the values from excel sheet into the variable Z %Plots height wrt time %Plot is labled with the following three commands ylabel('height') title('droplet Dynamics') figure, plot (Z(:,1),Z(:,2)) xlabel('time') %Plots spread wrt time %Plot is labled with the following three commands ylabel('spread') title('droplet Dynamics') 70
APPENDIX B Interactive tool for Analytical study: An interactive tool is designed using matlab to extract the required harmonic motion variables by observation from the available experimental data. The experimental data is made available from the matlab code which is explained in Appendix A. Below Figure shows the appearance of the interactive tool which has provisions for four input values. By varying these values, required curve can be generated whose values are used in regression analysis. 71