American Institute of Aeronautics and Astronautics 1 Preliminary Results of the Fluid Merging Viscosity Measurement Space Station Experiment Edwin C Ethridge * Marshall Space Flight Center, Huntsville, AL 35812 William Kaukler † University of Alabama in Huntsville, Huntsville, AL 35899 and Basil Antar ‡ University of Tennessee Space Institute, Tullahoma, TN 37388 During the Space Shuttle “down period” a call was put out for low upmass payloads. One of these “low up mass” International Space Station science experiments is the “Fluid Merging Viscosity Measurement”, FMVM investigation. The purpose of FMVM is to measure the rate of coalescence of two highly viscous liquid drops and correlate the results with the liquid viscosity and surface tension. The experiment takes advantage of the low gravitational free floating conditions in space to permit the unconstrained coalescence of two nearly spherical drops. The merging of the drops is accomplished by deploying them from a syringe and suspending them on 2 Nomex threads followed by the astronaut’s manipulation of one of the drops toward a stationary droplet till contact is achieved. Coalescence and merging occurs due to shape relaxation and reduction of surface energy, being resisted by the viscous drag within the liquid. The coalescence was recorded on video (ISS VTR) and some of the data was downlinked near real-time. A range of drop diameters, different liquids with differing viscosity and surface tensions should yield a large range of experimental parameters used to correlate with theory and to compare with numerical experiments. The results are important for a better understanding of the coalescence process. The experiment is also relevant to liquid phase sintering and is a potential new method for measuring the viscosity of viscous glass formers at low shear rates. I. Introduction ring the Space Shuttle “stand-down” period the Payloads Control Board put out a call to current microgravity science principal investigators for potential low upmass payloads for launch on 13P. In June 2003, the Microgravity Science & Applications Division at MSFC put forward FMVM as a candidate for this flight. These were “fast track” experiments since the payload was to be launched in about six months. The “fast track” schedule for operations required a streamlined approach to expedite payload development process. The direction was to utilize limited upmass (< 1 kg) and volume and to make maximum use of on-orbit resources. Upon its selection, the FMVM Development Team was quickly assembled to accomplish the design of the experiment, payload hardware, development of requirements, operations products, and assess the safety requirements. * Ceramics Engineer, EM40, AIAA nonmember. † Associate Professor, Department of Chemistry, nonmember. ‡ Professor, Department of Mechanical and Aerospace Engineering, AIAA Associate Fellow. D 44th AIAA Aerospace Sciences Meeting and Exhibit 9 - 12 January 2006, Reno, Nevada AIAA 2006-1142 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Transcript
American Institute of Aeronautics and Astronautics1
Preliminary Results of the Fluid Merging ViscosityMeasurement Space Station Experiment
Edwin C Ethridge *
Marshall Space Flight Center, Huntsville, AL 35812
William Kaukler †
University of Alabama in Huntsville, Huntsville, AL 35899
and
Basil Antar ‡
University of Tennessee Space Institute, Tullahoma, TN 37388
During the Space Shuttle “down period” a call was put out for low upmass payloads.One of these “low up mass” International Space Station science experiments is the “FluidMerging Viscosity Measurement”, FMVM investigation. The purpose of FMVM is tomeasure the rate of coalescence of two highly viscous liquid drops and correlate the resultswith the liquid viscosity and surface tension. The experiment takes advantage of the lowgravitational free floating conditions in space to permit the unconstrained coalescence of twonearly spherical drops. The merging of the drops is accomplished by deploying them from asyringe and suspending them on 2 Nomex threads followed by the astronaut’s manipulationof one of the drops toward a stationary droplet till contact is achieved. Coalescence andmerging occurs due to shape relaxation and reduction of surface energy, being resisted bythe viscous drag within the liquid. The coalescence was recorded on video (ISS VTR) andsome of the data was downlinked near real-time. A range of drop diameters, differentliquids with differing viscosity and surface tensions should yield a large range ofexperimental parameters used to correlate with theory and to compare with numericalexperiments. The results are important for a better understanding of the coalescenceprocess. The experiment is also relevant to liquid phase sintering and is a potential newmethod for measuring the viscosity of viscous glass formers at low shear rates.
I. Introductionring the Space Shuttle “stand-down” period the Payloads Control Board put out a call to current microgravityscience principal investigators for potential low upmass payloads for launch on 13P. In June 2003, the
Microgravity Science & Applications Division at MSFC put forward FMVM as a candidate for this flight. Thesewere “fast track” experiments since the payload was to be launched in about six months. The “fast track” schedulefor operations required a streamlined approach to expedite payload development process. The direction was toutilize limited upmass (< 1 kg) and volume and to make maximum use of on-orbit resources. Upon its selection, theFMVM Development Team was quickly assembled to accomplish the design of the experiment, payload hardware,development of requirements, operations products, and assess the safety requirements.
* Ceramics Engineer, EM40, AIAA nonmember.† Associate Professor, Department of Chemistry, nonmember.‡ Professor, Department of Mechanical and Aerospace Engineering, AIAA Associate Fellow.
D
44th AIAA Aerospace Sciences Meeting and Exhibit9 - 12 January 2006, Reno, Nevada
AIAA 2006-1142
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
American Institute of Aeronautics and Astronautics2
In response to the call for low upmass experiments, on of the authors (EE) the PI of the NRA “Mechanisms forthe Crystallization of ZBLAN” proposed the “Fluid Merging Viscosity Measurement” (FMVM) experiment.FMVM was a natural follow on experiment for examining the measurement of viscosity of highly viscous liquidsusing two drop coalescence. The main purpose of the experiment is to advance a totally new and different methodfor viscosity measurement. Currently the measurement of viscosity is limited for liquids that are susceptible tocrystallization. A method for determining the viscosity of highly viscous substances that are susceptible tocrystallization over the most important viscosity range for crystallization behavior does not exist. A glass that isfrequently used for glass crystallization studies is lithium di-silicate. The viscosity can be measured in the liquidand for a certain extent of undercooling below the melting temperature. It can also be measured around the glasstransition temperature. Figure 1 shows measured viscosity data illustrating the wide gap of 6 orders of magnitudethat cannot be measured by any method. This unmeasured region is also the most interesting temperature range forglass crystallization measurements.
Figure 1. Viscosity vs. Temperature for Lithium di-Silicate glass.
Viscosity is the only directly measurable kinetic parameter used in crystal nucleation and growth equations. Inthe classical treatment of crystallization, the nucleation rate, I, and crystal growth rate, U, are both inverselyproportional to viscosity, η, with the viscosity term appearing in the pre-exponential factor. 1
I = (kn/η) exp[-bα3βTm/T(1-Tr)2]
U = (k’n/η) [1-exp(-β(Tm-T/T)]
Where T is the absolute temperature, Tm is the melting temperature, and Tr is the reduced temperature. The kineticconstants kn and k’n, shape factor, b, and dimensionless parameters related to the liquid-crystal interfacial tension,α, and entropy of fusion, β. The fraction of glass crystallized, X, with time at a given temperature is a function ofthe rate of nucleation, the third power of the growth rate, and according to the above 2 equations the fraction of glasscrystallized is also inversely proportional to the fourth power of viscosity.
X = (π/3)(IU3t4)X ∝ 1/ η4
American Institute of Aeronautics and Astronautics3
This illustrates the importance of a new viscosity measurement method to the science of glass formation andcrystallization.
A method for measuring the viscosity of highly viscous substances that are susceptible to crystallization over themost important viscosity range is needed. A method suitable to containerless processing to avoid container walls toheterogeneous nucleation is indicated. The process should use relatively small samples for quickly getting thematerial to temperature and to quickly measure the viscosity. The relaxation of material to a sphere is one potentialmethod for determining the viscosity of very viscous liquids. The merging of two spheres of liquids to one spherewas selected as an executable experiment that can be used to verify shape relaxation models. The fluid flow isdriven by the reduction of surface energy but limited by the resistance to fluid flow by the liquid viscosity. The timeconstant of the experiment is proportional to the viscosity. The method is also interesting because it can be tested bycomputational fluid mechanics. Frenkel2 was the first to propose a model for the coalescence of two spheres.Others have numerically modeled the coalescence of two liquid spheres.3,4
FMVM was further supported by prior coalescence experiments on the low gravity KC-135 parabolic aircraftand computational fluid dynamical numerical experiments. Low gravity exploratory experiments were performedwith glycerin on KC-135 aircraft for short (10 sec) experiment times and the computational fluid dynamics programFIDAP was used for numerical modeling. It was shown that the rate of coalescence can be characterized by the rateof neck diameter growth. Two models were calculated. The 2 dimensional cylinder model was compared with a 3-D sphere model which illustrated slightly different neck diameter growth rates. Coalescing spheres grow slowerthan cylinders. The KC-135 experiments showed that the data fit the 3-D spherical coalescence model up to anormalized neck diameter of 0.5. The experiment constrained the drops keeping them from forming a sphere, thefinal shape being a cylindrical liquid bridge. Above the 0.5 normalized neck diameter, the data fit the 2-Dcylindrical model. The objective of FMVM is to examine 2 drop coalescence of viscous drops beyond thelimitations of the KC-135 experiments. The ISS provided the opportunity for much longer experiments, much largersamples, more viscous liquids, and unconstrained fluid motion in the weightless conditions of space. The goals ofthe experiment were to:
• Expand the experimental parameter space (larger drops, higher viscosity range, longer times, unconstrainedliquid motion) to validate the viscosity measurement method.
• Validate analysis of shape relaxation as a method of determining viscosities of highly viscous fluids in amicrogravity environment.
• Refine and calibrate mathematical models of calculation of fluid viscosities using the two drop coalescencetechnique.
• Test the 60 year old theory of Frenkel for liquid sintering.
II. Rapid Payload Deployment ISS Experiment
The FMVM experiment also provided the opportunity to demonstrate that International Space Station, ISS,experiments could be developed quickly. Another objective of the FMVM payload development was to develop theprocess for rapid payload deployment. The Payload Development Team was quickly assembled shortly after theFMVM experiment was selected to be flown. Table 1 lists the team members and shows how even a small, low-costexperiment involves dozens of people.
Table I. Payload Development Team
• Dr. E. Ethridge – Principal Investigator, MSFC• Dr. W. Kaukler – Co-Investigator, UAH• Dr. B. Antar – Co-Investigator, UTSI• D. Lehman – Graduate Student, UTSI• Dr. D. Gillies – Project Scientist, MSFC• J. Kennedy – Project Manager, TBE• L. Murphy – Project Manager, MSFC• J. Norris – Payload Integration Manager, United Space Alliance
American Institute of Aeronautics and Astronautics4
• J. Belt – Operations Lead, Sverdrup• G. Norris – Lead Increment 8 Payload Operations Director• J. Hardy – PRO / OC Team contact• M. Barnett – Payload Activities Requirements Coordinator, MSFC• J. Heinisch – Crew Procedures, TBE• T. Nordmann – Photo/Video, MSFC• B. Douglas – Sim. Engineer & Crew Training, TBE• R. Moore – Operations Safety, TBE• G. Davenport – Ground Data Services• M. Connolly – Project Safety, TBE
A number of ground rules were established to accomplish the task within the 6 month window of opportunity priorto the 13P Russian Progress resupply launch. The Payload Operations Ground rules included the development ofthe Crew Procedures. Training was accomplished via On-Board Training (OBT) on a CD. The timeline wasapproved through an Engineering Change Request, ECR, of the ISS Increment 8 Planning data Set. The SafetyProcess Ground rules included the Flight Safety Data Package, no formal MSFC Payload Safety Readiness ReviewBoard was deemed necessary, and a Russian Transportation Safety Data Package was developed. The MSADProcess Ground rules stated that a number of standard documents were not applicable. Documentation was notcontrolled through Control Board Process, Formal Risk Management was not to be conducted. Issues were to behandled only as necessary. A Flight Certification Review (FCR) was conducted within MSFC MSAD, and thepayload submitted to the CoFR process. The ISS Ground Rules included the Payload Integration Agreement (PIA)and CoFR Inputs were developed.
III. On-orbit Material and Hardware UtilizationDuring the development of crew procedures, a number of on-orbit materials and hardware were identified for
implementation of the experiment. The final list of items is shown in Table II.
Table II. On-Orbit Materials and Hardware
Mini Maglite / 528-20084-5 11.5V Battery Size AA / 528-41350-3 2Flexible Bracket / SEG33107630-301 1
Steel Ruler / YA120A 1Maintenance Work Area (MWA)P/N SEG33110270-301MWA Containment System /SEG33110290-301
1
MWA Utility Kit / SJG33110310-301 1Track Lock Assembly /SEG33108822-301
1
Bar Clamp Assembly /SEG33110167-301
1
8.50” Bar Shaft / SDG33110168-001 1
14” Track, Removable,/MWA/SEG33110152-301
1
1
1
American Institute of Aeronautics and Astronautics5
The ISS experiment hardware setup and crew procedures were validated at JSC in the Payload DevelopmentLaboratory, PDL, in the Space Station Mockup and Trainer Facility, SSMTF, of JSC's Space Vehicle MockupFacility, SVMF. The hardware was set up in the MWA and the crew procedures were tested by an astronaut. Thefollowing figures show the on board equipment used for the FMVM experiment.
Figure 2. The MWA containment.
Figure 3. The MWA utility kit.
These on-orbit supplies were configured to hold the background grid card, and provide support for the Nomexstrings that held the drops. A Fluke 190 Scopemeter with temperature probe was used to measure the temperatureof the liquid drops. A mini-Maglite provided illumination of the liquid drops, see Figure 4.
American Institute of Aeronautics and Astronautics6
Figure 4. FMVM setup in the MWA.
The on-board digital camcorder was set up to look through the window of the MWA containment to record theexperiment, see Figure 5.
Figure 5. Camcorder setup at the MWA.
American Institute of Aeronautics and Astronautics7
IV. FMVM Experiment Design
The experimental design parameters are quite simple. The experiment design required the measurement of theinitial diameters of the 2 drops and record the measurement of the time dependent change in the neck diameter withthe on-board digital camcorder. Several low toxicity viscous fluids spanning a 2 order of magnitude viscosity rangewith well characterized viscosity were utilized. The manifested fluids were contained in syringes also used todeploy the liquids for the experiment. On-orbit resources utilized to bring droplets together under controlledconditions were identified in the crew procedures controlled through crew training. Initially two 0.5 ml dropletswere deployed onto strings and coalesced to a single 1 ml drop. Next a 1 ml drop was deployed and coalesced withthe first 1 ml drop to form a single 2 ml drop. Next a 2 ml drop is deployed, the process repeated and followed by a4 ml drop. The final 8 ml drop was coalesced with a 0.5 ml drop. The temperatures of the drops were measuredwith on-board Scopemeter and temperature probe. The rate of shape change was recorded with a color video camera.Data was captured on lab camcorder and some transmitted real time to ground. Hi-8mm video tape was returnedfrom ISS and is to be used for detailed post flight analysis.
The liquids selected for FMVM had to meet certain criteria. The liquids had to span a large range of viscosities andbe nontoxic and nonirritant. Food items (liquids) were preferred since they are not characterized as chemicals. Itwas desired to have different liquids, with different surface tensions in the same viscosity range. It was highlydesirable for the liquids to have well characterized viscosity and it was very desirable to have a non-Newtonianliquid.
The initial selection of liquids included:
• Two syringes of Glycerin to compare with KC-135 data.• Two syringes of Silicone oil viscosity standard liquids, 2 different viscosities.• Two syringes of Honey, one syringe with Sue Bee Honey off the shelf and one syringe of Sue Bee Honey
cooked to remove water content to increase the viscosity.• Two syringes of Corn Syrup, one syringe with Kayro off the shelf and one syringe of Kayro cooked to
remove water content to increase the viscosity.• Two syringes of Mineral Oil with dissolved butylene, a highly non-Newtonian liquid (viscosity is a
function of shear rate). These liquids were not delivered for flight. There was a reaction with the syringeplunger causing it to swell.
The liquids were vacuum out gassed for 24 hours to remove dissolved gasses. They were then loaded into 10 ccmedical syringes that had been previously certified for flight. The materials manifested on 13P included 8 syringes,10 ml each of the first 4 sets of fluids above. One background grid card for droplet image calibration was producedfrom the same material and by the process for astronaut crew cue cards. It was also included in the manifest. Themanifest liquids were shipped to Russia for 13P. The Russian-built cargo freighter, Progress 13P, mounted atop anunmanned Soyuz rocket was launched from Baikonur Cosmodrome in Kazakhstan on January 29, 2004 to theInternational Space Station.
After delivery of the manifest materials, we undertook the characterization of the liquids. A Brookfield rotatingspindle viscometer was used to measure the viscosity of all the liquids. The shear dependence was also determinedfor liquids. Silicone oil viscosity standards, glycerin, and corn syrup are Newtonian liquids, viscosity is not afunction of shear rate for the rates measured. Mineral oil with butylenes and honey are non-Newtonian, see Figure6. The change in viscosity is directly related to the rate of shear. An order of magnitude increase of shear ratedecreases the viscosity by one order.
American Institute of Aeronautics and Astronautics8
Figure 6. Shear rate dependence of viscosity for honey.
Surface Tension of the flight liquids was determined both by the pendant drop method and the drop-weightmethods. The pendant drop method uses stationary equilibrium drop shape to determine surface tension. For thedrop-weight method, liquid contained in a syringe was slowly dripped through a known diameter orifice tube. Theweight of the liquid drop that causes the drop to break the liquid bridge holding the drop to the orifice is directlyrelated to the surface tension. A number of drops (20) were dropped to obtain an average drop weight.
The measured properties of the liquids used for ISS experiments is tabulated in Table III.
Table III. Properties of the FMVM liquids.
Density Surf Tension Viscosityg/cc erg/cmcm cPoise (Pa sec)
Glycerine A1 1.17 63 1490
B1 0.97 21.5 12500
B2 0.97 21.5 100000
Honey D1 1.45 90 12500
D2 1.47 88 42000
E1 1.41 83 2200
E2 1.41 90 15000
Corn Syurp
Thick Corn Syrup
FMVM Liquids
Silicone oil 12500
Silicone oil 100000
Thick Honey
Viscosity vs Shear Rate of Honeyat Room Temp 21.7-22.0 C
10 0 0 0
10 0 0 0 0
10 0 0 0 0 0
10 100 1000 10000 100000
Shear Rate 1/sec
Vis
cosi
ty
American Institute of Aeronautics and Astronautics9
IV. Ground Based Validation of the FMVM Experiment MethodsIn order to validate the flight experiment methods, desktop experiments were performed with the flight liquids.
Two Nomex threads were used to hold 2 small (1 mm) drops. A Sony Digital-8 Handycam digital camcorder with25X optical zoom was used to record the coalescence of the small drops. Uleadvideo Studio 7 and PIXELAsoftware was used to capture video AVI files and individual JPG images to be analyzed. Spotlight 16 imageanalysis software was used for digital measurements of the captured frames.
Recently a root time scaling law has been applied to the coalescence of two liquid drops. Based on physicalarguments, Eggers8 proposed that the droplet neck growth followed a square root time scaling law. This square rootcoalescence time dependence was confirmed experimentally in the inertial regime of inviscid liquid drops9.Ground based experiments were conducted with honey liquid spheres suspended on Nomex thread. Figure 7 showsa sequence of images from a recorded liquid coalescence experiment.
Figure 7. Sequence of frames for Drop coalescence in 1-g of honey drops (1 mm diameter).
The neck diameter was normalized to the initial drop diameter and plotted vs. time (Figure 8) and vs. square roottime (Figure 9). The coalescence process has a linear root time dependence over a rather large portion of thecoalescence process. This is confirmation of this scaling law with viscous drops.
Figure 8. Normalized neck diameter vs time for Figure 9. Normalized neck diameter vs root time forHoney coalescence. Honey coalescence.
Honey Coalescence
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American Institute of Aeronautics and Astronautics10
In June of 2004 roughly 1 year after starting the experiment development, FMVM was scheduled for operationsduring Space Station Increment 9. The investigators and payload developers supported the experiment from thetele-science center at MSFC.
Figure 10. FMVM team supporting the experiment from the Tele-science center at MSFC.
The experiment operations were observed real time during acquisition of signal, AOS. During loss of signal,LOS, from ISS, astronaut Mike Fincke proceeded along with the experiment operations with video data beingrecorded on board. Glycerin could not be deployed onto the Nomex thread. Operations with the 2 different siliconeoil liquids were completed. The samples of honey had crystallized and were stowed for later operations. Theoperations with the 2 different corn syrup liquids were also completed, but rapid rigid film formation caused anumber of drop coalescences to be compromised by the rigid film.
Procedures were developed to utilize the Space Station Food Warmer to heat the syringes with crystallizedhoney. Heating for an hour completely dissolved the crystals. In May 2005, experiment operations were scheduledfor the two honey liquids. Astronaut John Philips performed the experiments during Space Station increment 11.Unfortunately, there was extended loss of signal (LOS) during the operations. We are anxious to obtain the videotapes to see the results.
On his own initiative, astronaut Mike Fincke captured several of the drop coalescences as AVI files and sentthem down as digital files to the investigators. These high resolution video clips have been very useful forpreliminary data analysis. Figure 11 shows a representative still frame from the AVI video data.
American Institute of Aeronautics and Astronautics11
Figure 11. Two 4 ml silicone oil viscosity standard liquid drops inthe process of coalescing to a single spherical drop.
Prior to obtaining the high resolution video data tapes that were returned to earth by the return to flight SpaceShuttle mission, some preliminary examination of the data has been performed. Figure 12 shows the normalizedneck diameter vs. square root time for the 4 cc drop coalescence run shown in Figure 11. This plot shows the linearroot time dependence of coalescence with highly viscous, very large diameter liquid drops.
Figure 12. Normalized neck diameter vs. root time for 4 cc diameter silicone oil drops.
When the high resolution digital data tapes are obtained, all the ISS experiments will be examined according tothe coalescence models. This will provide a new insight into the liquid coalescence and sintering of spheresproblem. The data will also be used to evaluate the method for viscosity determination. Frenkel2 introduced theconcept of a characteristic coalescence time constant. The time constant is proportional to the liquid viscosity timesthe initial sphere diameter divided by the surface tension.
t = ( η D ) / σ
Initial evaluation of this simple concept indicates that by calculating the time constant for different viscosityliquids and experimental data with standard liquids, it will be possible to determine the viscosity from an unknownliquid’s measured time constant. Initially we are considering the time for the joining liquid neck of two coalescingspheres to reach 0.5 of the initial droplet diameter as the experimental time constant.
Silicone Oil 4 cc drops
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American Institute of Aeronautics and Astronautics12
V. Conclusions
• FMVM successfully examined the unconstrained fluid flow behavior of 2 coalescing drops in low gravityon the International Space Station with a number of different viscous liquids.
• The square root time coalescence dependence liquid sintering was observed in preliminary analysis of theISS data.
• A new method for determining the viscosity of highly undercooled liquids is being developed withparametric analysis of the data, the slope of the square root dependence, and the Frenkel time constant.
• The successful payload development, delivery to Russia, and launch within 6 months demonstrated a newpayload development process for rapid deployment of International Space Station experiments.
References
1D. Tucker, E. Ethridge, G. Smith, and G. Workman, "Effects of Gravity on ZBLAN Glass Crystallization" Ann. N.Y. Acad.
Sci. Vol. 1027:129-137 (2004).2Frenkel, J. “Viscous Flow of Crystalline Bodies Under the Action of Surface Tension,” J. Physics (Moscow), Vol. 9 (5),
385-391 (1945).3Jagota, A. and P. R. Dawson, “Micromechanical Modeling of Powder Compacts,” Acta Metall. Vol. 36, (9), 2551-2561
(1988).4Van de Vorst, G. A. L., “Numerical Simulation of Axisymetrical Viscous Sintering,” Eng. Annl. Boundary Elem. Vol. 14,
193-207 (1994).5Ethridge, E., B. Antar, and D. Maxwell ,"Viscosity Measurement of Highly Viscous Liquids Using Drop Coalescence in
Low Gravity", AIAA99-0708, 37th AIAA Aerospace Sciences Meeting and Exhibition, Jan 11-14, (1999).6Antar, B., E.C. Ethridge, and D. Maxwell, “Utilization of Low Gravity Environment for Measuring the Viscosity of Highly
Viscous Liquids,“ Advances in Space Research, 24:1289(1999).7Antar, B. N., E.C. Ethridge, and D. Maxwell, "Viscosity Measurement using Drop Coalescence in Microgravity",
Microgravity Sci. Tech. 14(1):9-19(2003).8Eggers, J., J. R. Lister and H. A. Stone, “Coalescence of Liquid Drops,” J. Fluid Mech., Vol. 401, 293 (1999).
9Wu, M., T. Cubaud, and C. M. Ho, “Scaling Law in Liquid Drop Coalescence Driven Surface Tension,” Phys. Fluids, Vol.
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