Research Article Heat Flux at the Surface of Metal Foil Heater under Evaporating Sessile Droplets Igor Marchuk, 1,2 Andrey Karchevsky, 1,3 Anton Surtaev, 2 and Oleg Kabov 2,4 1 Novosibirsk State University, Novosibirsk 630090, Russia 2 Kutateladze Institute of ermophysics SB RAS, Novosibirsk 630090, Russia 3 Sobolev Institute of Mathematics SB RAS, Novosibirsk 630090, Russia 4 Tomsk Polytechnic University, Tomsk 634050, Russia Correspondence should be addressed to Igor Marchuk; [email protected] Received 6 September 2015; Accepted 10 November 2015 Academic Editor: Corin Segal Copyright © 2015 Igor Marchuk et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Evaporating water drops on a horizontal heated substrate were investigated experimentally. e heater was made of a constantan foil with the thickness of 25 m and size of 42 × 35 mm 2 . e temperature of the bottom foil surface was measured by the infrared (IR) camera. To determine the heat flux density during evaporation of liquid near the contact line, the Cauchy problem for the heat equation was solved using the temperature data. e maximum heat flux density is obtained in the contact line region and exceeds the average heat flux density from the entire foil surface by the factor of 5–7. e average heat flux density in the region wetted by the drop exceeds the average heat flux density from the entire foil surface by the factor of 3–5. is fact is explained by the heat influx from the foil periphery to the drop due to the relatively high heat conductivity coefficient of the foil material and high evaporation rate in the contact line region. Heat flux density profiles for pairs of sessile droplets are also investigated. 1. Introduction Liquid droplets moving or spreading on the solid surface are widely distributed in nature, different areas of techniques, and technology. For example, they are observed in the coating technologies of solid surfaces by liquid films, during spraying of fertilizers and pesticides, in strengthening steel devices, in the spray cooling systems for electronic equipment, and others. Flow regime of liquid droplets is main in a two-phase mixture in rectangular channels of small height as shown in [1]. e angle between the solid surface and the tangent to the drop surface at the point of contact between the three phases (wetting angle) is a fundamental macroscopic characteristic of the contact line. is angle is determined by the equation of Young [2], as a result of the mechanical balance of the three surface tensions: liquid-gas, solid surface-gas, and solid surface -liquid. In the literature significant attention is paid to the study of various aspects of the droplets spreading over the solid surface. Only a few studies [2–8] have been performed for conditions with different level of gravity. Analysis of these papers shows that there is lack of understanding of the process of evaporation and wetting by droplets over solid surfaces in condition of changing gravity. Wetting of solid surface by droplets and processes in three-phase contact line gas-liquid-solid has a major influence on the droplets evaporation. Insufficient understanding of the properties of the three-phase contact line is currently holding back the development of mathematical models of evaporation of liquid droplets on solid surfaces. On the other hand, the liquid droplet on a solid surface is the most simple and convenient object for studying phenomena of wetting and spreading and properties of the three-phase contact line. In particular process of evaporation in the three-phase contact line can be studied on droplets. e authors along with research groups from Europe, Canada, China, and Japan are members in the preparation of the experiment “Drop Evaporation” on board the International Space Station. e transfer processes in the vicinity of a dynamic three- phase wall-liquid-vapor contact line belong to one of the most important physical problems not yet fully solved. Insufficient Hindawi Publishing Corporation International Journal of Aerospace Engineering Volume 2015, Article ID 391036, 5 pages http://dx.doi.org/10.1155/2015/391036
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Research ArticleHeat Flux at the Surface of Metal Foil Heater underEvaporating Sessile Droplets
Igor Marchuk12 Andrey Karchevsky13 Anton Surtaev2 and Oleg Kabov24
1Novosibirsk State University Novosibirsk 630090 Russia2Kutateladze Institute of Thermophysics SB RAS Novosibirsk 630090 Russia3Sobolev Institute of Mathematics SB RAS Novosibirsk 630090 Russia4Tomsk Polytechnic University Tomsk 634050 Russia
Correspondence should be addressed to Igor Marchuk marchukitpnscru
Received 6 September 2015 Accepted 10 November 2015
Academic Editor Corin Segal
Copyright copy 2015 Igor Marchuk et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Evaporating water drops on a horizontal heated substrate were investigated experimentally The heater was made of a constantanfoil with the thickness of 25 120583m and size of 42 times 35mm2 The temperature of the bottom foil surface was measured by the infrared(IR) camera To determine the heat flux density during evaporation of liquid near the contact line the Cauchy problem for the heatequation was solved using the temperature data The maximum heat flux density is obtained in the contact line region and exceedsthe average heat flux density from the entire foil surface by the factor of 5ndash7The average heat flux density in the region wetted by thedrop exceeds the average heat flux density from the entire foil surface by the factor of 3ndash5 This fact is explained by the heat influxfrom the foil periphery to the drop due to the relatively high heat conductivity coefficient of the foil material and high evaporationrate in the contact line region Heat flux density profiles for pairs of sessile droplets are also investigated
1 Introduction
Liquid droplets moving or spreading on the solid surface arewidely distributed in nature different areas of techniques andtechnology For example they are observed in the coatingtechnologies of solid surfaces by liquid films during sprayingof fertilizers and pesticides in strengthening steel devicesin the spray cooling systems for electronic equipment andothers Flow regime of liquid droplets is main in a two-phasemixture in rectangular channels of small height as shown in[1]The angle between the solid surface and the tangent to thedrop surface at the point of contact between the three phases(wetting angle) is a fundamental macroscopic characteristicof the contact line This angle is determined by the equationof Young [2] as a result of the mechanical balance of thethree surface tensions liquid-gas solid surface-gas and solidsurface -liquid In the literature significant attention is paid tothe study of various aspects of the droplets spreading over thesolid surface Only a few studies [2ndash8] have been performedfor conditions with different level of gravity Analysis of
these papers shows that there is lack of understanding of theprocess of evaporation and wetting by droplets over solidsurfaces in condition of changing gravity Wetting of solidsurface by droplets and processes in three-phase contactline gas-liquid-solid has a major influence on the dropletsevaporation Insufficient understanding of the properties ofthe three-phase contact line is currently holding back thedevelopment ofmathematicalmodels of evaporation of liquiddroplets on solid surfaces On the other hand the liquiddroplet on a solid surface is the most simple and convenientobject for studying phenomena of wetting and spreadingand properties of the three-phase contact line In particularprocess of evaporation in the three-phase contact line can bestudied on droplets The authors along with research groupsfrom Europe Canada China and Japan are members in thepreparation of the experiment ldquoDrop Evaporationrdquo on boardthe International Space Station
The transfer processes in the vicinity of a dynamic three-phasewall-liquid-vapor contact line belong to one of themostimportant physical problems not yet fully solved Insufficient
Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 391036 5 pageshttpdxdoiorg1011552015391036
2 International Journal of Aerospace Engineering
understanding of transfer processes in the region of the three-phase contact line impedes the development of models ofseveral phenomena such as boiling cavitation dropwisecondensation rivulet flows liquid filmbreak drop spreadingand evaporation [9ndash11] The liquid thickness in this zonewith the length of about several microns which is oftencalled the microregion decreases from 1ndash3 120583m to 10ndash20 nm(the adsorbed film) Significant curvature of the gas-liquidinterface in the microregion change in the thickness by 2-3orders ofmagnitude nonequilibriumeffects (the temperaturedrop at the interface) influence of van der Waals forcesand structure of the substrate cause considerable difficultiesfor both experimental and theoretical investigations of thisconfiguration
Theproblemof determining the evaporation rate and heatflux density in the heated liquid in the three-phase contactline region is actively studied in connection with importantpractical applications in power engineering medicine andchemical pharmaceutical and food industries [12ndash14] Boththeoretical and experimental studies prove that heat transferintensity in the contact line region can be more than an orderof magnitude higher than the average one and it causes alocal minimum in the temperature profile along the solidwall [15 16] However since it is impossible to perform thedirect measurements of the heat flux in the microregion(the length of which is estimated to be between 05 and 10ndash20120583m) various indirect methods and numerical proceduresare applied In Ibrahem et al [17] the meniscus of HFE7100liquid evaporated between two heated vertical plates of metalfoil The local heat flux to evaporating meniscus was cal-culated using the two-dimensional temperature distributionalong the outer foil edge obtained by the infrared scannerwith resolution of 148 microns It was found that the localheat flux density in the contact line region was 54ndash65 timeshigher than the average heat flux density on the surface
In the present study evaporation near contact lines isinvestigated using the sessile drop configuration There isa significant amount of literature on evaporating sessiledroplets [9 11 16] However the use of metal foil as aheater in droplet experiments is not common despite thefact that such configuration has potential for giving accuratemeasurements of the temperature field immediately underthe droplet and thus providing a tool for studies of localcoupling of the triple line motion and heat transfer inthe substrate with unprecedented accuracy We believe thatthis potential has not been fully realized For exampleSodtke et al [16] used a thin metal foil heater but reliedon thermochromic liquid crystals (TLCs) for temperaturemeasurementThe issues with TLCs are limited accuracy andlack of applicability outside of a relatively narrow temperaturerangeThe objective of the present study is to overcome thesedifficulties by using IR imaging Another limitation of thecurrent literature on droplets is that the vast majority of boththeoretical and experimental studies are focused on isolateddroplets while in applications one often encounters largearrays of dropletsThe issue of interaction of droplets remainspoorly understood and is discussed in the present study
CCD camera
Lenses
Droplet
Foil
MirrorPower supply
Light source
IR camera+ minus
Figure 1 The scheme of the experiment
2 Materials and Methods
21 Experiment The evaporating sessile water drops ona horizontal heated substrate were studied experimentallyusing the setup shown in Figure 1 The constantan foil(CuNi) of the thickness of 25 120583m size of 42 times 35mm2(119860 times 119861) and heat conductivity 120582 of 23WmK was used asthe substrate The substrate was heated by Joule heating Inexperiments the heating power 119876 varied from 025W to25WThe images of drop profile were obtained by the opticalshadow system with resolution of 8 120583mpx The geometricparameters of drops were determined using the images by theYoung-Laplace method in experiments they were as followsthe volume was 51ndash393 120583L base diameter (wetted spot)was 368ndash719mm height was 088ndash181mm and contactangle was 50ndash75∘ (average angle of 63∘) The temperatureof the bottom foil surface was measured by IR cameraTitanium 570M The thermographic study was performed atrecording frequency of 25Hz resolution of 640 times 512 pixels(108 120583mpx) and integration time of 805ms To increase theminimal resolvable temperature difference when using the IRcameras the reverse side of the foil was coated with a sootlayer The foil surface temperature measured in experimentswas in the range of 29ndash86∘C The average coefficient of heattransfer from the foil surface to surrounding air 120572av wasfound to be within 16ndash20Wm2K as seen in Figure 2 Thesemeasurements were performed without droplets on the foilsurface
22Theory and Computation To obtain accurate descriptionof heat transfer intensity near the contact line for evaporatingsessile droplets the Cauchy problem was formulated for theelliptic equation
120582Δ119879 + 119902V = 0 (1)
describing the heat conduction in the foil (ie the domaindefined by the equations 1198602 le 119909 le 1198602 1198612 le 119910 le1198612 and 0 le 119911 le 119867) Here 119879(119909 119910 119911) is temperature ofthe foil 120582 is (constant) thermal conductivity and 119902V is theconstant volume heat source On the bottom side of the foilthe temperature and the heat flux density were set as
119879 (119909 119910 0) = 119879119908(119909 119910)
120582120597119879
120597119911(119909 119910 0) = 120572av (119879119908 minus 119879119886)
(2)
International Journal of Aerospace Engineering 3
0 200 400 600 800 10000
5
10
15
20
25
120572(W
m2K)
qav (Wm2)
qav = W2S120572 = qav(Tw minus Ta)
Figure 2 The measured heat transfer coefficient for different heatflux densities 119878 area of one side of the foil 119882 power of electricheating of the foil 119879
119908 average measured temperature of the foil
surface and 119879119886 temperature of ambient air
The lateral walls were assumed adiabatic
120597119879
120597119909
10038161003816100381610038161003816100381610038161003816119909=plusmn1198602=120597119879
120597119910
10038161003816100381610038161003816100381610038161003816119910=plusmn1198612
= 0 (3)
The Cauchy problem for the Laplace equation or ellipticequations of the general form is one of the oldest ill-posedproblems [18ndash22] Its solution is unstable small variationsin the Cauchy data can lead to significant changes in thesolution If the existence of a bounded solution assumed itis possible to show the conditional stability of the problem[20] There are many different implementations of methodsof numerical solution to the Cauchy problem for ellipticequations (see eg [21ndash25]) Instability of solution to theCauchy problem becomes obvious when applying any ofnumerical methods to its solution In this work we use themethod that reduces the Cauchy problem solution to thesolution of the problem of moments [24 25] By a specialchoice of the boundary conditions in the conjugated problemcalculation of the heat flux value at the boundary partinaccessible for the measurements is reduced to summationof the corresponding series Simple formof the domain allowssolving the moment problem using the direct and inverseFourier transforms The regularizing procedure is used tosum the series of inverse Fourier transforms due to theinstability of the Cauchy problem The implemented methodof solution was applied to experimental data processing
3 Results and Discussion
With the use of the above method for solving the Cauchyproblem distributions of heat flux density on the foil surfacefrom the drop side were obtained from the thermographicmeasurements The initially measured temperature fieldcalculated distribution of heat flux density on the substrateand distribution of the local heat flux density along the linepassing through the central cross section of the drop areshown in Figure 3 The drop profile is also shown there The
results show that the maximum of the heat flux density isin the contact line region and it exceeds the average heatflux density on the entire foil surface by the factor of 5ndash7The average heat flux in the zone wetted by a drop exceedsthe average heat flux density on the entire foil surface bythe factor of 3ndash5 This can be explained by the heat influxfrom the foil periphery to the drop due to the relativelyhigh heat conductivity coefficient of the foil material andhigh evaporation rate in the contact line region Calculationof local heat flux density distribution without consideringredistribution of heat released in foil (line 2) is also shownin Figure 3 It can be seen that a negligence of substrateheat conductivity gives significantly underestimated heat fluxdensity and absence of peaks in the contact line region Dataare shown for two different sizes of drops at substantiallydifferent average heat flux density It can be seen that the ratioof maximal heat flux density in the contact line region andaverage heat flux density increases with a drop size decreaseit is equal to 37 for Figure 3(a) and 48 for Figure 3(b)
Previous results suggest that the heat flow from the drypart of the foil to the droplet plays an important role inthe overall heat transfer suggesting that evaporation ratesfor two or more droplets can be significantly different fromthe values predicted for a single droplet The results ofmeasurements and calculations for two sessile drops on thesubstrate are shown in Figure 4 the distance between thesedrops is 11mmThe symmetry conditions are broken for eachof these drops In the contact line region between drops wedo not observe the maxima of heat flux density because theheat influx from the periphery which is now split betweenthe two drops decreases for each of these dropsThemaximaof heat flux density are reached near the contact line at theouter edges of the two-droplet system the maximum value ishigher for the smaller droplet
The droplet on the heated substrate evaporates with timeThe evaporation rate can be evaluated using the available datathrough the heat flux density
119902ev = 119902119905 cos (120573) minus 119902conv (4)
Here 119902119905is the calculated heat flux density on the substrate
120573 is an angle of inclination of the drop surface to horizontalsubstrate and 119902conv is a convective heat flux from the dropletsurface
119902conv = 120572 (119879119904 minus 119879119886) (5)
It can be expressed through 119902119887 which is the heat flux from
the lower foil surface in the assumption that the temperatureof droplet surface slightly differs from the temperature ofsubstrate under it 119879
119904cong 119879119887 At that we should take into
account the difference between the area of droplet surface andsubstrate
119902conv = 119902119887 cos (120573) (6)
We obtain
119902ev = (119902119905 minus 119902119887) cos (120573) (7)
4 International Journal of Aerospace Engineeringq(W
m2)
1000
800
600
400
200
0
0 10 20 30
x (mm)
1
2
3
Thermal image1000
750
500
250
0
Heat flux
(Wm
2)
45
39
33
27
21
qt = 2qav minus qb
qb = 120572av(Tw minus Ta)
(∘C)
(a)
q(W
m2)
2000
1600
1200
400
800
0
0 10 20 30
x (mm)
1
2
3
Thermal image2000
1500
1000
500
0
Heat flux60
50
40
30
20
(Wm
2)
(∘C)
(b)
Figure 3 Distributions of heat flux density 1 Cauchy problem solution 2 calculation with neglecting of heat flux redistribution in the foil119902119905 3 on the bottom side of foil 119902
119887 (a) Drop 119863 = 66mm119867 = 182mm 119881 = 363 120583L 120579 = 65∘ and 119902av = 255Wm2 (b) Drop 119863 = 44mm
119867 = 115mm 119881 = 98 120583L 120579 = 59∘ and 119902av = 365Wm2
q(W
m2)
3600
2700
1800
900
0
x (mm)
66
59
52
45
38
0 10 200
1000
2000
3000
4000
2
Heat flux
1
Thermal image
(Wm
2)
(∘C)
Figure 4Distribution of heat flux density based onCauchy problemsolution for two drops 119902av = 653Wm2 (1) 119863 = 41mm 119867 =1085mm 119881 = 7120583L and 120579 = 58∘ (2) 119863 = 41mm119867 = 0903mm119881 = 573 120583L and 120579 = 52∘
The local evaporation rate can be calculated as
=119902ev119903lv=(119902119905minus 119902119887) cos (120573)119903lv
(8)
where 119903lv is the latent heat of vaporization Evaporationheat flux density and evaporation rate for the droplets ofdifferent diameters are shown in Figure 5 It can be seen thatevaporation rate near the contact line has a local maximumbut it is not so pronounced as heat flux density on thesubstrate in Figures 3 and 4
4 Conclusions
We have shown that the maximum of the heat flux densityis in the contact line region and it exceeds the average heatflux density on the entire foil surface by the factor of 5ndash7 In two-droplet system the maxima of heat flux densityare reached near the contact line at the outer edges of thetwo-droplet system the maximum value is higher for thesmaller droplet This can be explained by the heat influxfrom the foil periphery to the drop due to the relatively highheat conductivity coefficient of the foil material and highevaporation rate in the contact line region The possibility ofheat flux calculation by using infrared data gives interestingmethod of an experimental study of three-phase contact line
International Journal of Aerospace Engineering 5
2
1
1000
800
600
400
200
0
minus15 minus1 minus05 0 05 1 15
04
03
02
01
m (
gm
2s)
qev
(Wm
2)
2xD
Figure 5 Calculated distribution of evaporation heat flux densityand evaporation rate (1) drop from Figure 3(a) and (2) drop fromFigure 3(b)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The study was financially supported by the grant of theRussian Science Foundation (Project no 14-39-00003)
References
[1] E A Chinnov andO A Kabov ldquoTwo-phase flows in horizontalflat microchannelsrdquo Doklady Physics vol 57 no 1 pp 42ndash462012
[2] A Ababneh A Amirfazli and J A W Elliott ldquoEffect of gravityon the macroscopic advancing contact angle of sessile dropsrdquoCanadian Journal of Chemical Engineering vol 84 no 1 pp 39ndash43 2006
[3] D Brutin Z Zhu O Rahli J C Xie Q S Liu and LTadrist ldquoSessile drop in microgravity creation contact angleand interfacerdquoMicrogravity Science and Technology vol 21 no1 pp 67ndash76 2009
[4] G Abel G G Ross and L Andrzejewski ldquoWetting of a liquidsurface by another immiscible liquid inmicrogravityrdquoAdvancesin Space Research vol 33 no 8 pp 1431ndash1438 2004
[5] D Brutin Z-Q Zhu O Rahli J-C Xie Q-S Liu and LTadrist ldquoEvaporation of ethanol drops on a heated substrateunder microgravity conditionsrdquoMicrogravity Science and Tech-nology vol 22 no 3 pp 387ndash395 2010
[6] F Carle B Sobac and D Brutin ldquoHydrothermal waves onethanol droplets evaporating under terrestrial and reducedgravity levelsrdquo Journal of Fluid Mechanics vol 712 pp 614ndash6232012
[7] O A Kabov and D V Zaitsev ldquoThe effect of wetting hysteresison drop spreading under gravityrdquo Doklady Physics vol 58 no7 pp 292ndash295 2013
[8] F Carle B Sobac and D Brutin ldquoExperimental evidence ofthe atmospheric convective transport contribution to sessiledroplet evaporationrdquo Applied Physics Letters vol 102 no 6Article ID 061603 2013
[9] V K Dhir ldquoNumerical simulations of pool-boiling heat trans-ferrdquo AIChE Journal vol 47 no 4 pp 813ndash834 2001
[10] R J Braun ldquoDynamics of the tear filmsrdquoAnnual Review of FluidMechanics vol 44 pp 267ndash297 2012
[11] B Sobac and D Brutin ldquoTriple-line behavior and wettabilitycontrolled by nanocoated substrates influence on sessile dropevaporationrdquo Langmuir vol 27 no 24 pp 14999ndash15007 2011
[12] C L Moraila-Martınez M A Cabrerizo-Vılchez and M ARodrıguez-Valverde ldquoControlling the morphology of ring-likedeposits by varying the pinning time of driven receding contactlinesrdquo Interfacial Phenomena and Heat Transfer vol 1 no 1 pp195ndash205 2013
[13] A Kundan J L Plawsky and P C Wayner Jr ldquoThermophys-ical characteristics of a wickless heat pipe in microgravitymdashconstrained vapor bubble experimentrdquo International Journal ofHeat and Mass Transfer vol 78 pp 1105ndash1113 2014
[14] V S Ajaev E Y Gatapova and O A Kabov ldquoRupture of thinliquid films on structured surfacesrdquo Physical Review E vol 84no 4 Article ID 041606 2011
[15] SMoosman andGMHomsy ldquoEvaporatingmenisci of wettingfluidsrdquo Journal of Colloid and Interface Science vol 73 no 1 pp212ndash223 1980
[16] C Sodtke V S Ajaev and P Stephan ldquoDynamics of volatileliquid droplets on heated surfaces theory versus experimentrdquoJournal of Fluid Mechanics vol 610 pp 343ndash362 2008
[17] K Ibrahem M F Abd Rabbo T Gambaryan-Roisman and PStephan ldquoExperimental investigation of evaporative heat trans-fer characteristics at the 3-phase contact linerdquo ExperimentalThermal and Fluid Science vol 34 no 8 pp 1036ndash1041 2010
[18] J Hadamard Lectures on the Cauchy Problem in Linear Differen-tial Equations Yale University Press New Haven Conn USA1923
[19] L E Payne ldquoBounds in the Cauchy problem for the Laplaceequationrdquo Archive for Rational Mechanics and Analysis vol 5pp 35ndash45 1960
[20] M Lavrentev V Romanov and S Shishatskii Ill-Posed Problemsof Mathematical Physics and Analysis American MathematicalSociety 1986
[21] F Berntsson and L Eiden ldquoNumerical solution of a Cauchyproblem for the Laplace equationrdquo Inverse Problems vol 17 no4 pp 839ndash853 2001
[22] L Bourgeois and J Darde ldquoA duality-based method of quasi-reversibility to solve the Cauchy problem in the presence ofnoisy datardquo Inverse Problems vol 26 no 9 Article ID 0950162010
[23] C-L Fu H-F Li Z Qian and X-T Xiong ldquoFourier regular-ization method for solving a Cauchy problem for the Laplaceequationrdquo Inverse Problems in Science and Engineering vol 16no 2 pp 159ndash169 2008
[24] T Reginska and A Wakulicz ldquoWavelet moment method forthe Cauchy problem for the Helmholtz equationrdquo Journal ofComputational andAppliedMathematics vol 223 no 1 pp 218ndash229 2009
[25] A L Karchevsky ldquoReformulation of an inverse problem state-ment that reduces computational costsrdquo Eurasian Journal ofMathematical and Computer Applications vol 1 no 2 pp 5ndash202013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
2 International Journal of Aerospace Engineering
understanding of transfer processes in the region of the three-phase contact line impedes the development of models ofseveral phenomena such as boiling cavitation dropwisecondensation rivulet flows liquid filmbreak drop spreadingand evaporation [9ndash11] The liquid thickness in this zonewith the length of about several microns which is oftencalled the microregion decreases from 1ndash3 120583m to 10ndash20 nm(the adsorbed film) Significant curvature of the gas-liquidinterface in the microregion change in the thickness by 2-3orders ofmagnitude nonequilibriumeffects (the temperaturedrop at the interface) influence of van der Waals forcesand structure of the substrate cause considerable difficultiesfor both experimental and theoretical investigations of thisconfiguration
Theproblemof determining the evaporation rate and heatflux density in the heated liquid in the three-phase contactline region is actively studied in connection with importantpractical applications in power engineering medicine andchemical pharmaceutical and food industries [12ndash14] Boththeoretical and experimental studies prove that heat transferintensity in the contact line region can be more than an orderof magnitude higher than the average one and it causes alocal minimum in the temperature profile along the solidwall [15 16] However since it is impossible to perform thedirect measurements of the heat flux in the microregion(the length of which is estimated to be between 05 and 10ndash20120583m) various indirect methods and numerical proceduresare applied In Ibrahem et al [17] the meniscus of HFE7100liquid evaporated between two heated vertical plates of metalfoil The local heat flux to evaporating meniscus was cal-culated using the two-dimensional temperature distributionalong the outer foil edge obtained by the infrared scannerwith resolution of 148 microns It was found that the localheat flux density in the contact line region was 54ndash65 timeshigher than the average heat flux density on the surface
In the present study evaporation near contact lines isinvestigated using the sessile drop configuration There isa significant amount of literature on evaporating sessiledroplets [9 11 16] However the use of metal foil as aheater in droplet experiments is not common despite thefact that such configuration has potential for giving accuratemeasurements of the temperature field immediately underthe droplet and thus providing a tool for studies of localcoupling of the triple line motion and heat transfer inthe substrate with unprecedented accuracy We believe thatthis potential has not been fully realized For exampleSodtke et al [16] used a thin metal foil heater but reliedon thermochromic liquid crystals (TLCs) for temperaturemeasurementThe issues with TLCs are limited accuracy andlack of applicability outside of a relatively narrow temperaturerangeThe objective of the present study is to overcome thesedifficulties by using IR imaging Another limitation of thecurrent literature on droplets is that the vast majority of boththeoretical and experimental studies are focused on isolateddroplets while in applications one often encounters largearrays of dropletsThe issue of interaction of droplets remainspoorly understood and is discussed in the present study
CCD camera
Lenses
Droplet
Foil
MirrorPower supply
Light source
IR camera+ minus
Figure 1 The scheme of the experiment
2 Materials and Methods
21 Experiment The evaporating sessile water drops ona horizontal heated substrate were studied experimentallyusing the setup shown in Figure 1 The constantan foil(CuNi) of the thickness of 25 120583m size of 42 times 35mm2(119860 times 119861) and heat conductivity 120582 of 23WmK was used asthe substrate The substrate was heated by Joule heating Inexperiments the heating power 119876 varied from 025W to25WThe images of drop profile were obtained by the opticalshadow system with resolution of 8 120583mpx The geometricparameters of drops were determined using the images by theYoung-Laplace method in experiments they were as followsthe volume was 51ndash393 120583L base diameter (wetted spot)was 368ndash719mm height was 088ndash181mm and contactangle was 50ndash75∘ (average angle of 63∘) The temperatureof the bottom foil surface was measured by IR cameraTitanium 570M The thermographic study was performed atrecording frequency of 25Hz resolution of 640 times 512 pixels(108 120583mpx) and integration time of 805ms To increase theminimal resolvable temperature difference when using the IRcameras the reverse side of the foil was coated with a sootlayer The foil surface temperature measured in experimentswas in the range of 29ndash86∘C The average coefficient of heattransfer from the foil surface to surrounding air 120572av wasfound to be within 16ndash20Wm2K as seen in Figure 2 Thesemeasurements were performed without droplets on the foilsurface
22Theory and Computation To obtain accurate descriptionof heat transfer intensity near the contact line for evaporatingsessile droplets the Cauchy problem was formulated for theelliptic equation
120582Δ119879 + 119902V = 0 (1)
describing the heat conduction in the foil (ie the domaindefined by the equations 1198602 le 119909 le 1198602 1198612 le 119910 le1198612 and 0 le 119911 le 119867) Here 119879(119909 119910 119911) is temperature ofthe foil 120582 is (constant) thermal conductivity and 119902V is theconstant volume heat source On the bottom side of the foilthe temperature and the heat flux density were set as
119879 (119909 119910 0) = 119879119908(119909 119910)
120582120597119879
120597119911(119909 119910 0) = 120572av (119879119908 minus 119879119886)
(2)
International Journal of Aerospace Engineering 3
0 200 400 600 800 10000
5
10
15
20
25
120572(W
m2K)
qav (Wm2)
qav = W2S120572 = qav(Tw minus Ta)
Figure 2 The measured heat transfer coefficient for different heatflux densities 119878 area of one side of the foil 119882 power of electricheating of the foil 119879
119908 average measured temperature of the foil
surface and 119879119886 temperature of ambient air
The lateral walls were assumed adiabatic
120597119879
120597119909
10038161003816100381610038161003816100381610038161003816119909=plusmn1198602=120597119879
120597119910
10038161003816100381610038161003816100381610038161003816119910=plusmn1198612
= 0 (3)
The Cauchy problem for the Laplace equation or ellipticequations of the general form is one of the oldest ill-posedproblems [18ndash22] Its solution is unstable small variationsin the Cauchy data can lead to significant changes in thesolution If the existence of a bounded solution assumed itis possible to show the conditional stability of the problem[20] There are many different implementations of methodsof numerical solution to the Cauchy problem for ellipticequations (see eg [21ndash25]) Instability of solution to theCauchy problem becomes obvious when applying any ofnumerical methods to its solution In this work we use themethod that reduces the Cauchy problem solution to thesolution of the problem of moments [24 25] By a specialchoice of the boundary conditions in the conjugated problemcalculation of the heat flux value at the boundary partinaccessible for the measurements is reduced to summationof the corresponding series Simple formof the domain allowssolving the moment problem using the direct and inverseFourier transforms The regularizing procedure is used tosum the series of inverse Fourier transforms due to theinstability of the Cauchy problem The implemented methodof solution was applied to experimental data processing
3 Results and Discussion
With the use of the above method for solving the Cauchyproblem distributions of heat flux density on the foil surfacefrom the drop side were obtained from the thermographicmeasurements The initially measured temperature fieldcalculated distribution of heat flux density on the substrateand distribution of the local heat flux density along the linepassing through the central cross section of the drop areshown in Figure 3 The drop profile is also shown there The
results show that the maximum of the heat flux density isin the contact line region and it exceeds the average heatflux density on the entire foil surface by the factor of 5ndash7The average heat flux in the zone wetted by a drop exceedsthe average heat flux density on the entire foil surface bythe factor of 3ndash5 This can be explained by the heat influxfrom the foil periphery to the drop due to the relativelyhigh heat conductivity coefficient of the foil material andhigh evaporation rate in the contact line region Calculationof local heat flux density distribution without consideringredistribution of heat released in foil (line 2) is also shownin Figure 3 It can be seen that a negligence of substrateheat conductivity gives significantly underestimated heat fluxdensity and absence of peaks in the contact line region Dataare shown for two different sizes of drops at substantiallydifferent average heat flux density It can be seen that the ratioof maximal heat flux density in the contact line region andaverage heat flux density increases with a drop size decreaseit is equal to 37 for Figure 3(a) and 48 for Figure 3(b)
Previous results suggest that the heat flow from the drypart of the foil to the droplet plays an important role inthe overall heat transfer suggesting that evaporation ratesfor two or more droplets can be significantly different fromthe values predicted for a single droplet The results ofmeasurements and calculations for two sessile drops on thesubstrate are shown in Figure 4 the distance between thesedrops is 11mmThe symmetry conditions are broken for eachof these drops In the contact line region between drops wedo not observe the maxima of heat flux density because theheat influx from the periphery which is now split betweenthe two drops decreases for each of these dropsThemaximaof heat flux density are reached near the contact line at theouter edges of the two-droplet system the maximum value ishigher for the smaller droplet
The droplet on the heated substrate evaporates with timeThe evaporation rate can be evaluated using the available datathrough the heat flux density
119902ev = 119902119905 cos (120573) minus 119902conv (4)
Here 119902119905is the calculated heat flux density on the substrate
120573 is an angle of inclination of the drop surface to horizontalsubstrate and 119902conv is a convective heat flux from the dropletsurface
119902conv = 120572 (119879119904 minus 119879119886) (5)
It can be expressed through 119902119887 which is the heat flux from
the lower foil surface in the assumption that the temperatureof droplet surface slightly differs from the temperature ofsubstrate under it 119879
119904cong 119879119887 At that we should take into
account the difference between the area of droplet surface andsubstrate
119902conv = 119902119887 cos (120573) (6)
We obtain
119902ev = (119902119905 minus 119902119887) cos (120573) (7)
4 International Journal of Aerospace Engineeringq(W
m2)
1000
800
600
400
200
0
0 10 20 30
x (mm)
1
2
3
Thermal image1000
750
500
250
0
Heat flux
(Wm
2)
45
39
33
27
21
qt = 2qav minus qb
qb = 120572av(Tw minus Ta)
(∘C)
(a)
q(W
m2)
2000
1600
1200
400
800
0
0 10 20 30
x (mm)
1
2
3
Thermal image2000
1500
1000
500
0
Heat flux60
50
40
30
20
(Wm
2)
(∘C)
(b)
Figure 3 Distributions of heat flux density 1 Cauchy problem solution 2 calculation with neglecting of heat flux redistribution in the foil119902119905 3 on the bottom side of foil 119902
119887 (a) Drop 119863 = 66mm119867 = 182mm 119881 = 363 120583L 120579 = 65∘ and 119902av = 255Wm2 (b) Drop 119863 = 44mm
119867 = 115mm 119881 = 98 120583L 120579 = 59∘ and 119902av = 365Wm2
q(W
m2)
3600
2700
1800
900
0
x (mm)
66
59
52
45
38
0 10 200
1000
2000
3000
4000
2
Heat flux
1
Thermal image
(Wm
2)
(∘C)
Figure 4Distribution of heat flux density based onCauchy problemsolution for two drops 119902av = 653Wm2 (1) 119863 = 41mm 119867 =1085mm 119881 = 7120583L and 120579 = 58∘ (2) 119863 = 41mm119867 = 0903mm119881 = 573 120583L and 120579 = 52∘
The local evaporation rate can be calculated as
=119902ev119903lv=(119902119905minus 119902119887) cos (120573)119903lv
(8)
where 119903lv is the latent heat of vaporization Evaporationheat flux density and evaporation rate for the droplets ofdifferent diameters are shown in Figure 5 It can be seen thatevaporation rate near the contact line has a local maximumbut it is not so pronounced as heat flux density on thesubstrate in Figures 3 and 4
4 Conclusions
We have shown that the maximum of the heat flux densityis in the contact line region and it exceeds the average heatflux density on the entire foil surface by the factor of 5ndash7 In two-droplet system the maxima of heat flux densityare reached near the contact line at the outer edges of thetwo-droplet system the maximum value is higher for thesmaller droplet This can be explained by the heat influxfrom the foil periphery to the drop due to the relatively highheat conductivity coefficient of the foil material and highevaporation rate in the contact line region The possibility ofheat flux calculation by using infrared data gives interestingmethod of an experimental study of three-phase contact line
International Journal of Aerospace Engineering 5
2
1
1000
800
600
400
200
0
minus15 minus1 minus05 0 05 1 15
04
03
02
01
m (
gm
2s)
qev
(Wm
2)
2xD
Figure 5 Calculated distribution of evaporation heat flux densityand evaporation rate (1) drop from Figure 3(a) and (2) drop fromFigure 3(b)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The study was financially supported by the grant of theRussian Science Foundation (Project no 14-39-00003)
References
[1] E A Chinnov andO A Kabov ldquoTwo-phase flows in horizontalflat microchannelsrdquo Doklady Physics vol 57 no 1 pp 42ndash462012
[2] A Ababneh A Amirfazli and J A W Elliott ldquoEffect of gravityon the macroscopic advancing contact angle of sessile dropsrdquoCanadian Journal of Chemical Engineering vol 84 no 1 pp 39ndash43 2006
[3] D Brutin Z Zhu O Rahli J C Xie Q S Liu and LTadrist ldquoSessile drop in microgravity creation contact angleand interfacerdquoMicrogravity Science and Technology vol 21 no1 pp 67ndash76 2009
[4] G Abel G G Ross and L Andrzejewski ldquoWetting of a liquidsurface by another immiscible liquid inmicrogravityrdquoAdvancesin Space Research vol 33 no 8 pp 1431ndash1438 2004
[5] D Brutin Z-Q Zhu O Rahli J-C Xie Q-S Liu and LTadrist ldquoEvaporation of ethanol drops on a heated substrateunder microgravity conditionsrdquoMicrogravity Science and Tech-nology vol 22 no 3 pp 387ndash395 2010
[6] F Carle B Sobac and D Brutin ldquoHydrothermal waves onethanol droplets evaporating under terrestrial and reducedgravity levelsrdquo Journal of Fluid Mechanics vol 712 pp 614ndash6232012
[7] O A Kabov and D V Zaitsev ldquoThe effect of wetting hysteresison drop spreading under gravityrdquo Doklady Physics vol 58 no7 pp 292ndash295 2013
[8] F Carle B Sobac and D Brutin ldquoExperimental evidence ofthe atmospheric convective transport contribution to sessiledroplet evaporationrdquo Applied Physics Letters vol 102 no 6Article ID 061603 2013
[9] V K Dhir ldquoNumerical simulations of pool-boiling heat trans-ferrdquo AIChE Journal vol 47 no 4 pp 813ndash834 2001
[10] R J Braun ldquoDynamics of the tear filmsrdquoAnnual Review of FluidMechanics vol 44 pp 267ndash297 2012
[11] B Sobac and D Brutin ldquoTriple-line behavior and wettabilitycontrolled by nanocoated substrates influence on sessile dropevaporationrdquo Langmuir vol 27 no 24 pp 14999ndash15007 2011
[12] C L Moraila-Martınez M A Cabrerizo-Vılchez and M ARodrıguez-Valverde ldquoControlling the morphology of ring-likedeposits by varying the pinning time of driven receding contactlinesrdquo Interfacial Phenomena and Heat Transfer vol 1 no 1 pp195ndash205 2013
[13] A Kundan J L Plawsky and P C Wayner Jr ldquoThermophys-ical characteristics of a wickless heat pipe in microgravitymdashconstrained vapor bubble experimentrdquo International Journal ofHeat and Mass Transfer vol 78 pp 1105ndash1113 2014
[14] V S Ajaev E Y Gatapova and O A Kabov ldquoRupture of thinliquid films on structured surfacesrdquo Physical Review E vol 84no 4 Article ID 041606 2011
[15] SMoosman andGMHomsy ldquoEvaporatingmenisci of wettingfluidsrdquo Journal of Colloid and Interface Science vol 73 no 1 pp212ndash223 1980
[16] C Sodtke V S Ajaev and P Stephan ldquoDynamics of volatileliquid droplets on heated surfaces theory versus experimentrdquoJournal of Fluid Mechanics vol 610 pp 343ndash362 2008
[17] K Ibrahem M F Abd Rabbo T Gambaryan-Roisman and PStephan ldquoExperimental investigation of evaporative heat trans-fer characteristics at the 3-phase contact linerdquo ExperimentalThermal and Fluid Science vol 34 no 8 pp 1036ndash1041 2010
[18] J Hadamard Lectures on the Cauchy Problem in Linear Differen-tial Equations Yale University Press New Haven Conn USA1923
[19] L E Payne ldquoBounds in the Cauchy problem for the Laplaceequationrdquo Archive for Rational Mechanics and Analysis vol 5pp 35ndash45 1960
[20] M Lavrentev V Romanov and S Shishatskii Ill-Posed Problemsof Mathematical Physics and Analysis American MathematicalSociety 1986
[21] F Berntsson and L Eiden ldquoNumerical solution of a Cauchyproblem for the Laplace equationrdquo Inverse Problems vol 17 no4 pp 839ndash853 2001
[22] L Bourgeois and J Darde ldquoA duality-based method of quasi-reversibility to solve the Cauchy problem in the presence ofnoisy datardquo Inverse Problems vol 26 no 9 Article ID 0950162010
[23] C-L Fu H-F Li Z Qian and X-T Xiong ldquoFourier regular-ization method for solving a Cauchy problem for the Laplaceequationrdquo Inverse Problems in Science and Engineering vol 16no 2 pp 159ndash169 2008
[24] T Reginska and A Wakulicz ldquoWavelet moment method forthe Cauchy problem for the Helmholtz equationrdquo Journal ofComputational andAppliedMathematics vol 223 no 1 pp 218ndash229 2009
[25] A L Karchevsky ldquoReformulation of an inverse problem state-ment that reduces computational costsrdquo Eurasian Journal ofMathematical and Computer Applications vol 1 no 2 pp 5ndash202013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 3
0 200 400 600 800 10000
5
10
15
20
25
120572(W
m2K)
qav (Wm2)
qav = W2S120572 = qav(Tw minus Ta)
Figure 2 The measured heat transfer coefficient for different heatflux densities 119878 area of one side of the foil 119882 power of electricheating of the foil 119879
119908 average measured temperature of the foil
surface and 119879119886 temperature of ambient air
The lateral walls were assumed adiabatic
120597119879
120597119909
10038161003816100381610038161003816100381610038161003816119909=plusmn1198602=120597119879
120597119910
10038161003816100381610038161003816100381610038161003816119910=plusmn1198612
= 0 (3)
The Cauchy problem for the Laplace equation or ellipticequations of the general form is one of the oldest ill-posedproblems [18ndash22] Its solution is unstable small variationsin the Cauchy data can lead to significant changes in thesolution If the existence of a bounded solution assumed itis possible to show the conditional stability of the problem[20] There are many different implementations of methodsof numerical solution to the Cauchy problem for ellipticequations (see eg [21ndash25]) Instability of solution to theCauchy problem becomes obvious when applying any ofnumerical methods to its solution In this work we use themethod that reduces the Cauchy problem solution to thesolution of the problem of moments [24 25] By a specialchoice of the boundary conditions in the conjugated problemcalculation of the heat flux value at the boundary partinaccessible for the measurements is reduced to summationof the corresponding series Simple formof the domain allowssolving the moment problem using the direct and inverseFourier transforms The regularizing procedure is used tosum the series of inverse Fourier transforms due to theinstability of the Cauchy problem The implemented methodof solution was applied to experimental data processing
3 Results and Discussion
With the use of the above method for solving the Cauchyproblem distributions of heat flux density on the foil surfacefrom the drop side were obtained from the thermographicmeasurements The initially measured temperature fieldcalculated distribution of heat flux density on the substrateand distribution of the local heat flux density along the linepassing through the central cross section of the drop areshown in Figure 3 The drop profile is also shown there The
results show that the maximum of the heat flux density isin the contact line region and it exceeds the average heatflux density on the entire foil surface by the factor of 5ndash7The average heat flux in the zone wetted by a drop exceedsthe average heat flux density on the entire foil surface bythe factor of 3ndash5 This can be explained by the heat influxfrom the foil periphery to the drop due to the relativelyhigh heat conductivity coefficient of the foil material andhigh evaporation rate in the contact line region Calculationof local heat flux density distribution without consideringredistribution of heat released in foil (line 2) is also shownin Figure 3 It can be seen that a negligence of substrateheat conductivity gives significantly underestimated heat fluxdensity and absence of peaks in the contact line region Dataare shown for two different sizes of drops at substantiallydifferent average heat flux density It can be seen that the ratioof maximal heat flux density in the contact line region andaverage heat flux density increases with a drop size decreaseit is equal to 37 for Figure 3(a) and 48 for Figure 3(b)
Previous results suggest that the heat flow from the drypart of the foil to the droplet plays an important role inthe overall heat transfer suggesting that evaporation ratesfor two or more droplets can be significantly different fromthe values predicted for a single droplet The results ofmeasurements and calculations for two sessile drops on thesubstrate are shown in Figure 4 the distance between thesedrops is 11mmThe symmetry conditions are broken for eachof these drops In the contact line region between drops wedo not observe the maxima of heat flux density because theheat influx from the periphery which is now split betweenthe two drops decreases for each of these dropsThemaximaof heat flux density are reached near the contact line at theouter edges of the two-droplet system the maximum value ishigher for the smaller droplet
The droplet on the heated substrate evaporates with timeThe evaporation rate can be evaluated using the available datathrough the heat flux density
119902ev = 119902119905 cos (120573) minus 119902conv (4)
Here 119902119905is the calculated heat flux density on the substrate
120573 is an angle of inclination of the drop surface to horizontalsubstrate and 119902conv is a convective heat flux from the dropletsurface
119902conv = 120572 (119879119904 minus 119879119886) (5)
It can be expressed through 119902119887 which is the heat flux from
the lower foil surface in the assumption that the temperatureof droplet surface slightly differs from the temperature ofsubstrate under it 119879
119904cong 119879119887 At that we should take into
account the difference between the area of droplet surface andsubstrate
119902conv = 119902119887 cos (120573) (6)
We obtain
119902ev = (119902119905 minus 119902119887) cos (120573) (7)
4 International Journal of Aerospace Engineeringq(W
m2)
1000
800
600
400
200
0
0 10 20 30
x (mm)
1
2
3
Thermal image1000
750
500
250
0
Heat flux
(Wm
2)
45
39
33
27
21
qt = 2qav minus qb
qb = 120572av(Tw minus Ta)
(∘C)
(a)
q(W
m2)
2000
1600
1200
400
800
0
0 10 20 30
x (mm)
1
2
3
Thermal image2000
1500
1000
500
0
Heat flux60
50
40
30
20
(Wm
2)
(∘C)
(b)
Figure 3 Distributions of heat flux density 1 Cauchy problem solution 2 calculation with neglecting of heat flux redistribution in the foil119902119905 3 on the bottom side of foil 119902
119887 (a) Drop 119863 = 66mm119867 = 182mm 119881 = 363 120583L 120579 = 65∘ and 119902av = 255Wm2 (b) Drop 119863 = 44mm
119867 = 115mm 119881 = 98 120583L 120579 = 59∘ and 119902av = 365Wm2
q(W
m2)
3600
2700
1800
900
0
x (mm)
66
59
52
45
38
0 10 200
1000
2000
3000
4000
2
Heat flux
1
Thermal image
(Wm
2)
(∘C)
Figure 4Distribution of heat flux density based onCauchy problemsolution for two drops 119902av = 653Wm2 (1) 119863 = 41mm 119867 =1085mm 119881 = 7120583L and 120579 = 58∘ (2) 119863 = 41mm119867 = 0903mm119881 = 573 120583L and 120579 = 52∘
The local evaporation rate can be calculated as
=119902ev119903lv=(119902119905minus 119902119887) cos (120573)119903lv
(8)
where 119903lv is the latent heat of vaporization Evaporationheat flux density and evaporation rate for the droplets ofdifferent diameters are shown in Figure 5 It can be seen thatevaporation rate near the contact line has a local maximumbut it is not so pronounced as heat flux density on thesubstrate in Figures 3 and 4
4 Conclusions
We have shown that the maximum of the heat flux densityis in the contact line region and it exceeds the average heatflux density on the entire foil surface by the factor of 5ndash7 In two-droplet system the maxima of heat flux densityare reached near the contact line at the outer edges of thetwo-droplet system the maximum value is higher for thesmaller droplet This can be explained by the heat influxfrom the foil periphery to the drop due to the relatively highheat conductivity coefficient of the foil material and highevaporation rate in the contact line region The possibility ofheat flux calculation by using infrared data gives interestingmethod of an experimental study of three-phase contact line
International Journal of Aerospace Engineering 5
2
1
1000
800
600
400
200
0
minus15 minus1 minus05 0 05 1 15
04
03
02
01
m (
gm
2s)
qev
(Wm
2)
2xD
Figure 5 Calculated distribution of evaporation heat flux densityand evaporation rate (1) drop from Figure 3(a) and (2) drop fromFigure 3(b)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The study was financially supported by the grant of theRussian Science Foundation (Project no 14-39-00003)
References
[1] E A Chinnov andO A Kabov ldquoTwo-phase flows in horizontalflat microchannelsrdquo Doklady Physics vol 57 no 1 pp 42ndash462012
[2] A Ababneh A Amirfazli and J A W Elliott ldquoEffect of gravityon the macroscopic advancing contact angle of sessile dropsrdquoCanadian Journal of Chemical Engineering vol 84 no 1 pp 39ndash43 2006
[3] D Brutin Z Zhu O Rahli J C Xie Q S Liu and LTadrist ldquoSessile drop in microgravity creation contact angleand interfacerdquoMicrogravity Science and Technology vol 21 no1 pp 67ndash76 2009
[4] G Abel G G Ross and L Andrzejewski ldquoWetting of a liquidsurface by another immiscible liquid inmicrogravityrdquoAdvancesin Space Research vol 33 no 8 pp 1431ndash1438 2004
[5] D Brutin Z-Q Zhu O Rahli J-C Xie Q-S Liu and LTadrist ldquoEvaporation of ethanol drops on a heated substrateunder microgravity conditionsrdquoMicrogravity Science and Tech-nology vol 22 no 3 pp 387ndash395 2010
[6] F Carle B Sobac and D Brutin ldquoHydrothermal waves onethanol droplets evaporating under terrestrial and reducedgravity levelsrdquo Journal of Fluid Mechanics vol 712 pp 614ndash6232012
[7] O A Kabov and D V Zaitsev ldquoThe effect of wetting hysteresison drop spreading under gravityrdquo Doklady Physics vol 58 no7 pp 292ndash295 2013
[8] F Carle B Sobac and D Brutin ldquoExperimental evidence ofthe atmospheric convective transport contribution to sessiledroplet evaporationrdquo Applied Physics Letters vol 102 no 6Article ID 061603 2013
[9] V K Dhir ldquoNumerical simulations of pool-boiling heat trans-ferrdquo AIChE Journal vol 47 no 4 pp 813ndash834 2001
[10] R J Braun ldquoDynamics of the tear filmsrdquoAnnual Review of FluidMechanics vol 44 pp 267ndash297 2012
[11] B Sobac and D Brutin ldquoTriple-line behavior and wettabilitycontrolled by nanocoated substrates influence on sessile dropevaporationrdquo Langmuir vol 27 no 24 pp 14999ndash15007 2011
[12] C L Moraila-Martınez M A Cabrerizo-Vılchez and M ARodrıguez-Valverde ldquoControlling the morphology of ring-likedeposits by varying the pinning time of driven receding contactlinesrdquo Interfacial Phenomena and Heat Transfer vol 1 no 1 pp195ndash205 2013
[13] A Kundan J L Plawsky and P C Wayner Jr ldquoThermophys-ical characteristics of a wickless heat pipe in microgravitymdashconstrained vapor bubble experimentrdquo International Journal ofHeat and Mass Transfer vol 78 pp 1105ndash1113 2014
[14] V S Ajaev E Y Gatapova and O A Kabov ldquoRupture of thinliquid films on structured surfacesrdquo Physical Review E vol 84no 4 Article ID 041606 2011
[15] SMoosman andGMHomsy ldquoEvaporatingmenisci of wettingfluidsrdquo Journal of Colloid and Interface Science vol 73 no 1 pp212ndash223 1980
[16] C Sodtke V S Ajaev and P Stephan ldquoDynamics of volatileliquid droplets on heated surfaces theory versus experimentrdquoJournal of Fluid Mechanics vol 610 pp 343ndash362 2008
[17] K Ibrahem M F Abd Rabbo T Gambaryan-Roisman and PStephan ldquoExperimental investigation of evaporative heat trans-fer characteristics at the 3-phase contact linerdquo ExperimentalThermal and Fluid Science vol 34 no 8 pp 1036ndash1041 2010
[18] J Hadamard Lectures on the Cauchy Problem in Linear Differen-tial Equations Yale University Press New Haven Conn USA1923
[19] L E Payne ldquoBounds in the Cauchy problem for the Laplaceequationrdquo Archive for Rational Mechanics and Analysis vol 5pp 35ndash45 1960
[20] M Lavrentev V Romanov and S Shishatskii Ill-Posed Problemsof Mathematical Physics and Analysis American MathematicalSociety 1986
[21] F Berntsson and L Eiden ldquoNumerical solution of a Cauchyproblem for the Laplace equationrdquo Inverse Problems vol 17 no4 pp 839ndash853 2001
[22] L Bourgeois and J Darde ldquoA duality-based method of quasi-reversibility to solve the Cauchy problem in the presence ofnoisy datardquo Inverse Problems vol 26 no 9 Article ID 0950162010
[23] C-L Fu H-F Li Z Qian and X-T Xiong ldquoFourier regular-ization method for solving a Cauchy problem for the Laplaceequationrdquo Inverse Problems in Science and Engineering vol 16no 2 pp 159ndash169 2008
[24] T Reginska and A Wakulicz ldquoWavelet moment method forthe Cauchy problem for the Helmholtz equationrdquo Journal ofComputational andAppliedMathematics vol 223 no 1 pp 218ndash229 2009
[25] A L Karchevsky ldquoReformulation of an inverse problem state-ment that reduces computational costsrdquo Eurasian Journal ofMathematical and Computer Applications vol 1 no 2 pp 5ndash202013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 International Journal of Aerospace Engineeringq(W
m2)
1000
800
600
400
200
0
0 10 20 30
x (mm)
1
2
3
Thermal image1000
750
500
250
0
Heat flux
(Wm
2)
45
39
33
27
21
qt = 2qav minus qb
qb = 120572av(Tw minus Ta)
(∘C)
(a)
q(W
m2)
2000
1600
1200
400
800
0
0 10 20 30
x (mm)
1
2
3
Thermal image2000
1500
1000
500
0
Heat flux60
50
40
30
20
(Wm
2)
(∘C)
(b)
Figure 3 Distributions of heat flux density 1 Cauchy problem solution 2 calculation with neglecting of heat flux redistribution in the foil119902119905 3 on the bottom side of foil 119902
119887 (a) Drop 119863 = 66mm119867 = 182mm 119881 = 363 120583L 120579 = 65∘ and 119902av = 255Wm2 (b) Drop 119863 = 44mm
119867 = 115mm 119881 = 98 120583L 120579 = 59∘ and 119902av = 365Wm2
q(W
m2)
3600
2700
1800
900
0
x (mm)
66
59
52
45
38
0 10 200
1000
2000
3000
4000
2
Heat flux
1
Thermal image
(Wm
2)
(∘C)
Figure 4Distribution of heat flux density based onCauchy problemsolution for two drops 119902av = 653Wm2 (1) 119863 = 41mm 119867 =1085mm 119881 = 7120583L and 120579 = 58∘ (2) 119863 = 41mm119867 = 0903mm119881 = 573 120583L and 120579 = 52∘
The local evaporation rate can be calculated as
=119902ev119903lv=(119902119905minus 119902119887) cos (120573)119903lv
(8)
where 119903lv is the latent heat of vaporization Evaporationheat flux density and evaporation rate for the droplets ofdifferent diameters are shown in Figure 5 It can be seen thatevaporation rate near the contact line has a local maximumbut it is not so pronounced as heat flux density on thesubstrate in Figures 3 and 4
4 Conclusions
We have shown that the maximum of the heat flux densityis in the contact line region and it exceeds the average heatflux density on the entire foil surface by the factor of 5ndash7 In two-droplet system the maxima of heat flux densityare reached near the contact line at the outer edges of thetwo-droplet system the maximum value is higher for thesmaller droplet This can be explained by the heat influxfrom the foil periphery to the drop due to the relatively highheat conductivity coefficient of the foil material and highevaporation rate in the contact line region The possibility ofheat flux calculation by using infrared data gives interestingmethod of an experimental study of three-phase contact line
International Journal of Aerospace Engineering 5
2
1
1000
800
600
400
200
0
minus15 minus1 minus05 0 05 1 15
04
03
02
01
m (
gm
2s)
qev
(Wm
2)
2xD
Figure 5 Calculated distribution of evaporation heat flux densityand evaporation rate (1) drop from Figure 3(a) and (2) drop fromFigure 3(b)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The study was financially supported by the grant of theRussian Science Foundation (Project no 14-39-00003)
References
[1] E A Chinnov andO A Kabov ldquoTwo-phase flows in horizontalflat microchannelsrdquo Doklady Physics vol 57 no 1 pp 42ndash462012
[2] A Ababneh A Amirfazli and J A W Elliott ldquoEffect of gravityon the macroscopic advancing contact angle of sessile dropsrdquoCanadian Journal of Chemical Engineering vol 84 no 1 pp 39ndash43 2006
[3] D Brutin Z Zhu O Rahli J C Xie Q S Liu and LTadrist ldquoSessile drop in microgravity creation contact angleand interfacerdquoMicrogravity Science and Technology vol 21 no1 pp 67ndash76 2009
[4] G Abel G G Ross and L Andrzejewski ldquoWetting of a liquidsurface by another immiscible liquid inmicrogravityrdquoAdvancesin Space Research vol 33 no 8 pp 1431ndash1438 2004
[5] D Brutin Z-Q Zhu O Rahli J-C Xie Q-S Liu and LTadrist ldquoEvaporation of ethanol drops on a heated substrateunder microgravity conditionsrdquoMicrogravity Science and Tech-nology vol 22 no 3 pp 387ndash395 2010
[6] F Carle B Sobac and D Brutin ldquoHydrothermal waves onethanol droplets evaporating under terrestrial and reducedgravity levelsrdquo Journal of Fluid Mechanics vol 712 pp 614ndash6232012
[7] O A Kabov and D V Zaitsev ldquoThe effect of wetting hysteresison drop spreading under gravityrdquo Doklady Physics vol 58 no7 pp 292ndash295 2013
[8] F Carle B Sobac and D Brutin ldquoExperimental evidence ofthe atmospheric convective transport contribution to sessiledroplet evaporationrdquo Applied Physics Letters vol 102 no 6Article ID 061603 2013
[9] V K Dhir ldquoNumerical simulations of pool-boiling heat trans-ferrdquo AIChE Journal vol 47 no 4 pp 813ndash834 2001
[10] R J Braun ldquoDynamics of the tear filmsrdquoAnnual Review of FluidMechanics vol 44 pp 267ndash297 2012
[11] B Sobac and D Brutin ldquoTriple-line behavior and wettabilitycontrolled by nanocoated substrates influence on sessile dropevaporationrdquo Langmuir vol 27 no 24 pp 14999ndash15007 2011
[12] C L Moraila-Martınez M A Cabrerizo-Vılchez and M ARodrıguez-Valverde ldquoControlling the morphology of ring-likedeposits by varying the pinning time of driven receding contactlinesrdquo Interfacial Phenomena and Heat Transfer vol 1 no 1 pp195ndash205 2013
[13] A Kundan J L Plawsky and P C Wayner Jr ldquoThermophys-ical characteristics of a wickless heat pipe in microgravitymdashconstrained vapor bubble experimentrdquo International Journal ofHeat and Mass Transfer vol 78 pp 1105ndash1113 2014
[14] V S Ajaev E Y Gatapova and O A Kabov ldquoRupture of thinliquid films on structured surfacesrdquo Physical Review E vol 84no 4 Article ID 041606 2011
[15] SMoosman andGMHomsy ldquoEvaporatingmenisci of wettingfluidsrdquo Journal of Colloid and Interface Science vol 73 no 1 pp212ndash223 1980
[16] C Sodtke V S Ajaev and P Stephan ldquoDynamics of volatileliquid droplets on heated surfaces theory versus experimentrdquoJournal of Fluid Mechanics vol 610 pp 343ndash362 2008
[17] K Ibrahem M F Abd Rabbo T Gambaryan-Roisman and PStephan ldquoExperimental investigation of evaporative heat trans-fer characteristics at the 3-phase contact linerdquo ExperimentalThermal and Fluid Science vol 34 no 8 pp 1036ndash1041 2010
[18] J Hadamard Lectures on the Cauchy Problem in Linear Differen-tial Equations Yale University Press New Haven Conn USA1923
[19] L E Payne ldquoBounds in the Cauchy problem for the Laplaceequationrdquo Archive for Rational Mechanics and Analysis vol 5pp 35ndash45 1960
[20] M Lavrentev V Romanov and S Shishatskii Ill-Posed Problemsof Mathematical Physics and Analysis American MathematicalSociety 1986
[21] F Berntsson and L Eiden ldquoNumerical solution of a Cauchyproblem for the Laplace equationrdquo Inverse Problems vol 17 no4 pp 839ndash853 2001
[22] L Bourgeois and J Darde ldquoA duality-based method of quasi-reversibility to solve the Cauchy problem in the presence ofnoisy datardquo Inverse Problems vol 26 no 9 Article ID 0950162010
[23] C-L Fu H-F Li Z Qian and X-T Xiong ldquoFourier regular-ization method for solving a Cauchy problem for the Laplaceequationrdquo Inverse Problems in Science and Engineering vol 16no 2 pp 159ndash169 2008
[24] T Reginska and A Wakulicz ldquoWavelet moment method forthe Cauchy problem for the Helmholtz equationrdquo Journal ofComputational andAppliedMathematics vol 223 no 1 pp 218ndash229 2009
[25] A L Karchevsky ldquoReformulation of an inverse problem state-ment that reduces computational costsrdquo Eurasian Journal ofMathematical and Computer Applications vol 1 no 2 pp 5ndash202013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 5
2
1
1000
800
600
400
200
0
minus15 minus1 minus05 0 05 1 15
04
03
02
01
m (
gm
2s)
qev
(Wm
2)
2xD
Figure 5 Calculated distribution of evaporation heat flux densityand evaporation rate (1) drop from Figure 3(a) and (2) drop fromFigure 3(b)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The study was financially supported by the grant of theRussian Science Foundation (Project no 14-39-00003)
References
[1] E A Chinnov andO A Kabov ldquoTwo-phase flows in horizontalflat microchannelsrdquo Doklady Physics vol 57 no 1 pp 42ndash462012
[2] A Ababneh A Amirfazli and J A W Elliott ldquoEffect of gravityon the macroscopic advancing contact angle of sessile dropsrdquoCanadian Journal of Chemical Engineering vol 84 no 1 pp 39ndash43 2006
[3] D Brutin Z Zhu O Rahli J C Xie Q S Liu and LTadrist ldquoSessile drop in microgravity creation contact angleand interfacerdquoMicrogravity Science and Technology vol 21 no1 pp 67ndash76 2009
[4] G Abel G G Ross and L Andrzejewski ldquoWetting of a liquidsurface by another immiscible liquid inmicrogravityrdquoAdvancesin Space Research vol 33 no 8 pp 1431ndash1438 2004
[5] D Brutin Z-Q Zhu O Rahli J-C Xie Q-S Liu and LTadrist ldquoEvaporation of ethanol drops on a heated substrateunder microgravity conditionsrdquoMicrogravity Science and Tech-nology vol 22 no 3 pp 387ndash395 2010
[6] F Carle B Sobac and D Brutin ldquoHydrothermal waves onethanol droplets evaporating under terrestrial and reducedgravity levelsrdquo Journal of Fluid Mechanics vol 712 pp 614ndash6232012
[7] O A Kabov and D V Zaitsev ldquoThe effect of wetting hysteresison drop spreading under gravityrdquo Doklady Physics vol 58 no7 pp 292ndash295 2013
[8] F Carle B Sobac and D Brutin ldquoExperimental evidence ofthe atmospheric convective transport contribution to sessiledroplet evaporationrdquo Applied Physics Letters vol 102 no 6Article ID 061603 2013
[9] V K Dhir ldquoNumerical simulations of pool-boiling heat trans-ferrdquo AIChE Journal vol 47 no 4 pp 813ndash834 2001
[10] R J Braun ldquoDynamics of the tear filmsrdquoAnnual Review of FluidMechanics vol 44 pp 267ndash297 2012
[11] B Sobac and D Brutin ldquoTriple-line behavior and wettabilitycontrolled by nanocoated substrates influence on sessile dropevaporationrdquo Langmuir vol 27 no 24 pp 14999ndash15007 2011
[12] C L Moraila-Martınez M A Cabrerizo-Vılchez and M ARodrıguez-Valverde ldquoControlling the morphology of ring-likedeposits by varying the pinning time of driven receding contactlinesrdquo Interfacial Phenomena and Heat Transfer vol 1 no 1 pp195ndash205 2013
[13] A Kundan J L Plawsky and P C Wayner Jr ldquoThermophys-ical characteristics of a wickless heat pipe in microgravitymdashconstrained vapor bubble experimentrdquo International Journal ofHeat and Mass Transfer vol 78 pp 1105ndash1113 2014
[14] V S Ajaev E Y Gatapova and O A Kabov ldquoRupture of thinliquid films on structured surfacesrdquo Physical Review E vol 84no 4 Article ID 041606 2011
[15] SMoosman andGMHomsy ldquoEvaporatingmenisci of wettingfluidsrdquo Journal of Colloid and Interface Science vol 73 no 1 pp212ndash223 1980
[16] C Sodtke V S Ajaev and P Stephan ldquoDynamics of volatileliquid droplets on heated surfaces theory versus experimentrdquoJournal of Fluid Mechanics vol 610 pp 343ndash362 2008
[17] K Ibrahem M F Abd Rabbo T Gambaryan-Roisman and PStephan ldquoExperimental investigation of evaporative heat trans-fer characteristics at the 3-phase contact linerdquo ExperimentalThermal and Fluid Science vol 34 no 8 pp 1036ndash1041 2010
[18] J Hadamard Lectures on the Cauchy Problem in Linear Differen-tial Equations Yale University Press New Haven Conn USA1923
[19] L E Payne ldquoBounds in the Cauchy problem for the Laplaceequationrdquo Archive for Rational Mechanics and Analysis vol 5pp 35ndash45 1960
[20] M Lavrentev V Romanov and S Shishatskii Ill-Posed Problemsof Mathematical Physics and Analysis American MathematicalSociety 1986
[21] F Berntsson and L Eiden ldquoNumerical solution of a Cauchyproblem for the Laplace equationrdquo Inverse Problems vol 17 no4 pp 839ndash853 2001
[22] L Bourgeois and J Darde ldquoA duality-based method of quasi-reversibility to solve the Cauchy problem in the presence ofnoisy datardquo Inverse Problems vol 26 no 9 Article ID 0950162010
[23] C-L Fu H-F Li Z Qian and X-T Xiong ldquoFourier regular-ization method for solving a Cauchy problem for the Laplaceequationrdquo Inverse Problems in Science and Engineering vol 16no 2 pp 159ndash169 2008
[24] T Reginska and A Wakulicz ldquoWavelet moment method forthe Cauchy problem for the Helmholtz equationrdquo Journal ofComputational andAppliedMathematics vol 223 no 1 pp 218ndash229 2009
[25] A L Karchevsky ldquoReformulation of an inverse problem state-ment that reduces computational costsrdquo Eurasian Journal ofMathematical and Computer Applications vol 1 no 2 pp 5ndash202013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Related Documents
