International Journal of Heat and Mass Transfer 127 (2018) 660–664

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Technical Note

Suppression of the condensational growth of droplets of a levitatingcluster using the modulation of the laser heating power

https://doi.org/10.1016/j.ijheatmasstransfer.2018.07.0550017-9310/� 2018 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: ldombr@yandex.ru (L.A. Dombrovsky).

Alexander A. Fedorets a, Nurken E. Aktaev a, Leonid A. Dombrovsky a,b,⇑aUniversity of Tyumen, 6 Volodarskogo St, Tyumen 625003, Russiab Joint Institute for High Temperatures, 17A Krasnokazarmennaya St, Moscow 111116, Russia

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 June 2018Received in revised form 6 July 2018Accepted 10 July 2018

Keywords:Droplet clusterLevitationEvaporationCondensationCluster stabilization

The relatively stable clusters of regularly positioned small droplets formed over the locally heated watersurface have been studied starting from the early paper by the first author. The life of a cluster appearedto be not long because of growing of droplets due to condensation of steam and the final coalescence oflarge droplets with the substrate layer of water. However, further laboratory studies of biochemical pro-cesses in single droplets will be possible only in the case of their longer life. One way to stabilize the clus-ter suggested recently by the authors is an external infrared heating which prevents growing of droplets.The present paper is focused on another way to stabilize the cluster. This is an expected rebuilding of dro-plet clusters at periodic changes in power of the heating laser beam. The frequency of these oscillations ismore important than their amplitude because of the resonance nature of the phenomenon. The experi-ments showed that in the simplest variant of alternating constant power values with the same durationof heating, the droplet growth rate is approximately twice reduced with a power modulation period ofabout 0.9 s. It means that such a ‘‘rejuvenescence” procedure can be used at laboratory conditions to pro-long the life of levitating droplet clusters.

� 2018 Elsevier Ltd. All rights reserved.

1. Introduction

A wide variety of processes in the atmosphere and also indiverse engineering applications are related with a behaviour ofliquid aerosols which can form sprays, mists, and clouds. In manycases, it is important to study physical, chemical or bio-chemicalprocesses in small single droplets of size about several microme-tres. It is interesting that specific conditions inside small dropletsincluding the processes induced by possible external irradiationand also relative independence of these internal processes ofthermo-chemical conditions and flow field in a surrounding gasmedium enable one to consider these droplets as unique micro-reactors. It is expected that even well-studied chemical and bio-chemical processes have unusual features in these reactors. Thekey problem of any laboratory work with relatively stable smalldroplets is their spatial localization which takes place in self-assembled regular structures of the so-called droplet clusters levi-tating above the locally heated surface of a liquid [1].

The levitating droplet clusters have been observed anddescribed for the first time in early papers [2,3]. The behaviour ofthe levitating single droplets and droplet clusters in the upcoming

flow of vapour of various liquids and entrained air has been studiedexperimentally. The detailed laboratory observations and theoret-ical analysis appeared to be useful for understanding this fascinat-ing phenomenon [4–12]. The experiments with clusters of waterdroplets showed that the life of a cluster appeared to be usuallynot long because of growing of droplets due to condensation ofsteam and the final coalescence of large droplets with the substratelayer of water.

An original method for stabilization of the cluster has been sug-gested by the authors in paper [13]. Theoretical predictions on pos-sible stabilization of levitating droplet clusters with the use of aninfrared irradiation based on estimates of infrared absorption bysemi-transparent water droplets [14,15] were qualitatively con-firmed in the laboratory experiments. It was proven experimen-tally that an external infrared heating of the cluster preventsgrowing of droplets and can be really used to stabilize droplet clus-ters due to a desirable balance between evaporation and condensa-tion over the droplet surface. The required infrared radiationpower obtained in the experiments appeared to be relatively smalland directly proportional to the laser power used for heating thewater layer [13]. Unfortunately, the power of compact sources ofthe near-infrared radiation is rather low and they have to be usedin the forced mode, which negatively affects the resource of work.

Fig. 2. Schematics of a side view of the laboratory set-up: 1 – droplet cluster,2 – water layer, 3 – sitall substrate with a black bottom surface, 4 – laser beam,5 – MEMS mirror, 6 – lens of microscope/thermal imaging.

Nomenclature

d thicknessS area of droplet surface_S rate of increase of droplet surface areaT temperaturet current timeW laser power

Greek symbolss characteristic time period

Subscripts and superscriptsmax maximummin minimumrel relaxationw power0 average value

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This is an additional motivation of a search for alternative methodsfor the cluster stabilization.

Another idea of possible stabilization of droplet clusters hasbeen also considered by the authors. However, this alternativeapproach was not confirmed in preliminary experiments becauseof insufficient information on dynamics of a response of dropletclusters to periodic changes in power of the heating laser beam.It is expected that the resulting partial rebuilding at every changeof laser irradiation power can be used to slow or even suppress thegrowth of droplets. One cannot exclude that an oscillating power ofthe heating laser is a way to stabilize the cluster. The period ofthese oscillations is expected to be the most important parameterof the problem, whereas a small amplitude is not so important. Thepresent paper is concerned with a particular study on behaviour ofdroplet clusters to clarify the above assumed possibility.

It should be recalled that recent experiments of [16] showedthat a step-wise variation of the heating laser power can be usedto generate the clusters with the expected (and relatively small)number of monodisperse droplets (see Fig. 1) In the present paper,the authors focused on reaction of condensational rate of growth ofthe cluster droplets to the periodic time variation of the heatinglaser power with a relatively small amplitude. This laboratorystudy was undertaken for the first time.

2. Experimental procedure

The laboratory equipment described in paper [12] has beenadditionally modified. The new powerful laser Omicron LaserageBrixX� 808-800HP (Germany) and a special MEMS mirror SercaloMM 2536 (Switzerland) with variable tilt angle (see Fig. 2) wereused in a combination with an appropriate present-day software.

Fig. 1. Droplet clusters containing (a) 11 an

The main characteristics of the single-mode laser: the wavelengthis 808 ± 5 nm, the maximum power is 800 mW, the beam diameteris 0.76 mm, and the beam angular divergence 1.02 mrad.

These modifications enable us to vary accurately the laserpower and also to move the laser beam along the substrate surface,if necessary [18]. The current power of laser was measured by thedevice PM200 complete with sensor S401 (Thorlabs, USA). Thispower was periodically changed with the use of a computer codethat controls the laser. The period of laser power variation, sw,was measured in the experiments.

d (b) 16 identical water droplets [16].

662 A.A. Fedorets et al. / International Journal of Heat and Mass Transfer 127 (2018) 660–664

The thickness of distilled water layer (containing naturaladmixtures of surfactants) was equal to 400 ± 2 lm in all experi-ments. Since special measures for obtaining ultrapure water werenot undertaken, the water contained uncontrolled micro-impurities of surfactants. Their concentration ensured completesuppression of a thermo-capillary flow in the layer. The thicknessof water layer was controlled using the confocal chromatic sensorIFC2451 made by the company Micro-Epsilon (USA), the cuvettetemperature was equal to 20 ± 0.5 �C. The temperature field ofwater surface was measured by thermal imaging Flir A655sc(USA) with the lens Close-up IR 2.9�. The main parameters of thisdevice are: the spectral range of 7.5–14 µm and the matrix of 640� 480 pixels at the pixel size of 50 � 50 µm. Video images of thecluster were taken using stereomicroscope Zeiss AXIO Zoom. V16(Germany) and high-speed (1000 frames per second) video-camera PCO.EDGE 5.5C (Germany) with spatial resolution of 0.6µm. The average speed of increasing the surface area of the five lar-gest droplets in the central part of the cluster was measured fromvideo recordings (a procedure of these measurements wasdescribed in [11]).

Fig. 4. ‘‘Breathing” droplet cluster at sw = 1.8 s; Tmax,0 and W0 are the averagevalues.

3. Experimental results

The simplest step-wise periodic variation of laser power likethat shown in Fig. 3 with constant average laser power W0 = 182mW was used in the experiments. The measured variation of themaximum temperature, Tmax, of the open surface of water layerat Wmin = 0.75Wmax is also presented in Fig. 3. The conventionalvalue of Tmax was determined as an average temperature of thecentral surface area of 0.2 mm in diameter. The estimates showedthat the temperature difference in this small area is less than 0.1 K.Obviously, the period of temperature variation is the same as thatof laser power, whereas the temperature curves are smooth andsimilar to the harmonic ones.

A periodic change in the temperature of the water surface leadsto a corresponding change in the velocity of the ascending gas flow,which leads to the characteristic ‘‘breathing” of the cluster – a peri-odic change in the distance between the drops and the size of thecluster (Fig. 4). The image shown in Fig. 4 is a superposition of twophotographs taken at different times with a half-period time inter-val between them. One can see that large central droplets remainpractically immobile, but radial displacements of smaller dropletson the periphery are well noticeable.

It is known that the rate of droplet growth due to condensationis usually well described by the d-squared law [19] or its modifica-tion called the elliptic law [20]. Therefore, according to previouspapers by the authors, the value of the surface area rate of growth,

Fig. 3. Typical periodic time variation of the current laser power and the maximumtemperature of water layer surface; Tmax,0 and W0 are the average values.

_S, is used to analyze the effect of variable laser power. It is naturalto assume that a definite frequency of the laser power variation isthe most favorable to suppress the growth of droplets in the clus-ter. This idea was examined using the experiments with a periodicvariation of laser power shown in Fig. 3. The experimental resultsfor the growth rate of water droplets are presented in Fig. 5, where

the reference value of _S0 corresponds to the constant value W0 oflaser power. The minimum growth rate is observed at sw � 0:9 s,

and the minimum value of _S is approximately twice less than _S0.Note that this effect takes place at very small amplitude of temper-ature oscillations at the water layer surface (less than 1 K).

Obviously, the value of _S obtained at very fast variation of thepower is the same as that observed at the constant and equal to_S0 observed at the average power of laser. A similar situation takesplace in the limit of a very slow variation of laser power. Therefore,it is natural that there is an optimum value of the power oscillationperiod, which is the best one to prevent the droplet growth due tosteam condensation.

The thermal diffusivity of sitall is much greater than that ofwater [17]. Therefore, at the same thickness of the substrate andwater layer, the thermal relaxation time, srel, of the working partof laboratory equipment is determined by a thermal inertia of

Fig. 5. Effect of the period of laser power variation on the growth rate of waterdroplets.

0.6

0.8

A.A. Fedorets et al. / International Journal of Heat and Mass Transfer 127 (2018) 660–664 663

water layer. The value of srel can be easily estimated from the con-dition of Fo ¼ 1 for the Fourier number. As one can expect, thisestimate gives the value of srel about 1 s, i.e. approximately thesame as the optimum period of laser power variation. It is impor-tant to note that the relaxation time is directly proportional to thesquare of the water layer thickness.

0.5 1.0 1.50.0

0.2

0.4

measurements calculations

T max

, K

w, s

Fig. 7. Effect of power variation period on the amplitude of water temperatureoscillations.

4. Some computational results for water temperature

An ordinary transient axisymmetric conduction problem can besolved to determine the temperature field in the substrate plateand water layer in the quasi-steady thermal regime of oscillatinglaser power. As in paper [17], the experimental data for tempera-ture profile of open water surface at several constant values oflaser power were used to retrieve the parameters of a third-kindboundary condition on the surface of evaporating water. Theknown temperature dependences of volumetric heat capacity andthermal conductivity of water were taken into account in thecalculations.

As one can expect, the complete numerical calculations aremore time-consuming than those of paper [17], but even the over-all computational time appeared to be about one second with theuse of the ordinary laptop because of the optimised computationalprocedure based on the early studies [21–23] and the so-calledalternating-direction implicit finite-difference procedure of secondorder. A simple shape of the computational region determined thechoice of this finite-difference method (FDM) instead of the widelyemployed finite element method (FEM), which is more general andflexible [24,25]. Our experience showed that the computationaltime with the use of FDM is about six times less than that forFEM calculations at approximately the same discretization of thecomputational region. The latter makes the finite-difference homecode to be applicable as a regular tool for in-situ processing theexperimental data.

The typical numerical data for the quasi-steady periodic timevariation of the maximum surface temperature of water layer areshown in Fig. 6. The quasi-steady regime is the most interestingone in the present study. Therefore, the initial period of water heat-ing is not presented in this figure. In addition, the temperaturescale is not shown in Fig. 6 because the variation of the amplitudeof temperature oscillations is given in Fig. 7. The calculationsshowed the perfect harmonic behaviour of water temperature,and the amplitude of the temperature oscillations increases mono-tonically in the range of sw under consideration.

A comparison of computational predictions and laboratorymeasurements of the amplitude of oscillations of the maximum

Fig. 6. Harmonic behaviour of the maximum temperature of water surface atvarious periods of a step-wise laser power variation.

temperature at the open surface of water layer is presented inFig. 7. It is clear that there is good agreement between numericaland experimental data. This can be considered as a confirmationof sufficiently good accuracy of the computational model. Note thatthe natural pulsations of water temperature with the amplitude upto 0.08 K are observed even in the case of constant radiation power.It means that the smallest amplitude of temperate oscillations atsw ¼ 0:25 s should not be considered.

Of course, it would be good to use the computational analysis toguide the experiments rather than just to compare the results ofcalculations with experimental data. However, our computationalmodel includes transient temperature of water layer only whereasthe main physical problem of a relation between periodic variationof surface temperature of the water layer and the growth rate ofwater droplets is still waiting for the solution. The latter is consid-ered as the most important task of the forthcoming theoreticalresearch.

5. Conclusions

An alternative method (aside of the earlier suggested infraredheating) to stabilize levitating self-arranged clusters of water dro-plets is examined. This method is based on changes of the mainparameters of local laser heating to find an acceptable ‘‘rejuvenes-cence” procedure to prolong the life of levitating droplet clusters inthe laboratory. The latter is important for further studies of bio-chemical processes in single water droplets.

The behaviour of droplet clusters at periodic changes of laserpower has been studied in a series of the laboratory experiments.The optimum period of laser power variation was found todecrease significantly (by about 50%) the unfavourable condensa-tional growth of water droplets. This period appeared to be equalabout 0.9 s at the particular conditions of the laboratory experi-ment. The latter value is in good agreement with theoretical andcomputational estimates of transient conductive heat transferthrough a substrate plate and water layer in the laboratoryexperiments.

The results obtained are promising for developing an experi-mental procedure for possible use in studies of biochemical pro-cesses in stabilized water droplets.

Conflict of interests

None declared.

664 A.A. Fedorets et al. / International Journal of Heat and Mass Transfer 127 (2018) 660–664

Acknowledgements

The authors are grateful to the Russian Ministry of Educationand Science (project no. 3.8191.2017/<X) and also to the RussianFoundation for Basic Research (project no. 18-38-00232_vok_a)for the financial support of the present study.

References

[1] A.A. Fedorets, L.A. Dombrovsky, Self-assembled stable clusters of droplets overthe locally heated water surface: milestones of the laboratory study andpotential biochemical applications, in: Proc. 16th Int. Heat Transfer Conf.(IHTC-16), August 10–15, 2018, Beijing, China, keynote lecture IHTC16-KN17.

[2] A.A. Fedorets, Droplet cluster, JETP Lett. 79 (8) (2004) 372–374.[3] A.A. Fedorets, On the mechanism of non-coalescence in a drop cluster, JETP

Lett. 81 (9) (2005) 437–441.[4] E.A. Arinstein, A.A. Fedorets, Mechanism of energy dissipation in a droplet

cluster, JETP Lett. 92 (10) (2010) 658–661.[5] A.A. Fedorets, I.V. Marchuk, O.A. Kabov, Role of vapor flow in the mechanism of

levitation of a droplet cluster dissipative structure, Tech. Phys. Lett. 37 (3)(2011) 116–118.

[6] A.A. Fedorets, Mechanism of stabilization of location of a droplet cluster abovethe liquid–gas interface, Tech. Phys. Lett. 38 (11) (2012) 988–990.

[7] A.A. Fedorets, L.A. Dombrovsky, A.M. Smirnov, The use of infrared self-emission measurements to retrieve surface temperature of levitating waterdroplets, Infrared Phys. Technol. 69 (2015) 238–243.

[8] T. Umeki, M. Ohata, H. Nakanishi, M. Ichikawa, Dynamics of microdroplets overthe surface of hot water, Sci. Rep. 5 (2015) (paper 8046).

[9] A.V. Shavlov, V.A. Dzhumandzhi, S.N. Romanyuk, Sound oscillation of dropwisecluster, Phys. Lett. A 376 (28–29) (2012) 2049–2052.

[10] O.A. Kabov, D.V. Zaitsev, D.P. Kirichenko, V.S. Ajaev, Interaction of levitatingmicrodroplets with moist air flow in the contact line region, NanoscaleMicroscale Thermophys. Eng. 21 (2017) 60–69.

[11] A.A. Fedorets, L.A. Dombrovsky, P.I. Ryumin, Expanding the temperature rangefor generation of droplet clusters over the locally heated water surface, Int. J.Heat Mass Transfer 113 (2017) 1054–1058.

[12] A.A. Fedorets, M. Frenkel, E. Shulzinger, L.A. Dombrovsky, E. Bormashenko, M.Nosonovsky, Self-assembled levitating clusters of water droplets: Pattern-formation and stability, Sci. Rep. 7 (2017) (article no. 1888).

[13] L.A. Dombrovsky, A.A. Fedorets, D.N. Medvedev, The use of infrared irradiationto stabilize levitating clusters of water droplets, Infrared Phys. Technol. 75(2016) 124–132.

[14] L.A. Dombrovsky, Absorption of thermal radiation in large semi-transparentparticles at arbitrary illumination of the polydisperse system, Int. J. Heat MassTransfer 47 (25) (2004) 5511–5522.

[15] L.A. Dombrovsky, D. Baillis, Thermal Radiation in Disperse Systems: AnEngineering Approach, Begell House, New York, 2010.

[16] A.A. Fedorets, M. Frenkel, E. Bormashenko, M. Nosonovsky, Small levitatingordered droplet clusters: stability, symmetry, and Voronoi entropy, J. Phys.Chem. Lett. 8 (22) (2017) 5599–5602.

[17] A.A. Fedorets, L.A. Dombrovsky, Generation of levitating droplet clusters abovethe locally heated water surface: a thermal analysis of modified installation,Int. J. Heat Mass Transfer 104 (2017) 1268–1274.

[18] A.A. Fedorets, L.A. Dombrovsky, D.V. Shcherbakov, New experimental resultson dynamics of droplet clusters levitating over the locally heated watersurface, in: Proc. 16th Int. Heat Transfer Conf., August 10–15, 2018, Beijing,China, paper IHTC16-22228.

[19] W.A. Sirignano, Fluid Dynamics and Transport of Droplets and Sprays,Cambridge Univ. Press, Cambridge, 1999.

[20] L.A. Dombrovsky, S.S. Sazhin, A simplified non-isothermal model for dropletheating and evaporation, Int. Comm. Heat Mass Transfer 30 (6) (2003) 787–796.

[21] N. Yanenko, The Method of Fractional Steps, Springer-Verlag, Heidelberg,1971.

[22] A.R. Mitchel, D.F. Griffiths, The Finite-Difference Method in Partial DifferentialEquations, Wiley, Chichester (UK), 1980.

[23] G.D. Smith, Numerical Solution of Partial Differential Equations: FiniteDifference Methods, third ed., Clarendon Press, Oxford (UK), 1986.

[24] Z. Chen, The Finite Element Method: Its Fundamentals and Applications inEngineering, World Sci. Publ., Singapore, 2011.

[25] O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis andFundamentals, seventh ed., Elsevier, New York, 2013.