Physics Letters A 376 (2012) 3185–3191

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Physics Letters A

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Velocity and rotation measurements in acoustically levitated droplets

Abhishek Saha a, Saptarshi Basu b, Ranganathan Kumar a,∗a University of Central Florida, Orlando, FL 32816, United Statesb Indian Institute of Science, Bangalore 560012, India

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

Article history:Received 26 July 2012Accepted 9 August 2012Available online 14 August 2012Communicated by C.R. Doering

The velocity scale inside an acoustically levitated droplet depends on the levitator and liquid properties.Using Particle Imaging Velocimetry (PIV), detailed velocity measurements have been made in a levitateddroplet of different diameters and viscosity. The maximum velocity and rotation are normalized usingfrequency and amplitude of acoustic levitator, and droplet viscosity. The non-dimensional data are fittedfor micrometer- and millimeter-sized droplets levitated in different levitators for different viscosity fluids.It is also shown that the rotational speed of nanosilica droplets at an advanced stage of vaporizationcompares well with that predicted by exponentially fitted parameters.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Evaporation of pure and binary liquid droplets is of interestin thermal sprays and spray drying of food, ceramics and phar-maceutical products. Understanding the rate of heat and masstransfer in any drying process is important not only to enhanceevaporation rate or vapor-gas mixing, but also to predict andcontrol the final morphology and microstructure of the precipi-tates. In drying research, liquid precursor droplets are convertedinto fine particles or powders [1–4] of unique agglomeration mor-phologies and microstructures in an acoustic single-axis levita-tor. In nanoparticle suspension based precursor solutions, viscosityincreases with nanoparticle concentration and changes the mor-phology. Various techniques have been adopted to study dropletdrying such as crosswire, glass filament, micro needle etc. [5,6].These techniques, however, are not free from surface contamina-tion. Moreover, the solid–liquid contact does not allow the dropletto rotate. For droplets with suspension, the problems become se-vere as particles start to deposit along the solid surface. One of thealternative ways of conducting drying research is to use an acous-tic levitator to suspend sub-millimeter size droplet [2,4,7]. We cangain understanding of the fundamental mechanisms by examin-ing the vaporization characteristics of isolated levitated dropletsin an acoustic field. Studies at the level of individual droplets canreveal precipitation mechanisms and important physics related tothermophysical transformations that lead to unique morphologicalstructures.

Several of our group’s recent studies have been in dropletscontaining pure fluids and nanosuspensions [2–4,8,9]. In suspen-

* Corresponding author.E-mail address: ranganathan.kumar@ucf.edu (R. Kumar).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2012.08.013

sions, we delineated different stages of evaporation and agglom-eration/precipitation. A levitated cerium nitrate [Ce(NO3)3, 6H2O]droplet under external heating shows initial faster evaporationleading to precipitation, followed by a thermal decomposition ofcerium nitrate forming nanoceria [2,10]. The final precipitate isporous in nature as the gas liberated during thermal decom-position left vents on the surface. On the other hand, dropletswith nanosilica form a bowl-shaped structure after the initialevaporation stage. The bowl-shaped structure is the outcome ofpreferential migration of the nanoparticles to the bottom of thedroplet, followed by agglomeration. At high particle concentrations,the bowl-shaped structure further transforms into a horizontal ringshaped morphology, which subsequently reorients itself to a ver-tical ring. The initial bowl formation is dependent on the flowstructure within the droplet in addition to the pressure imbal-ance due to density stratification [8]. Levitation also facilitates alab-on-a-drop type of studies on evaporation characteristics of newbiofuels. Our previous work with biofuels [11,12] showed that thestudy of a single droplet of bio-derived fuels in a levitator could beused to predict vaporization rate and evaporation characteristics.

Droplet drying in a levitator is controlled by the acousticstreaming-induced motion in the droplet. In order to understandthe heat and mass transfer inside a levitated droplet undergoingdrying, it is important to determine the velocity field inside thedroplet triggered by the flow field in the ambient as a result ofacoustic streaming. However, maximum velocity in the droplet canchange due to the relative strength of the acoustic streaming andpressure amplitude in addition to the size and thermo-physicalproperties of the droplet. Therefore, the measured velocity pro-files inside the droplets in different levitators should be scaledusing appropriate parameters to obtain a generalized expressionin terms of normalized size and viscosity. This Letter addressesthese issues by measuring the velocity in levitated droplets of size

3186 A. Saha et al. / Physics Letters A 376 (2012) 3185–3191

Fig. 1. (a) Experimental setup, (b) acoustic levitator with suspended droplet (adapted from Saha et al. [3]).

300 μm to 1 mm. In order to mimic the viscosity of nanofluids atdifferent concentrations, solvents such as water, cerium nitrate so-lution and glycerol have been used. The specific objectives of thisLetter are to understand the fundamental effects of internal circu-lation inside a droplet for different viscosity fluids using ParticleImage Velocimetry (PIV) and compare the result with levitatorsof different acoustic frequencies and amplitudes in terms of non-dimensional parameters.

There have been several studies on different levitation pro-cesses using magnetic, non-magnetic material and live animals[13,14]. The pressure difference between two poles of a levitateddroplet counteracts gravity. Acoustic streaming affects aspect ra-tio and vaporization characteristics of a levitated droplet [1,15,16].The properties of a levitator are such that as the levitator fre-quency increases, the droplet diameter that can be levitated willgo down. Hasegawa et al. [17] and Yan et al. [18] used frequen-cies ∼ 20 kHz to levitate 4 mm size droplets. Yarin et al. [1] andSaha et al. [3] levitated 1.6 mm and 500 μm drops at 56 kHzand 100 kHz respectively. While Yarin et al.’s [1] work showedtwo toroidal cells, Hasegawa et al. [16], Yan et al. [18] and Sahaet al. [3] showed the presence of a strong single recirculation cell.This Letter discusses the single cell versus the double cell pat-tern and will make a case that for high SPL (Sound Pressure Level)needed in single axis levitators, a complete control of droplet rota-tion is unavoidable, however, it is predictable. This droplet rotationlends its way toward various morphological changes such as bowland ring shapes, which would not be possible in the absence of ro-tation. This work will utilize the data from the literature and thedata generated in this Letter to develop an expression using expo-nential curve fit for velocity and rotation in a single vortex thatcan be applied to any levitator. The expression (curve fit) can beused for modeling where liquid phase velocity needs to be spec-ified in order to compute vaporization rate, precipitation kineticsand particle agglomeration.

2. Experimental setup

The experimental setup involves a 100 kHz acoustic levita-tor operated at a sound pressure level (SPL) of ∼ 164 ± 1 dB.The droplet is held in position as a result of multiple reflectionsbetween an ultrasonic radiator and a solid, flat or concave re-flector, which is adjusted concentrically so that a standing wavewith equally spaced nodes and antinodes is generated. Solid orliquid samples with effective diameters of 300 μm to 1 mm canbe levitated without contact below the pressure nodes as a re-sult of the axial radiation pressure and radial Bernoulli stress.More details of the experimental setup have been reported by Sahaet al. [3].

A dual-pulsed 532 nm Nd-YAG laser with beam diameter of4 mm was used for PIV. Using a set of cylindrical and sphericallenses, the circular beam was converted into a ∼ 65 μm thick lasersheet with a height of 4 mm (Fig. 1(a)). The laser was placed insuch a way that the vertical light sheet passes through the centerof the droplet. The levitator was stationed on X–Y–Z table in orderto achieve minute control over the light sheet location with re-spect to the suspended droplet. A CCD camera (pixel resolution of1376 × 1040) along with a zoom lens was placed perpendicular tothe light sheet. The liquid was seeded by adding polystyrene mi-crospheres (diameter ∼ 1 μm) with a concentration of less than 2%of volume. The images were analyzed using an edge detectionmethod to calculate an equivalent volume of the droplet, fromwhich the diameters of the droplets were measured [3].

The laser pulses and the camera frames were synchronized us-ing an external timing unit. The pulse duration of the laser ison the order of 10 ns. The whole setup was operated at 5 fps.The time lag between two pulses (�t) was varied to see the effecton the velocity field. After a few iterations it was determined that200 μs is optimum for the flow field. A set of 100 image-pairs wasaveraged to get the PIV vectors [3]. This number was determinedto be sufficient to minimize the uncertainty from grid generation,correlation and peak validation.

The random error associated with this statistical method ofcross-correlation peak in PIV to determine displacement vectorsdecreases with the ratio of maximum displacement to linear sizeof interrogation window [3]. In our 32 × 32 window size, thisrandom error is limited to 1%. The maximum displacement waskept limited to 15 pixels [3]. The uncertainty from inherent gridgeneration, correlation, peak validation becomes insignificant withincrease in number of image pairs. A hundred image pairs wereused to achieve good quality vectors. In our PIV measurements,the camera records digital images of a droplet with a resolu-tion of 1376 × 1040 pixel and each image corresponds to an areaof 1650 × 1300 μm, i.e., the area of each pixel is ∼ 1.5 μm2.The seeding particles with maximum diameter of 1 μm, occu-pies approximately 4 pixels. Thus, the uncertainty involved withindisplacement of each particle is around 0.1 μm. Current studyinvolves 200 μs pulse separation time on average, which leadsto a velocity uncertainty of 0.5 mm/s [3]. The response time isestimated to be 10−7 s [3]. The separation used between twopulses is 200 μs, which is three orders higher than the responsetime. Therefore, the particles have enough time to follow theflow. The corrections due to droplet curvature were carried outusing algorithms prescribed by Minor et al. [19]. They used raytracing method to calculate the optical distortion and proposedan algorithm for its correction. This algorithm is used to cor-rect for the velocity vectors inside the levitated droplet. More

A. Saha et al. / Physics Letters A 376 (2012) 3185–3191 3187

details about the experimental setup has been provided by Sahaet al. [3].

3. Results

In a levitator, a droplet is suspended by acoustic forces.The pressure distribution between the transducer and the reflectoris shown in Fig. 1(b). The acoustic pressure is symmetric aroundthe pressure nodes as shown in Fig. 1(b). For a stable levitation, thepressure (acoustic) force around the droplet balances the weightand surface tension forces of the droplet. Levitated droplets ina zero gravity environment would remain at the pressure nodeas shown in Fig. 1(b). However, under gravity the droplet shiftsdownwards so that the pressure distribution around the dropletcan balance the weight of the droplet. When a droplet is levitated,it tends to take an ellipsoidal shape due to the asymmetric pres-sure distribution around the droplet. Our previous works [3,8] haveshown that the shape and location of the droplet depend on thedroplet size, density, viscosity and sound pressure level (SPL).

3.1. Single cell vs. toroidal cell

In a perfectly controlled experiment, the flow structure insidea levitated, non-rotating droplet will have two toroidal cells. Thiswas observed by Yarin et al. [1] in a single axis levitator. Trinhand Wang [20] ultrasonically levitated a droplet in an immisci-ble liquid and observed shape oscillations that resembled a twotoroidal cell mode (quadrupole). However, Trinh and Wang alsoreported that when the shape oscillations reached threshold am-plitude, a rotational instability would set in and possibly break thesymmetry of these torroidal cells. In most single axis levitators,small misalignments are inevitable which leads to droplet rotationabout the horizontal axis and hence break the symmetry of thetorroidal cells leading to a single cell forced vortex type structure.In addition to our current and previous work, many articles havereported this single cell flow structure within the droplets [17,18].Saha et al. [3] reported that the droplet rotates about the verticalaxis of the levitator as well. These multi-axial droplet rotations de-stroy the symmetry of the internal flow field. These rotations areconsistent and repeatable for the same levitator and flow condi-tions.

Thus, the droplets in single axis levitator exhibit rotationsabout both horizontal and vertical axes mainly due to the acous-tic torque arising out of small misalignments/perturbations, com-positional inhomogeneity and non-spherically symmetric externalheating. Smaller droplet size, as in the current experiment, alsoaugments this rotational rate. The rotation, in essence, merges thetwo toroidal cells normally expected into a single cell structure.For any single axis levitator involving a multi-component droplet,this behavior can be expected, unless the environment is com-pletely controlled with low sound pressure level and partial vac-uum. As shown in our previous papers [3,8], when droplets withnanosuspensions are levitated in an acoustic field, it may be in ourbest interest to allow such rotations in droplets in order to obtaindesired morphology of the final dried particle. Therefore, a singlevortex as found by us and others is inevitable and later in Sec-tion 3.5, we provide expressions using data fit for the maximumvelocity and rotation that can be expected in any levitator oper-ated at any frequency.

3.2. Nanosuspensions in droplets

We have observed multicomponent levitated droplets likenanosilica [3,4,8] and cerium nitrate solution [2], exhibit twomodes of rotations, one about the horizontal axis and another

Fig. 2. Streak images of rotating droplet, (a)–(d): frontal view, (e)–(h): side view.

about the vertical axis of the levitator. The pure liquid droplets pri-marily rotate about a horizontal axis as shown in Fig. 2. When theliquid content of the droplet with nanosilica or precursor solutionis significantly higher compared to the solid particles (or precipi-tate), the droplets primarily rotate about the horizontal axis, whilethe motion about the vertical axis is very slow (less than one hun-dredth of a revolution per second). However, when the droplet isheated, it evaporates and undergoes heterogeneous precipitation oragglomeration between the top and bottom halves (compositionalinhomogeneity), leading to stronger rotation about the verticalaxis and correspondingly weaker motion about the horizontal axis.Upon complete precipitation of salts (cerium nitrate) or agglom-eration of nanoparticles (nanosilica), the droplet generally takesa heterogeneous bowl-shaped structure [2–4] which rotates onlyabout the vertical axis. With external heating, the droplets withhigh concentration of nanosilica can form a ring shaped structurewhich rotates about the vertical axis at an even faster rate. Fur-ther heating results in the reorientation of the ring in the verticalaxis when the ring rotates about both horizontal and vertical axes;however the rotation about vertical axis still dominates [2–4]. It isclear that the internal circulation in the liquid phase of the dropletcauses such structural and morphological changes. Therefore it isimportant to determine the flow patterns and rotational speedsprior to such morphological changes.

3.3. Qualitative visualization of rotation

Initially for qualitative understanding, we used long exposureimaging technique to capture the flow structure inside liquiddroplets. Droplets were seeded with 8–10 μm glass beads and anLED light source was used to illuminate the droplet at an angleof 30◦ with the camera. The light scattered from the glass beadswas captured using the Phantom V12 camera in conjunction withNavitar zoom lens. Streaklines are imaged when the camera sen-sor is exposed for 0.02 seconds. The long exposure time limits thecamera framing rate to 50 fps. Hence, external heating was absentfor this set of experiments. The droplet vaporized naturally over25 minutes and the evaporation process was captured to observethe evolution of the flow structure with time. Fig. 2 shows a setof the streak images. As the acoustically levitated droplet showssmall oscillations about the axis of the levitator, the long exposedimages look slightly blurry. However a qualitative understandingof the flow pattern can still be obtained. From Fig. 2(a)–(d), a sin-gle vortex cell rotating about a horizontal axis can be observed.When the camera is set parallel to the horizontal plane (Fig. 2(a)–(d)), concentric circles are first seen. In time, when the dropletrotates about 90◦ (Fig. 2(e)–(h)), the camera sees parallel lines,showing the trajectories of the particles on the surface. The ro-tational speed about the vertical axis is weak and is measured tobe around 0.0167 rpm (Fig. 2(e)–(h)).

Longer exposed high resolution images with glass beads (streakimages) in Fig. 2, clearly show that the predominant liquid motion

3188 A. Saha et al. / Physics Letters A 376 (2012) 3185–3191

Fig. 3. Different stages of evaporation of nanosilica and cerium nitrate droplets (adapted from Saha et al. [2] and Kumar et al. [4]).

is in the vertical plane. If there was any strong motion in thehorizontal plane (i.e., if the droplet was rotating about a verticalaxis), then the streak images (Fig. 2(e)–(h)) would not show par-allel rings in vertical plane. These parallel rings in vertical planeconfirm that the predominant rotation is about the horizontal axisonly. However, as stated before, the rotation about the vertical axisincreases significantly for nanosilica around the structure forma-tion time. The rotational speed depends on the concentration ofnanosilica in water. For example, for 3% concentration, the rotationspeed about the vertical axis is 900 rpm (15 rps) after the bowlformation. For higher concentrations of nanosilica, further heatingresults in a horizontal ring formation and the speed increases to4200 rpm (70 rps). Later, when the horizontal ring reorients tobecome a vertical ring, the speed further increases to 7200 rpm(120 rps). This change of the rotational speed about the verticalaxis is shown using real images schematically in Fig. 3. To investi-gate the rotational speed about the horizontal axis, Particle ImageVelocimetry (PIV) has been performed.

PIV studies were conducted on droplets with three different liq-uids and three different droplet diameters. The properties of these

Table 1Properties of the liquids.

Liquid Density(kg/m3)

Dynamic viscosity(Pa-s)

Water 998 0.73×10−3

Cerium nitrate–water solution (0.5 M) 1090 1.33×10−3

Glycerol–water solution (3 : 2 v/v) 1120 11.10×10−3

liquids are given in Table 1. These liquids are chosen such that theymimic the viscosity levels of nanosuspensions at different concen-trations [3,21–23].

3.4. Quantitative flow structure using PIV

In this section, two-dimensional velocity measurements usingPIV are shown in a vertical plane passing through the origin. Onlythe flow field in the vertical plane (parallel to levitator axis) hasbeen resolved. PIV vectors on levitated droplets with different di-ameter and viscosity show a similar flow structure as observed

A. Saha et al. / Physics Letters A 376 (2012) 3185–3191 3189

Fig. 4. Velocity vectors in a vertical plane passing through droplet center (adaptedfrom Saha et al. [3,8]).

during streak imaging. This structure resembles a forced vortexflow around the center of the droplet. At any point inside thedroplet, the flow is tangential, and velocity is maximum at theouter radius and zero at the center. The strength of the flow de-creases with increase in viscosity and diameter. Fig. 4 shows thevectors for all three liquids at three different diameters. Prelimi-nary velocity vectors in a levitated droplet were earlier reportedby Saha et al. [3]. Fig. 5 shows the velocity profile along the radiusfor different fluids. Vector fields from PIV also show that the pri-mary velocity field is tangential in nature. It can also be inferredthat the velocity changes linearly along the radius. This confirmsthat the velocity field inside the droplet resembles a forced vortexor rotating droplet, where the slope of each curve in Fig. 5 repre-sents the rotational speed.

The important inference can be drawn by combining the streakimages and the PIV vectors that the pure liquid droplet essen-tially contains rotation about a horizontal axis. As discussed earlier,PIV measurements in the meridional plane (on a vertical planewhich intersects the droplet center) show single cell structures.From the streak images it is now clear that the single vortex aboutthe horizontal axis is maintained, as the droplet slowly rotatesabout the vertical axis. The rotational speed about the vertical axisis so small that it does not have any significant effect on the singlecell structure. These streak images also show that even after thedroplet size reduces significantly due to natural evaporation, theflow structure remains unaltered. However, if the droplet containsa secondary species which may precipitate or agglomerate duringevaporation, this flow structure will be disturbed and will decayrapidly.

3.5. Normalized maximum velocity and rotation

The velocity vectors show that the maximum velocity (V max)occurs at the equator or at the radius of the droplet. Also, we have

Fig. 6. Change in maximum velocity (V max) with droplet diameter and viscosity.

seen that this V max varies with droplet radius and viscosity of theliquid. Fig. 6 shows how V max changes with droplet size at threedifferent viscosities, namely for water, cerium nitrate and dilutedglycerol. The properties of these liquids are provided in Table 1.

Although there is some scatter in the V max data for differentdiameter and viscosity, they follow a trend. We see that V maxreduces with both size and viscosity of the droplet. In order tounderstand the dependence of V max on the droplet diameter andviscosity of the liquid, we first performed a dimensional analysis.Three non-dimensional parameters are introduced in the analysisnamely D∗ (normalized diameter), μ∗ (normalized dynamic vis-cosity), V ∗

max (normalized maximum velocity inside the droplet).The diameter of the droplet is normalized by the wave num-ber (k0), and the liquid viscosity and V max are normalized by thesurrounding fluid (air) viscosity and velocity scale of the surround-ing medium in the absence of a droplet.

D∗ = k0 D (1)

μ∗ = μl/μg (2)

V ∗max = V max/Uscale(D) (3)

where μl is the dynamic viscosity of the liquid and μg is the dy-namic viscosity of the gas phase surrounding the droplet. Uscale isthe velocity scale which depends on the levitator and the sur-rounding medium in the absence of a liquid droplet and is derivedfrom the theory of standing waves. According to acoustic the-ory [24], this gas velocity is

U (r, z) = Umax cos(kz z) J0(krr) (4)

where r and z are measured in radial and vertical coordinates re-spectively with respect to the pressure node. The wave numbersin the radial and axial directions, kr , kz and k0 are related byk0 =

√k2

r + k2z and q = kr/kz . Using these relations, we can rewrite,

Fig. 5. Velocity distribution along horizontal radius of the droplet (a) water, (b) cerium nitrate–water solution (0.5 M), (c) glycerol–water solution (3 : 2 v/v).

3190 A. Saha et al. / Physics Letters A 376 (2012) 3185–3191

Fig. 7. Proposed correlation closely matches with the experimental data from Fig. 6and data from Hasegawa et al. [17] and Yan et al. [18].

U (r, z) = Umax cos

(k0z√1 + q2

)J0

(qk0r√1 + q2

)(5)

Here, Umax = Ma · c0 and SPL[dB] = 197 + 20 log(Ma) [24]. Ma isacoustic Mach number, c0 is speed of sound and SPL is sound pres-sure level of the levitator in dB. This equation for SPL is valid forunperturbed acoustic wave without any levitated droplet. The gasphase velocity scales are generally computed based on the max-imum velocity attained in unperturbed standing wave [1,16,24],and are related to SPL through this equation. Yarin et al. [1,16]showed that the presence of levitated droplet alters the SPL due toits ellipsoidal shape. As the use of SPL in the current study is lim-ited only to determine the velocity scale of the unperturbed field,this variation is not considered. Thus, the velocity scale U (r, z) isvalid only when there is no liquid droplet present in the levitator.This velocity scale will be substantially altered in the presence ofa liquid droplet.

Since the gas velocity at the location of droplet radius definesa limit for the maximum velocity (V max) within the liquid droplet,Uscale = U (r = R, z = 0) was used where R = radius of the droplet(= D/2). Thus,

Uscale(D) = Umax J0

(qD∗

2√

1 + q2

)(6)

V ∗max, D∗ , μ∗ contain information about the levitator frequency,

SPL and properties of the liquid and the external gas field. Notethat V ∗

max is a quantity that is not equal to 1 since the velocityscale Uscale(D) is only defined when no liquid droplet is present.In essence, Uscale(D) is the most generic velocity scale that is onlydependent on the frequency and amplitude of the levitator in theabsence of any droplet.

From the normalized data,

V ∗max = f

[(D∗)n(

μ∗)m](7)

The best fit is obtained with n = 1 and m = 1/4. Fig. 7 shows theplot of V ∗

max vs. [(D∗) · (μ∗)0.25]. The blue symbols represent thecurrent experiments, which are fitted using an exponential fit ofthe form, V ∗

max = exp(C D∗(μ∗)0.25). V ∗max varies as μ∗ 0.25 in this

relationship, while Yarin et al. [1] obtained a μ∗ 0.5 dependencefrom the shear balance at the interface. Yarin et al. used an axi-symmetric model while in the current experiment, the observedflow is a forced vortex. For best fit, C was found to be −0.4. Wesee from the fitted equation that V ∗

max → 1, i.e., the liquid viscositywould approach the surrounding gas viscosity, when the drop sizebecomes negligibly small. In addition, in the other limit when thedroplet approaches a solid body (μ∗ → ∞), the flow within thedroplet would approach zero.

Fig. 8. Comparison of rotational speeds (ωx) from experimental data and from theexponential curve fit in Fig. 7 for three different droplet diameters.

The data points from Yan et al. [17] and Hasegawa et al. [16]for different levitator settings are normalized and plotted withthe current data in Fig. 7. Yan et al. [17] studied a water dropletof 3.72 mm diameter suspended using a levitator with 22 kHz fre-quency and reported a maximum velocity of 70 mm/s. Hasegawaet al. [16] worked with 4 mm diameter water and glycerol dropletsin a levitator with 19.4 kHz frequency. For a glycerol droplet, theyreported that they did not observe any velocity in vertical plane.Thus, the curve fit of the normalized data predicts the data fromother levitators as well as for much larger droplets with differentviscosities.

As seen in Figs. 4 and 5, the flow inside the droplet resem-bles a forced vortex, therefore, V max = ωx(D/2), where ωx is ro-tational speed. If we define non-dimensional rotational speed asω∗

x = V ∗max/(D∗/2), then, ω∗

x = ω/(k0Uscale). Applying these re-lations, non-dimensional rotational speed can be estimated froman exponential fit (Fig. 7) using droplet diameter and viscosity.

Converting this non-dimensional rotational speed to dimen-sional rotational speed, one can now plot the rotational speed (ωx)in rpm with respect to dynamic viscosity as shown in Fig. 8. In thisfigure, dotted lines represent the change in rotational speed withviscosity for three diameters of droplets, 300, 700 and 1000 μm.The profiles developed in this manner match the experimentaldata very well (Fig. 8). The small variations can be attributed to theuncertainty in droplet diameter during the experiments. The varia-tion in droplet diameter was found to be ±5% which yields a maxi-mum ±10% variation in rotational speed within the viscosity rangeof these droplets (0.5–5 cP).

Our previous work [3] showed rotation of droplet contain-ing nanosilica suspensions captured through high speed images.Fig. 9(a) shows viscosity profiles in terms of particle concentrationfor nanosilica (10 nm) and nanoalumina (50 nm). From Fig. 9(a)we can see that for the same concentration of 3%, nanosilica andnanoalumina have different viscosities of 1.55 and 1.85 cP respec-tively. In this work, the nanosuspensions are considered to beNewtonian, and any possible non-Newtonian effect was not con-sidered. The PIV measurements were principally carried out inpure liquids with analogous viscosity behavior as nanosuspensions.It can be seen that at this concentration for the same diame-ter, nanosilica droplet would rotate faster than the nanoaluminadroplet. Rotational speed is plotted versus viscosity for a droplet of450 μm diameter in Fig. 9(b). From this figure, we can confirm thatfor 3% concentration of nanoparticles, the nanosilica droplet rotatesat ∼ 9000 rpm and nanoalumina rotates at ∼ 6000 rpm. The exter-nal heating promotes faster vaporization of the droplet and it goesthrough a structure formation stage. The rotation captured through

A. Saha et al. / Physics Letters A 376 (2012) 3185–3191 3191

Fig. 9. (a) Viscosity of nanosuspensions (nanosilica 10 nm and nanoalumina 50 nm) at different nanoparticle concentration, (b) rotational speed (ωx) for a 450 μm droplet asa function of viscosity, (c) high speed images of 3% nanosilica droplet under laser irradiation: snapshots near the liquid depletion point.

high speed images shown in Fig. 9(c) is for a nanosilica structurewhen most of the solvent is depleted. The rotation speed is calcu-lated through careful frame by frame investigation of high speedimages.

As shown in Fig. 9(c), the angular displacement of a natu-ral marker on the droplet surface such as surface irregularity, ismeasured between few frames. In Fig. 9(c) this displacement isshown as (θ2 − θ1). Knowing the time interval between those twoframes (t2 − t1), we can calculate the rotational speed (ωx) as(θ2 − θ1)/(t2 − t1). This would mean that the nanosilica concentra-tion would have become very high, although there is no other wayof measuring viscosity at this stage. Image analysis shows that themeasured rotation for this condition for nanosilica is 400–600 rpm.For the measured range of rotation (400–600 rpm), the viscosity ofthe colloidal suspension is expected to be very high, and is esti-mated from Fig. 9(b) to be in the range 500–600 cP. This showsthat after structure formation, the droplet is almost devoid of liq-uid.

4. Summary

Internal flow structures in micron-sized droplets are difficult tomeasure, but are important for the understanding of drying pro-cesses of nanoparticle suspensions. Viscosities of nanosuspensionsare significantly higher than that of the base fluid, and increasefurther during vaporization. In this work, using Particle ImagingVelocimetry (PIV), detailed velocity measurements have been madein a levitated droplet of different diameters and viscosity. Differentviscosities were chosen to mimic the fluid motion inside a precur-sor droplet with suspended nanoparticles. The motion inside thedroplet is that of a forced vortex which has also been reported indifferent levitators in two other papers. Exponential curve fit for

maximum velocity and rotation have been obtained which follownot only the current data but also from two other papers whichused levitators of different amplitudes and frequencies, and largerdrop sizes. The rotational speed of nanosilica droplets at an ad-vanced stage of vaporization were also obtained from high speedimages.

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