Identification of Nonlinear Processesin Microfluidic Bubble Flow

Florinda Schembri, Francesca Sapuppo, Luigi Fortuna and Maide Bucolo

Abstract— An approach based on nonlinear dynamical sys-tems theory is used in this work to identify the complextemporal patterns in air bubbles flow carried by water ina snake microfluidic channel with two inlets. Air and waterwere pumped in with periodic flow. Different experimentalcampaigns have been designed varying the frequency of theflow rate alternatively for the water and for the air andmaintaining fixed the other fluid flow. Microfluidic bubble flowswere optically acquired by means of a photodiode-based systemand converted into time series. In relation to the input controlparameters (flow rate, frequency), the diversity of bubbles’temporal dynamic patterns was identified through nonlinearmethodologies. Relationships between nonlinear parameters,volume fraction of fluids and capillary number were foundsuggesting the chaotic behavior of the system. This work is afundamental step toward the control of bubble based operationsin microfluidics.

I. INTRODUCTION

The possibility to produce passive and active chaoticflow in microfluidics has been theoretically investigated byOttino and Wiggins [1]. A practical application of chaoticstreamlines in micro-droplets has been the improvementof chemical kinetics of reagents mixed within plugs [2].Thus the manipulation of small droplets and bubbles inmicrofluidic systems are very important because they couldhave an efficient use in lab-on-chip (LOC) applications: fromsequential micro-reactors [2] to the mixing in the carrierliquid using gaseous plugs [3] with the advantages of improv-ing reaction processes in the chemical, cosmetic and phar-maceutical industries. Moreover the controlled movementsof microfluidic bubbles have been recently used to performlogic operations in the perspective of digital microfluidicprocessors [4].In particular the study of two phase flow in microfluidicshas showed the importance of the input flow parameters forcontrolling droplets’ characteristics and dynamics (frequencyof emission, size, composition, speed) [5][6][7].The aim of this study is to investigate on the complextemporal dynamics of bubbles created at the y-junction ofa snake micro-channel for different input periodic flow con-ditions. In order to achieve this task time series representativeof bubbles’ temporal dynamics have been experimentallyextracted and then analyzed through nonlinear analysis meth-ods. In particular nonlinear time series analysis [8] have beenexploited to identify the occurrence of nonlinear behaviorin the temporal dynamic of air bubbles [9] as the periodic

F. Schembri, F. Sapuppo, L. Fortuna and M. Bucolo are with Dipartimentodi Ingegneria Elettrica Elettronica e dei Sistemi (DIEES), University ofCatania, Catania, Italy.mbucolo@diees.unict.it

Fig. 1. Microfluidic snake mixer and the experimental setup

input (flow rate, frequency) changes. Dimensionless numbersassociated to input flow (capillary number and air fraction)have been related to nonlinear indicators (Largest LyapunovExponentλmax and d-infinited∞). The methodology hereproposed is independent from the fluid, geometry of themicrochannel and the forces used to generate the two-phaseflow.

II. TWO PHASE FLOW IN MICROCHANNEL ANDTHE EXPERIMENTAL SETUP

The COC (cyclic oleifin copolymer) serpentine micro-channel (SMS0104,Thinxxs), here considered, belongs to theclass of passive snake microchannel with two inlets. Thesection of the microchannel is S=640µm and the internalradius of curvature is R=1.28 mm. The periodic input flowrates (Vair, Vwater) for both air and water were obtainedthrough piezoelectric twin pumps (TPS1304, Thinxxs) con-trolled by an electronic pump control (EDP0704, Thinxxs).Such control is actuated through the frequencies (fair, fwater)that set the actuator vibration of the piezo-driven microp-umps’diaphragm generating input flow rates for both air andwater. The micro-channel’s y-junction creates a two-phasebubble flow. The bottom inset in Fig.1 shows the frames ofa bubble passage through the micro-serpentine captured bymeans of a CCD camera. Bubbles’ dynamics were acquiredin a specific area of the snake mixer by means of photodiodeintegrated in an ad hoc electro optic instrumentation (Fig.1).Details of the experimental set-up were described previously[10].

Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 • 5–9 July, 2010 • Budapest, Hungary

ISBN 978-963-311-370-7 241

Four experimental campaigns have been designed (Table I)in order to investigate on the relationship betweenfair andfwater, and the nonlinear dynamics of bubbles.

In all the campaigns one flow rate respectively air or waterwas kept constant at the frequency of 5 Hz and at the sametime the other flow was varied in one case in the range ofmultiple values between 5 Hz to 60 Hz (experimental cam-paign 2 and 4) and in the second case in the range between 7Hz to 37 Hz of not multiple values (experimental campaign1 and 3). A total of 32 couples of inputs (fair, fwater)have been generated. The four experimental campaigns aresummarized in Table I.

TABLE I

EXPERIMENTAL CAMPAIGNS

Experimental Water Frequency Air FrequencyCampaign and Flow Rate and Flow rate

1-Fixed (fair, Vair) f not multiple of 5 Hz f=5 HzV=1.57-5 ml/min V=1.20 ml/min

2-Fixed (fair, Vair) f multiple of 5 Hz f=5 HzV=1.57-5 ml/min V=1.20 ml/min

3-Fixed (fwater, Vwater) f=5 Hz f not multiple of 5 HzV=1.57 ml/min V=1.2-8,46 ml/min

4-Fixed (fwater, Vwater) f=5 Hz f multiple of 5 HzV=1.57 ml/min V=1.2-8,46 ml/min

III. DIMENSIONLESS PARAMETERS INTWO-PHASE FLOW

Input control parameters as capillary number (Ca) (1) andair fraction (AF)(2) have been calculated:

Ca =µv

γ(1)

AF =V air

V air + V water(2)

whereµ is the viscosity of the liquid,v the mean velocity,γ is the interfacial tension between the two fluid and V isthe volumetric flow rate. It is well known that Ca and AFare important parameters for bubble’s formation and dynamicpatterns diversification. In particular Ca is very important intwo phase microfluidics because indicates the competitionsbetween viscous forces to surface tension forces, so thecapillary number is a relevant criterion to predict liquidthread breakup. When it becomes less than a critical valuethe surface tension forces break the air filament into bubbles,minimizing the surface energy; this phenomenon is usuallycalled the Rayleigh-Plateau instability [11].

IV. NONLINEAR TIME SERIES ANALYSIS

The time series from the experiments have been filteredwith a low pass filter at 60 Hz and Notch at 50 Hz, in orderto reduce unwanted components of environmental light andelectric noise. Examples of filtered time series are shown inFig.2.Nonlinear time series analysis methods have been thenapplied for system’s identification and for the extraction

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Fig. 2. Experimental time series in a time window of 2s for the experiment:(a) fair=5 Hz fwater=17 Hz (experimental campaign 1) (b) fair=5 Hzfwater=25 Hz (experimental campaign 2) (c) fair=17 Hz fwater=5 Hz(experimental campaign 3) (b) fair=25 Hz fwater=5 Hz (experimentalcampaign 4).

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(a)

(b)

Fig. 3. Largest Lyapunov Exponents and capillary number vs waterfrequency for the experimental campaign 1-2 (fair =5Hz fixed) (a) fairand fwater not multiple (b) fair and fwater multiple.

of meaningful indicators of nonlinearity, as the LargestLyapunov Exponent (λmax) and the d-infinite (d∞).The nonlinear dynamic analysis has been carried out bymeans of the software TISEAN [12] integrated in a Matlabprogram in ordert to perform automatic calculation andvisualization.Important parameters such as embedding dimension, timedelay,λmax, dj ,d∞ [13], characterize and quantify the dy-namics of a nonlinear time series. The Lyapunov Exponents,in particular, quantify the dependence on initial condition ofthe system. If the system has at least one positive LyapunovExponent then system is chaotic. The larger the positiveexponent the more chaotic the system will be.In order to reconstruct the phase-space portraits from theexperimental time series and to determine the embeddingdimensiond, the method of false nearest has been here used.Taking into account the delayτ , auto-mutual information hasbeen used in choosing the optimal value of the delay. Theformula of the prediction error between very close trajec-tory was used to calculate theλmax [14]. The asymptoticvalue of the divergence curved∞ [13], that determines theaverage divergence between trajectories starting from closeinitial conditions, is considered a parameter sensitive to bothstretching and folding mechanisms.

V. RESULTS AND DISCUSSION

Meaningful results for all the experimental campaigns(Table I) are here presented. The bubble dynamics showqualitative and quantitative differences in their behaviors asthe water or air frequency is respectively not multiple or mul-tiple of the fixed frequency (5 Hz). For all the experimentsthe Largest Lyapunov Exponents are positive indicating a

(a)

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Fig. 4. d∞ and air fraction with state space reconstruction for theexperimental campaign 1-2 (fair=5Hz, fixed) (a) fair and fwater not multiple(b) fair and fwater multiple.

chaotic behaviour. For the experimental campaigns 1 and 2(Table I) The Largest Lyapunov Exponents and the capillarynumber (Ca) versus the water frequency are show in Fig.3.In details Fig.3(a) is related to the campaign in which thefrequency of water flow rate is not multiple of the frequencyof air flow rate (fixed at 5 Hz) whereas in the Fig.3(b) thewater input frequency is multiple of the air frequency. Thedynamic properties of the whole system identified by meansof the Largest Lyapunov Exponents can be confirmed alsoby visual inspection of the phase space trajectories in Fig.4.The phase portraits have been connected to the values of thed∞ and AF curve both for water frequency multiple and notmultiple of the fixed air frequency (Fig.4).It is clear that if the frequency of the carrier fluid flowincreases with multiple values of the fixed air frequency,the nonlinear properties of the microfluidic two phase flowsystem increase as well.The results for the experimental campaigns 3 and 4 (TableI) show different behavior with respect to the experimental

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(a)

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Fig. 5. Largest Lyapunov Exponents and capillary number vs air frequencyfor the experimental campaign 3-4 (fwater =5Hz fixed) (a) fair and fwaternot multiple (b) fair and fwater multiple.

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Fig. 6. d∞ and air fraction with state space reconstruction for theexperimental campaign 3-4 (fwater=5Hz, fixed) (a) fair and fwater notmultiple (b) fair and fwater multiple.

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campaigns associated to water frequency changes (experi-mental campaign 1 and 2). In particular the Ca curve showsa linear trend with respect to air frequency and almost allthe values of the Largest Lyapunov Exponents in Fig.5arelower than one. Moreover for the experimental campaigns 3the system exhibits a maximum peak in theλmax graph at 32Hz while in the experimental campaign 4 at 5 Hz (Fig.5). Thestate space representation associated tod∞ and AF versusair frequency is shown in Fig.6.

Comparing the results of experimental campaigns 1-2 withexperimental campaigns 3-4 it is to notice that a markednonlinear temporal dynamic of bubbles is achieved whenthe air is fixed and the water flow rate is varied. A proofof this result is that most Largest Lyapunov Exponents inthe experimental campaign 1 and 2 are greater than in theexperimental campaign 3 and 4 (table I).Moreover the study of time series through Fourier spectrumshows that a phenomenon of frequency modulation, in avariegated spectrum, take place when fwater≤ fair: fair=5Hz fwater=17 Hz (Fig.7(a)), in the other case for fair=5Hz fwater=25 Hz is also clearly visible (Fig.7(b)) becausethe peaks are not entire multiple of a fundamental peak. Itis a significant result that the frequency modulation is notachieved in the complementary case (experimental campaign3 and 4) whenfair=17 Hz or fair=25 Hz and fwater is fixedat 5 Hz (Fig.8). Comparing the spectrum in Fig.7 and Fig.8a variegated spectrum is pronounced whenfair and fwaterare not multiple.Frequency modulation is very important in communicationsystems and could have an efficient use also in digital

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microfluidics for encoding and decoding information carriedby water and air.

VI. CONCLUSIONS

The identification of two phase flow systems, with periodicflow as inputs, through nonlinear analysis methods showsthe importance of the frequency of the flow rate in thediversification of the chaotic temporal dynamic of air bubblescarried by water. In particular the frequency of the carrierfluid (water) is a useful tuning and control parameter forthe nonlinear temporal pattern of bubbles in microchannels.The possibility to have frequency modulation of the signalrelated to two phase flow lead to the suggestion of usingmicrofluidic two phase flow systems also for communicationpurpose using water and air bubbles flow as informationcarrier. A future study would be the exploration of two-phaseflow dynamics generated by continuous input flow rate forboth air and water in order to define the effects of the flowrate’s amplitude in the nonlinear dynamics of bubbles.

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[11] H. A. Stone, Dynamics of Drop Deformation and Breakup in ViscousFluids, Annual Review of Fluid Mechanics, vol. 26, 1994, pp 65-102.

[12] R. Hegger, H. Kantz and T. Schreiber, Practical Implementation ofnonlinear Time Series Methods: The TISEAN packageCHAOS, vol.9, 1999, pp 413-435.

[13] A. Bonasera, M. Bucolo, L. Fortuna, M. Frasca and A. Rizzo, A NewCharacterization of Chaotic Dynamics: the d-infinite ParameterNonlinear Phenomena in Complex Systems, vol. 6, 2003, pp 779-786.

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