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Colloids and Surfaces A: Physicochem. Eng. Aspects 309 (2007) 132–136

Foam drainage with twin inputs in two dimensions

Jin Huang, Qicheng Sun ∗Institute of Process Engineering, Chinese Academy of Sciences, Beijing, China

Received 19 July 2006; received in revised form 16 March 2007; accepted 8 April 2007Available online 13 April 2007

bstract

Forced foam drainage is the flow of constantly input liquid through the network of interstitial channels between bubbles in foam under actions ofravity and capillarity. When two streams of liquid are input with twin narrow nozzles separated with a distance, they usually merge into a singularrainage wave to propagate in foams. This work reports results of behaviours of the merged drainage wave. We find that (1) along the vertical

irection beneath one nozzle, the liquid spreading is well described by power law with exponents greater than that of individual drainage. (2) Alonghe vertical direction down the center of the two nozzles, we find the spreading speed is nearly constant for a fixed separation. As the separationistance increases six times, the speed increases only about 14%. This result would benefit designing of micro-mixers and micro-reactors, basedn foam-like channels, for liquid/liquid systems widely used in pharmaceutical industry. 2007 Elsevier B.V. All rights reserved.

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eywords: Foam drainage; Twin inputs; Experiment; Simulation

. Introduction

In liquid foams, a small amount of surfactant solution existsn interstitial channels between bubbles. Due to large densityifference from gas, liquid drains out of foams under the action ofravity. This process is called drainage, which is directly relatedo formation and evolution of liquid foams. Studies on foamrainage are not only of theoretical value, but also contribute tomprovements of the efficiency of industrial processes, such asil recovery, ore refining and production of metallic foams.

Over last decades, theoretical investigations on one-imensional foam drainage have made great progress. For forcedrainage, Weaire et al. found the scaling law between the velocityf drainage solitary wave υ and input flow rate Q: υ ∝ Q1/2 [1],hile Koehler et al. obtained υ ∝ Q1/3 [2]. This conflict is mainly

ttributed to differences of chemical properties of surfactantssed in their experiments, which cause gas/liquid interfaces toe either rigid or mobile, and eventually lead to difference in flowypes within Plateau borders: Poiseuille flow or plug flow [3].

he corresponding models have been remarkably successful.

In one-dimensional drainage experiments, several techniquesave been developed to calibrate liquid fraction φ with easily

∗ Corresponding author. Tel.: +86 10 8262 7076; fax: +86 10 6255 8065.E-mail address: sunq@home.ipe.ac.cn (Q. Sun).

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927-7757/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.colsurfa.2007.04.012

easured physical quantities. Saint-James and Langevin usedight transmission to determine φ in thick enough foam sam-les [4]. Koehler et al. used the fluorescence dye and UV lightechnique to measure φ in nearly dry foams with φ < 0.05 [5,6].utzler et al. used ac capacitance measurements and found a

east square fit between the measured capacitance and observediquid fraction [7]. Hutzler and Wang et al. devised a novel tech-ique to visualize drainage propagation by adding dark colourood dye into detergent solution [8,9], and obtained the calibra-ion relation between light intensity andφ. Because the measuredight intensity is dependent on experimental set-up, such as lightource properties, distance between the digital camera and Hele-haw cell, the uniformity of light intensity along foams is hard

o achieve which would lead to limited accuracy of the calibra-ion relation. In this work, we proposed to calibrate φ with theransmission ratio of light, T, while T is calculated to be theatio of transmitted light intensity through foams to transmit-ed intensity of light through the Hele-Shaw cell without foams.y doing this, influences of light intensity variation would bereatly reduced.

The history of study on two-dimensional foam drainage isuch short. Koehler et al. made theoretical estimations based

n the plug flow assumption, and carried out pulsed drainagexperiments [6]. Hutzler and Wang et al. experimentally studiedorced drainage with a single input into a 2D foam confined in aele-Shaw cell [8,9]. They also extended the drainage equation

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J. Huang, Q. Sun / Colloids and Surfaces A:

o two-dimensions. Valuable insights have already been pro-ided into liquid spreading both vertically due to gravity, andorizontally due to capillary force. They found that the progres-ion of the liquid in time in both horizontal and vertical directions described by power laws, which are strongly dependent on theatio of input width to the width of the Hele-Shaw cell.

In this paper, we report results about the drainage procedure ofwin streams of liquid injected onto dry foams. At the beginningtage, the twin input liquid spread independently both verticallynd horizontally, behaving in the same way as the single-inputorced drainage observed by Hutzler et al. [8]. However, oncehe two drainage waves meet in the foam between two noz-les, they emerge into a singular solitary wave and propagate ashown in Fig. 1. The merging of liquid fronts will affect drainageehaviour in other regions, such as along the vertical line beneathhe two input points. What we want to understand is that, afterwo drainage liquid merge, (1) how drainage waves evolve bothlong the vertical position down the center of the two input pointsnd along the vertical position underneath an input nozzle, and2) how much the influence of separation distance between thewo input nozzles on both drainage evolutions is. Another pur-ose is that this study would benefit the design of micro-mixershich have found many promising application in engineeringelds and life sciences. Micro-mixers are a kind of minitypeachine to mix reagent, which are usually made by manufac-

uring hundreds of thousand micro channels or micro nozzlesn silicon and plastic substrate. Due to the dominating laminarow on the microscale, mixing in micro-mixers relies mainly

n molecular diffusion and chaotic advection [10]. Foam-likehannels have high surface area and where chaotic advection isikely to be realized by manipulating the laminar flow, result-ng in improved safety and low consumption of the valuable

ig. 1. Experimental set-up for forced drainage with two narrow input nozzles.ark foaming solution is injected onto the dry foam. The foam with a higher

iquid faction appears darker. The liquid fraction profiles and their evolutionslong the two vertical dotted lines are measured, respectively.

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cochem. Eng. Aspects 309 (2007) 132–136 133

eagents used. Therefore, it is necessary to quantitatively studywin inputs (and multiple inputs in further work) liquid spreadinghrough foams, which not only has great theoretical importanceor the development of foam physics, but also be beneficial forngineering practice such as in the field of fine chemistry, foodrocessing and pharmacy.

We use the technique proposed by Hutzler et al. to visualizerainage wave by adding food dye into the foaming solution8], and calibrate the transmission ratio of light with liquid frac-ion. After the drainage waves merge in the region betweenwo nozzles, we track the positions of drainage wave fronts inertical directions both beneath one input point and down theenter of the two inputs. The influence of distance separatinghe twin inputs is then discussed. We further carry out numericalimulations with a two-dimensional drainage equation.

. Experimental set-up and technique

Our experiments are conducted in a Hele-Shaw cell. The cellonsists of two parallel glass plates of a height of 30 cm and aidth of 20 cm, separated by a depth of D = 3.0 mm, as illus-

rated in Fig. 1. The foaming solution is made of de-ionizedater, Goldfish detergent liquid and dark red food dye liquidith a volumetric ratio of 40:1:10. Foam is produced in the celly blowing nitrogen through a syringe needle into solution. Aicro-flow needle value is used to precisely control the flow

ate of nitrogen, and thus the bubble size in our experimentss well distributed within d = 1.1 ± 0.1 mm. Twin nozzles withame opening size of 3 mm are placed with a separation of Sn the surface of dry foams. Foaming solution is input at a flowate of 0.0026 ml/s for each nozzle, which is provided with twondividual syringe pumps of the same model. The subsequentrainage wave can be easily observed due to the sharp contrastetween dark colour of input solution and bright white back-round of dry foam. To capture the drainage images, we usehigh-speed CCD camera which is mounted with a long focus

ens and could provide high quality pictures with 2 million pixelst 10 frames/s.

Calibration procedure is necessary to relate the grey value ofhe digital images to the local liquid fraction. These includene-dimensional forced drainage experiments in a slenderele-Shaw cell which has a rectangular cross-section of arearec = 3 mm × 6 mm = 18 mm, filled with gas bubbles of theame diameter d = 1.1 mm as in two-dimensional drainage exper-ments. From the profiles of light intensity versus verticalosition of the resulting drainage wave over time, we can deter-ine the moving wave’s velocity υ. Because the time required

or the added liquid to reach the cell walls was less than 0.5 s,uring which the wave will propagate downwards by muchess distance which is negligible compared to the length of theell, the drainage wave can be regarded as one-dimensional.nce this constant moving wave has reached the bottom of

he foam column, the liquid fraction φ is constant through-

ut and is obtained from Q = φArecυ. Fig. 2 shows the relationetween the experimentally determined liquid fraction φ andhe measured transmission ratio of light intensity T. Then in theollowing two-dimensional drainage experiments, we have used

134 J. Huang, Q. Sun / Colloids and Surfaces A:

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ig. 2. Calibration between liquid fraction φ and measured transmission ratiof light intensity T. The experimental data can be well described by a fit function= 0.293 + 1.038 e−29.39φ .

his relationship to transform the light intensity into local liquidraction.

We notice that as liquid fraction less than 0.01, T goes up to.0. It indicates that the optical technique in this work does notrovide higher sensitivity to recognize dryer foams.

In the following experiments, input flow rate of 0.0026 ml/snd bubble size of 1.1 mm are constant. We just vary separationistances to observe how the two drainage waves evolve. Liq-id fraction is converted from measured transmission ratio by

sing the calibration relations. In one-dimensional experiments,e obtained a scaling law between the velocity of the drainage

olitary wave υ and input flow rate Q: υ ∝ 3.442Q0.5 ∝ Q0.5,hich indicates Poiseuille flow within Plateau borders. Thus,

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ig. 3. Forced drainage with two narrow input nozzles. Separation distances are: (aiquid spreading into region between two nozzles and merging into a unity wave to pr

Physicochem. Eng. Aspects 309 (2007) 132–136

he following simulations are conducted based on the border-ominated drainage model.

. The two-dimensional drainage equation

Based on the assumption of Poiseuille flow in Plateau borders,he drainage equation for one dimension is derived [11,12], andas extended to two-dimensional form by Hutzler et al. [8],

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here A is the cross-sectional area of the Plateau border,hich can be related to liquid fraction φ; C a geometric con-

tant related to the shape of a Plateau border, and is giveny C = (31/2 – π/2)1/2, for a concave triangular-shaped one; ρ,

and γ the density, viscosity and surface tension of surfac-ant solution, respectively; f represents the surface mobility. Foroiseuille flow condition, f is taken as 50. g is the gravity. Theimensionless form of Eq. (1) is written as

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here ξ = z/x0, η = x/x0, τ = t/t0, A = φx20. The scale x0 is

he length of a single Plateau border, expressed as x0 =7/1212−1/2V

1/3b (Vb is bubble volume); time scale t0 is

50μx0/Cγ . The Bond number B0 = ρgx20/Cγ is proposed to

easure the relative importance of gravity over surface tensionn the drainage. In our simulations, properties of washing-upiquid solution are taken as those of pure water at 20 ◦C, i.e.

) 2 cm and (b) 4 cm, respectively. (c) The simulation result. The processes ofopagate downwards are displayed.

Physicochem. Eng. Aspects 309 (2007) 132–136 135

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Fig. 4. (a) Along the vertical direction in the halfway between the two noz-zles, the wave-front position is nearly linearly increasing with time. It indicatesdrainage moves downwards at a constant speed. Note that after drainage startedaround 8 s, two waves begin to merge. (b) Along the vertical direction beneathoce

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J. Huang, Q. Sun / Colloids and Surfaces A:

= 1 × 10−3 kg/s m, ρ = 1 × 103 kg/m3. γ is 2.5 × 10−2 kg s−2

hich is about one third of surface tension of water [3].In one-dimensional forced drainage, the flow rate is found

o be proportional to the square of liquid fraction based onoiseuille flow type assumption. For 2D drainage, boundaryondition of constant input flow rate is an awkward feature,hich is usually represented by imposing a greater liquid frac-

ion φl along the input width. In the two-dimensional drainagequation, the relation between dimensionless input flow ratesim and experimental flow rate Qexp is clearly defined bysim = Qexpt0/(x2

0D). And Qsim = 2φ2B0ω/δη, where φ is theesulting liquid fraction in forced drainage, ω is the dimension-ess total input width and δη is the spatial step size used inhe simulation, that is in grid definition for the finite differencepproximations in our simulations. However, an opening prob-em exists about how to match Qsim to the liquid fraction φlmposed along the input width. One possible solution may behat we simply run a drainage simulation with a φl for a period ofime t. The volume of input solution could be obtained by inte-rating area by local liquid over all the area passed by solution,nd the flow rate of Qsim is calculated to be the volume over theime t. By repeating the above procedure with a series of φl, weould calibrate Qsim, Qexp with φl respectively. In this work, weet up φ = 0.0005 as the initial background liquid fraction of thery foam. Along the input width of the nozzles, we ran manyimulations with a range of φl from 0.02 to 0.35. By comparingxperimental results with a flow rate of 0.0026 ml/s and a sep-ration distance of 4 cm, we find that φl = 0.23 is a right value.hus, in our simulations, we settle the liquid fraction φl = 0.23s a constant value along the input width, and just simply varyhe separation distances.

. Results

Fig. 3 shows liquid spreading in both vertical and horizontalirections. The separation distances are: (a) 2 cm and (b) 4 cm,espectively. Fig. 3c is the simulated result that shows a goodimilarity with Fig. 3b.

At the beginning stage, two drainage liquids spread individu-lly and this process has been well studied by Hutzler et al. [8].hey found that positions of the drainage wave in both verticalnd horizontal directions proceed in the form of power laws. Inertical direction, we got the exponent of 0.71 ± 0.01, while it isround 0.72 in the work by Hutzler et al. Along with time increas-ng, drainage liquids proceed closer in the region between twoozzles and unify to a singular wave propagating downwards.

We propose a dimensionless separation distance χ = S/d toeasure the relative separation S to bubble size d. For example,= 36.36 corresponds to S = 4 cm in Fig. 4. The position where

oam has a liquid fraction value of 0.02 is used to track theotion of drainage wave. Fig. 4 shows positions of wave front

n the vertical directions. We start to continuously take photos

at a rate of 3 pictures/s) about 5 s before introducing solution athe top of the foam. It is simple to recognize a figure showingery beginning of the propagating drainage wave. In this figure,he origin of time, i.e. t = 0, for forced drainage is then set by the

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ne nozzle, position and time are depicted on double logarithmic scales. It indi-ates that a power law could describe the data well. The flow rate is 0.0026 ml/sach nozzle and the separation distance is 4 cm.

rst photo that shows the drainage red liquid. Along the verticalirection down the center of the two nozzles (see Fig. 4a), theave-front position is about linearly increasing with time, which

ndicates drainage moves downwards at a constant speed. We get.50 and 2.35 mm/s in experiment and simulation, respectively.long the vertical direction beneath a nozzle (see Fig. 4b), we

an see that the position could be well described by a poweraw, z = �tβ, with exponents 0.770 and 0.746 in experiment andimulation, respectively. Simulation results, based on φl = 0.23long the input width, are consistent with experiment results.ote that because the foam between two input nozzles is wetter

han the background dry foam, the spreading into the region,riven by capillary forces, is slowed down, resulting in moreiquid propagates into outer regions in the foam. Therefore, thexponent describing the vertical spreading of drainage liquidlong the vertical direction beneath a nozzle will be greater than= 0.71 ± 0.01 for individual drainage wave.We carried out a series of experiments with a range of sepa-

ations: 2, 3, 4, 5, 6 and 7 cm, respectively. Along the direction

nderneath one input point, as χ is increased, the exponent inhe power law for wave-front position over time is decreased,s shown in Fig. 5a. We can see that β is always greater than.71 ± 0.01. The prefactor, by contrast, is increased along with

136 J. Huang, Q. Sun / Colloids and Surfaces A: Physicochem. Eng. Aspects 309 (2007) 132–136

Fig. 5. Along the vertical direction beneath one nozzle, (a) variation of exponentsiwi

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n the power law with dimensionless separation S/d. (b) Variation of prefactorith S/d. It converges to a value around 7.0 as S/d becomes larger. The flow rate

s fixed at 0.0026 ml/s for each nozzle.

ncreasing χ as seen in Fig. 5b. When χ > 40, the curve developsat. Simulation results show the same tendency and qualitativelygreement with experiments.

Along the vertical position in the middle line between the twonputs, the speed is measured by tracking the moving wave front.s indicated in Fig. 6, the speed is decreased with increased χ.owever, its variation is only about 14% as χ increases six times.

. Outlook

Bubble coarsening and film rupture are noticeable in ourxperiments. How to estimate their effects on the obtained results

n this work? If using different flow rates, exponents in Fig. 5a

ay be held, while prefactors in Fig. 5b and speed in Fig. 6ill vary. How much are the variations? The two questions wille further investigated quantitatively so that we could use as a

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ig. 6. Along the vertical direction in the halfway between the two nozzles,ariations of drainage speed with S/d. Despite a small discrepancy exits betweenimulations and experiments, they are agreeable qualitatively. The flow rate isxed at 0.0026 ml/s for each nozzle.

uideline to make foam-like channels applicable in the designf micro-mixers.

cknowledgements

This work has financial support from the National Scienceoundation of China (Grant Nos.: 20336040; 20490201) and

he Scientific Research Foundation of the State Human Resourceinistry of China.

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[3] S.J. Cox, G. Bradley, S. Hutzler, D. Weaire, J. Phys. Condens. Matter 13(2001) 4863–4869.

[4] A. Saint-James, D. Langevin, J. Phys.: Condens. Matter 14 (2002)9397–9412.

[5] S.A. Koehler, S. Hilgenfeld, H.A. Stone, Langmuir 16 (2000) 6327–6341.[6] S.A. Koehler, S. Hilgenfeld, H.A. Stone, Europhys. Lett. 54 (2001)

335–341.[7] S. Hutzler, G. Verbist, D. Weaire, J.A. van der Steen, Europhys. Lett. 31

(1995) 497–502.[8] S. Hutzler, S.J. Cox, G. Wang, Colloid Surf. A 263 (2005) 178–183.[9] G. Wang, Forced foam drainage in two dimensions, Master Thesis, Trinity

College, University of Dublin, 2004.10] N.T. Nguyen, Z. Wu, Micromixers—a review, J. Micromech. Microeng. 15

(2005) R1–R16.11] D. Weaire, S. Hutzler, The Physics of Foams, Oxford University Press,

England, 1999.12] G. Verbist, D. Weaire, A.M. Kraynik, J. Phys. Condens. Matter 8 (1996)

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