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Contents lists available atScienceDirect

Flow Measurement and Instrumentation

journal homepage:www.elsevier.com/locate/flowmeasinst

Computational study of droplet breakup in a trapped channel con fi guration using volume of fl uid method

Taw fi q Cheki fi

a,b,⁎

aUniversity Tahri Mohamed Bechar, ENERGARID laboratory, 08000 Bechar, Algeria

bResearch Center in Industrial Technologies CRTI, P.O. Box 64, Cheraga, Algiers

A R T I C L E I N F O

Keywords:

Droplet breakup CFD

VOF (volume-of-fluid) method Multiphaseflow and trapped channel

A B S T R A C T

Computational Fluid Dynamics is performed to numerically investigate the droplet breakup of water in oil in trapped channel configuration. The volume-of-fluid (VOF) method based the commercial code FLUENT is adopted to track the interface. Two designs are suggested to study the effect offlow conditions parameters and outer channel size on the droplet breakup mode, droplet generation frequency and size. As a function of the velocity ratio, droplets are formed in two modes, dripping mode: droplets were generated closed to the nozzle, it was identified at low capillary number (Ca< 0.005) and jetting mode: droplets were produced after a long jet, where the capillary number Ca varies from 0.01 to 0.025. The numerical results indicated the collection channel diameter plays potential role in the determination of droplet size and droplet generation frequency, the shear forces excreted by the continuous phase on the dispersed thread are reduced in the wider model leading to have droplets much bigger than the narrow model, the latter produced small droplets due the high shear stress generated in the confinement region. Furthermore, the droplet frequency and size are found to be strongly dependent on the capillary velocity ratio. However, increasing theflow velocity ratio in both models leads droplets to be generated in high frequency, while the droplet length was decreasing. This work also demonstrates that the VOF method is an effective way to simulate the droplets breakup in trapped channel geometry.

1. Introduction

Droplet generation in immiscible fluids has applications in many fields including pharmaceuticals [1–4], biology [5–7], medicine [8–10], foods[11–13]and cosmetics[13–15]. Different microchannel configurations have been used to generate droplet, such as T-junction [16–22], flow focusing [23–29], cross junction [30–32] trapped channel (called also co flowing channel) [33–37]. In these devices, droplets are generated by the shear stress applied by the continuous phase to the dispersed phase [34,38–42], the droplet size and the generation frequency can be controlled by adjusting theflow conditions of the two phases[23,39,41–47]. Many works have been made to study the droplet generation by trapped channel configuration [34–37,48,49]. Umbanhowar et al. [49], described an experimental technique using trapped channel device, the phase to be dispersed is introduced into a trapped channel, surfactant-laden continuous phase via a tapered capillary, Droplet size is found to be function of the ca- pillary tip diameter, the velocity of the continuous phase, the extrusion rate, and the viscosities and interfacial tension of the two phases.

produced droplet sizes ranging from 2 to 200 µm have been produced

using this technique. Using trapped channel configuration for thefirst time, Cramer et al.[34], experimentally investigated the effect offlow rates of both liquids, viscosity ratio and interfacial tension on the drop formation was studied. In addition, the transition point between drip- ping and jetting was focused. Sunflower oil was used as continuous phase and either an aqueous solution ofκ-Carrageenan or an aqueous solution of polyethylene glycol was used as disperse phase. In one of the earliest studies conducted on the trapped channel device, Pingan et al.

[50]performed experimental investigation of droplet generation in co- flow microfluidics, a mechanical vibration in theflow rate of innerfluid is applied, where the frequency of droplet generation is synchronized with the vibration frequency of innerfluid tube, thereby, the vibration is found to be influencing droplet breakup regime (dripping/jetting), where the mechanical vibration enabled the dripping regime to be oc- curred in a wide range offlow rates.

Computational fluid dynamics (CFD) studies provides many in- formation onflow details such as pressures and velocities that are dif- ficult to measure experimentally in some complicated devices [13,44,51–58]. Recently, different techniques were developed for in- terface tracking [43,46,59–66]. Traditionally, the interface that

https://doi.org/10.1016/j.flowmeasinst.2017.11.013

Received 23 September 2016; Received in revised form 11 October 2017; Accepted 12 November 2017

Correspondence to: ENERGARID, Mechanical Engineering Department, University Tahri Mohamed Bechar, Bp. 417, route de Kenadsa, 08000 Bechar, Algeria.

E-mail address:chekifi.tawfiq@gmail.com.

Available online 06 December 2017

0955-5986/ © 2017 Elsevier Ltd. All rights reserved.

T

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separates the immiscible phases can be solved using volume-tracking method[67–73], front-tracking method[74–79], fractional volume of fluid (VOF)[61,62,80–84], phase-field method[85–90], lattice Boltz- mann method (LBM)[14,17,30,47,91–93], the level-set method (LSM) [63,75,94–101], and many others. Slavov and Dimova[102], classified these methods into two forms of tracking: explicit tracking and implicit tracking methods. The Explicit tracking includes: boundary integral method, front tracking method and immersed interface method. While the Implicit one includes: phase-field method and level-set method.

Various numerical studies have been performed to investigate the droplet breakup in microchannels. The theoretical model on breakup of droplets in a microfluidic T-junction at low Capillary number as pre- sented in the work of Leshansky et al.[81]was verified by Afkhami et al.[81]using two-dimensional VOF simulations. Wang et al.[49]

used the VOF method based on Fluent, to study the generation of monodisperse bubbles and droplets. using trapped channel configura- tion, two periodical Taylor modes on bubble formation processes were characterized and focused; shearing and wetting mode, in addition, the effects of gas/liquidflow rate and gas nozzle injection length on the bubble formation process, the bubble/droplet length and the bubbling frequency are found to be strongly dependent on the penetration size of middle tube. Using front tracking method, Jinsong et al.[17]numeri- cally investigated the mechanism of droplet generation in a trapped channel device, the droplet formation mode (dripping/ jetting) was described. Moreover, a transition mode from dripping to jetting regime was focused, this passage was found to be occurred when the jet inertia exceeds the capillary pressure. Suryo et al. [37] numerically in- vestigated the droplet formation from the tip of a tube into a trapped channel immiscible, incompressible Newtonian fluid, finite element method for spatial discretization and an adaptive finite difference method for time integration was performed, transient droplet shapes, fluid velocities and pressures were focused. His numerical results showed that increasing of theflow rate ratio reduces the volume of droplets. Thus, the authors identified a criticalflow rate ratio for the transition from the dripping regime to the tip streaming regime, while for low and middle viscosity ratio, the criticalflow rate varies inversely with viscosity ratio forfixed capillary number. Jaewon et al.[62], used

effect on droplet formation in a trapped channel microfluidic device.

The authors described the preferable conditions that give polydisperse droplets. In addition, four drops pattern was identified; monodisperse droplets with very small droplet satellite, Polydisperse droplets, La- minarflows are formed and only monodisperse droplets pattern. fur- thermore, droplet size is found to be either approximately independent of or strongly dependent on theflow rate ratio. Depending on the re- lation between strain rate and droplet length.

In trapped channel configuration, the dispersed phase channel is located inside the largest channel of continuous phase, as a result, uniform shear stress applied by the continuous phaseflow velocity is surrounded the dispersed phase, at the end of water channel. The competition between the shear forces and surface tension results in droplet formation[85]. Although many studies on droplet generation by trapped channel devices have been carried out with both experi- mental and numerical approaches, a fundamental understanding of the droplet breakup and non-breakup, that account for the effect of the geometry of the devices and simulation of these phenomenon, using VOF method based CFDfluent are still missing. This method has been proved to be a powerful method to simulate immisciblefluidsflows, droplet breakup[62]. In the present paper, we study the droplet gen- eration by trapped channel device at low Reynolds and Capillary numbers. We determine the droplet size, generation frequency as function offlow conditions (velocities ratio, viscous effects) and geo- metrical characteristics. In this section, droplet size depends on velo- cities ratio, viscosities ratio and outer channel diameter. In addition, droplet flow regime (breakup and non-breakup) is focused. The nu- merical results have been compared with experimental and numerical data for validation.

2. Model configuration

Fig. 1presents the geometrical characteristics used for simulation of droplet breakup in trapped channel device; Three main parts of the device can be distinguished as inner channel (includes three sub- channels), Confinement region (it links the inner and the collection Nomenclature

C the volume fraction of onefluid (%)

D diameter (mm)

F the surface tension force fr frequency (Hz)

h grid size (mm)

n unit vector normal to the interface.

U velocity (m/s) ρ density (kg/m3)

µ dynamic viscosity (kg/m−1.s−1) α phase fraction

Β incline angle of the straight attachment wall

σ the surface tension coefficient

τ time (s)

CFD Computational Fluid Dynamics VOF Volume-of-Fluid

HF height function Subscript

Atm atmosphere

O oil

Out outlet

W water

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continuous phase, at the end of Confinement region (nozzle). The continuous phase pressure and stress force the dispersed phase into a confined thread. Then, the dispersed phase breaks inside of the outer channel, at that moment a droplet is formed at the end of the dispersed phase thread. Moreover many important variables can affect droplet breakup mode (jetting dripping) and size.

3. The multiphaseflow model

The VOF (volume offluid) method uses the volume fraction of one fluid phase or component (denoted asC) to characterize the interface [43,58,61,83,84,103–107]. In the bulk phase, in multi-fluid computa- tional cells, 0 < C < 1. In general, the VOF model uses phase aver- aging to define the amount of continuous and dispersed phase in each cell. A variable,α, was defined as[33]:

α= 1 when the cell is 100%filled with continuous phase

α= 0 when the cell is 100%filled with dispersed phase

0 <α< 1 when the cell contains an interface between the two phases.

The densityρ, and viscosityμ, for both phases (water and oil) are calculated using a linear dependence: The subscript 1 is chosen for the continuous liquid (primary) phase, while the subscript 2 for the discrete phase (droplet)[82,108]

= + −

ρ ρ α1 ρ2(1 α) (1)

= + −

µ µ α1 µ (12 α) (2)

4. Numerical solution

The solutions of the velocityfield and pressure are calculated using a body-force-weighted discretization scheme for the pressure, the Pressure-Implicit with Splitting of Operators (PISO) scheme for the pressure velocity [105,109,110]. The body-force-weighted scheme works well with the VOF model, and the PISO scheme is chosen to improve the efficiency of the calculation of the momentum balance after the pressure correction equation is solved[61,111]. The governing equations are the mass conservation equation for each phase and the momentum equation[82]:

tc+u∇ =c 0 (3)

Where the velocity is given by u. In addition, a single momentum equation is used for two-phase-fluid described by[112]:

⃗ ⃗ ⃗ ⃗

∂ + ∇ + ∇ ∇ = −∇ − t(ρ u) .(ρ u u) u. [ ]μ P Fsur

(4) whereF=σҡ (x)n, ҡ is the surface tension force and n is a unit

vector normal to the interface. σ is the surface tension coefficient [28,108]

= ∇

F σ ρ+

ρ ρ

2 α

sur 1 1

1 2

ҡ

(5) κ1is the interface curvature computed as[28]:

= −∇ ∇. ( α /∇α )

1 1 1

ҡ (6)

The CFD software Fluent uses a control-volume-based technique to convert the governing equations into algebraic equations that to be solved numerically. The governing equation used was the mass con- servation equation for each phase and the momentum equation [112,113]. Theflows are taken to be two-dimensional and laminar. The fluid density and viscosity are assumed to be constant for each phase.

Both models of trapped channel geometry and meshes were generated using Gambit. A quadratic mesh was used on all geometry, that the calculations of surface tension effects are more accurate with this mesh than with a triangular or tetrahedral mesh[114,115]. In order to in- crease the resolution of the droplet formation area, Iso-value of water phase inside channel was performed. The inlet of both phases is defined as velocity inlet, with no viscous stress boundary condition, and the pressure of outlet is the same as the atmospheric pressure, with no-slip boundary conditions at all the walls. The second-order upwind scheme is used for discretization of the momentum equation. The SIMPLE scheme is taken as the pressure-velocity coupling scheme, while the PRESTO! is taken as the pressure discretization scheme. The geometric reconstruction scheme is used for volume fraction discretization. The calculations were converged for (10−5) of time steps to generate a database. The residual was 10−3for the continuity and velocity com- ponents. In order to check the grid independence, different grid sizes were tested, the suitable grid sizes allows to have the same results of Cramer et al. [34] for droplet breakup regime (dripping/jetting), however the intervals mesh sizes we used for thefirst model ish = 0.008 mm, while h = 0.012 mm for the second model. A grid in- dependence study has been checked by varying the number of cells in the domain. A grid having 10496 cells forh= 0.008mmwas sufficient as increasing of refinement didn’t produce any change in velocity and pressure profiles, at the confinement and the main channel. Velocities corresponding to Capillary number ranging from 0.01 to 0.025 are used in the simulation.

5. Results and discussion

In the present work, the effects offlow conditions (theflow velocity and the viscosity ratio) and geometrical size on the droplet breakup process are investigated.

Fig. 2.Flow regime evolution as function of both phases velo- cities (Nb: non-breakup).

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5.1. Influence offlow velocity

Droplet breakup is affected by theflow dynamics of both the dis- perse and the continuous phase, Cramer et al. experimentally presented droplet generation mode in a trapped channel device, while Jinsong et al.[48]used front tracking method to numerically investigate the breakup in trapped channel liquid in microchannel. First simulations have been conducted for a range of flow velocity for both phases to determine the droplet breakup interval for the small model, withfixed viscosity for both phases (µw = 0.001 kg/m.s−1, µoil = 0.0083 kg/

m s−1), the values of velocities are taken referring to the work of Jaewon et al.[62].

The Fig. 2shows the dependence of the flow mode on the flow velocity of both phases and the collection channel diameter. Two re- gimes of droplet formation were observed, thefirst is dripping regime;

in which the droplet is generated closed to the nozzle, when the velocity of the dispersed phase thread is fast enough, the viscous stress of con- tinuous phase exceeds the interfacial tension, the liquid thread start to tapered at the point where it has the highest velocity to conform the principle of mass conservation [48], a high capillary pressure is pro- duced at the end of liquid thread leading the dispersed phase to be broken down. consequently, the dispersed phase at the end of thread forms the droplet. This mode is occurred for both models with low velocity of water. The increasing of water flow velocity leads to the second mode is jettingflow regime, in which, the droplet is formed at the end of jetflow of dispersed phase. Here, the dispersed phase thread grows into a jet due to the highflow inertia of the dispersed phase at the upstream, the breakup of the jet into droplet occurs due to the jet in- stability. However the droplet breakup in trapped channel configura- tion in our simulations agree with experimental observation of Carmer et al.[34]and numerical analysis of Jinsong et al.[48]. A continuous flow is identified when theflow velocity is much high (Uw> 0.5 m/s), where the interfacial tension is much important compared with shear stress. For the reverse case, under a very low water velocity, the high shear stress contracts the dispersed phase to be stopped into the water capillary, only the oil stream is observed in the output of the config- uration.

5.2. Geometry effects

In this subsection, computations are performed to investigate the effect of the collection channel diameter on the droplet breakup. to single out the geometrical parameter, allflow conditions are kept the same for both models, the densities of oil and water respectively are 830 kg/m3and 998.2 kg/m3. the viscosities are: µw= 0.001 kg/m.s−1, µoil= 0.0083 kg/m.s−1, the droplet breakup process for both config- urations are presented in theFig. 3.

TheFig. 3show the droplet breakup process in both models, three stages could be distinguished to reach the droplet breakup; In thefirst stage, a thread of water came out of the nozzle to be injected into the collection channel (Fig. 4. A and a′); In the second stage, the thread develops under the pressure of continuous phase to become longer and

tension, results in the reduction of the shear stress effect because of the collection channel enlargement. As a consequence, the dispersed phase grows enormously (Fig. 3a, b and c) into the collection channel, where the downstream of the thread tends to be reduced because of the ac- cumulation of pressure, since the thread head of dispersed phase has reached the maximum in term of pressure velocity at this moment, the downstream of the thread has reached the minimum pressure, here the shear force is able to break the thread. Subsequently, the droplet gets generated into the collection channel (Fig. 3d), and the thread of dis- persed phase relaxes then takes its initial position. So the collection channel diameter is strongly affects the droplet size in trapped channel configuration. Additionally, it is clear that a small satellite droplet is formed and dragged after the in both models after the main droplet (called also mother droplet), due to the breakup of the cone head during the coming up to the initial stage. Based on these results, it can be deduced that the collection channel diameter influences on the gener- ated droplet size, droplets of a smaller size are formed in the smaller device. While, droplet with bigger size are generated in the larger model (Fig. 3d).

5.3. Viscous effects

In this parametric study, computations have been conducted for only the small model (Dout = 0.4 mm), in order to characterize the dependence of the continuous phase viscosity on the droplet breakup process, two different viscosities ratio are used for oil;μ* = 3.31 andμ*

= 8.29 (where the viscosity of water wasfixed onμ= 0.001 kg/m-s).

TheFig. 4shows droplet breakup process in larger trapped channel configuration model, before droplet get generated, the process takes place in the same three stagesFig. 4, that are described in geometry effects sub-section, in which the dispersed phase thread grows to take the conical shape, after that the shear stress applied by the continuous phase enforces the thread, after a short time the head of the thread breaks up to give birth offirst droplet, the latter is followed by droplet satellite with small size, the effect of continuous phase viscosity could be observed on the droplet size, forμ* = 8.29 (Fig. 4) the droplets seem to be a bit bigger. Increasing the viscosity of oil means to increase the

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inertia force applied on the dispersed phase, which allows to exceeds the surface tension force and breaks up the dispersed thread faster (Fig. 4. d′), leading to generate a small droplet size. On the other hand, decreasing the viscous stress of continuous phaseμ* = 3.31 (Fig. 4) allows to approximately reduce the shear forces, results in slower rupture of the dispersed thread, consequently larger droplet is formed into the collection channel. Compared with geometry effect, here, droplets satellite are generated after the mother droplet with bigger size, after the breaks up of the droplet, the downstream of the thread takes a long formFig. 4d and d′, during the coming up to the initial position, the shear forces ruptures the head to a small droplet. More- over, an interesting result could be deduced is that the droplet size is approximately dependent on the viscosity ratio of both phases.

5.4. Phase diagram

A phase diagram is shown inFig. 5. to characterize the different flow patterns in both models, several simulations are carried out for two speeds of the continuous phase (U = 0.7 m/s and U = 0.9 m/s) with a variation of theflow velocity of the dispersed phase, the viscosity of both phases are; µw= 0.001 kg/m.s−1and µoil= 0.0033 kg/m.s−1. except theflow velocity, allflow parameters are kept the same in this section.

Having analyzed the influences offlow conditions and geometrical size separately, moving now to map the overall effects of droplet gen- eration in co-flow configuration. All collected data are plotted to make a phase diagram (Fig. 5). As droplet breakup was found to be strongly depend onflow velocity of both phases,flow modes are presented in (Fig. 5) by velocity ratio and capillary number of dispersed phaseCa = ηU/σ.

The non-dimensionless number is usually utilized to present the relative importance of viscous effect and interfacial tension forces in order to analyze theflow mechanism inside channels. Here,Cais cal- culated at the droplet breakup zone, beside the nozzle. Threeflow re- gimes are observed; dripping, jetting and continuousflow. Simulation have been made for wide range of capillary number (Ca) for both models. At low capillary number (Ca< 0.005), droplets are pinched off beside the nozzle in dripping regime (DP), where the thread of the dispersed phase takes a conical shape, in this mode the shear stress is dominated the interfacial tension force. However increasing of con- tinuous phase velocity (0.01 <Ca< 0.025) allows to increase the ca- pillary pressure produced at the end of the thread leading the thread to take a long form, the instability of this long jet induces the head to detach and form a droplet in jetting mode (JT). The non dimensionless numberCain the experimental work can be determined by varying the velocity[34]or the viscosity[116]. For all models, a continuousflow (CF) regime is observed with high velocity ratio (Ca > 0.03), the jet grows and extends till the output if the configuration, however the shear forces exerted by the continuous phase fails to break the dispersed phase stream. consequently, a parallel streams (called also slugflow) of both phases is observed in the collection channel. Another interesting interval was identified in the phase diagramFig. 5, it's a mixed zone observed between (DP) and (JT) for 0.005 <Ca< 0.01, droplets in this stage are generated in both mode dripping and jetting. As well as be- tween jetting (JT) and continuous flow (CF) where 0.0025 <Ca< 0.03, both continuousflow and jetting mode are ob- served in this phase.

Fig. 4.Droplet breakup process as function of viscosity ratio of bothfluids forDout= 0.4 mmchannel,μ* =μoilw, Uoil/Uw = 13, the otherflow conditions are the same in both models.

Fig. 5.Phase diagram offlow regime for both configuration; U*;

Uo/Uw, D* = Dout/Dwater, ((DP): Dripping, (JT): Jetting and (CF):

continuousflow) and two mixed zone.

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5.5. Droplet generation frequency and size

The droplet generation frequency and size are very important parameters, for many applications of droplet in microfluidic, it de- termines the number of droplets produced per second, it is given by:fr

= 1 /Δt; whereΔt represents the time difference between two droplets successive. Here the satellite droplets are not considered in term of droplet generation frequency, However, many variables can effects both important parameters, using the same configurations presented previously, computations have been conducted for dripping mode, to characterize the impact of the velocityflow ratio, the collection channel diameter on the droplets generation and its sizes. simulations have been performed for Reynolds number (Re) ranges from 34 to 231.

Fig. 6. shows the evolution of droplet generation frequency (fr) and the non-dimensional droplet length as a function of theflow velocity ratio of both phases, with the collection channel size (D* = 2 and D* = 4) and both phases viscosities. it is clearly observed that both of the frequency and the size can be controlled by the flow condition para- meters and configuration size. The increase of velocity ratio of both phases (U* = 12.5 with constant velocity of water) means to increase the shear force applied by the continuous phase, which allows to cut the dispersed phase thread head faster, however, the time of droplet de- tachment is shorter, which means to produce many droplets in very short time (for D* = 2,μ* = 3.31 and fr was 4210 Hz), the high shear force affected also the droplet size, where the compression of small thread results in the generation of small droplets. Another interesting thing which is the dependence of the frequency and droplet size on the collection channel diameter, for the sameflow velocity ratio and visc- osity ratio the frequencies obtained by the narrow model (D* = 2) are much bigger, in this device the pressure of continuous phase stream in the confinement region is bigger compared of the large model, which leads to easily and quickly pinch offthe thread head of water to form the droplet, this process takes long time in the second model where the

droplet breakup in trapped channel configuration has been numerically investigated. The Volume of Fluid Method was chosen for conducting the parametric study in this device. In this device, droplets of water in oil are generated based the shear stress excreted by the continuous phase (oil) on the dispersed phase (water). Two regimes of droplet breakup were identified as function of flow fluids proprieties and configuration size. Dripping mode; droplets are formed closed to the nozzle and Jetting mode; droplets are generated after a long jet. An interval of mixed mode has been identified, where droplets are pro- duced in both regimes. On comparison of configuration size effects, simulations revealed that droplet size are much bigger for the wider model (with collection channel diameter of 0.8 mm). For the narrow model (D = 0.4 mm), droplets are found to be smaller, due the high shear stress forcing the dispersed phase thread. Additionally, the visc- osity ratio slightly affected the droplet breakup regime, droplets gen- erated with low viscosity ratio (μ* = 3.31) were much bigger then that generated with high viscosity ratio (μ* = 8.29). It is also observed that at low capillary number (Ca< 0.005), droplets are generated in drip- ping regime, while increasing of this number leads the droplets to be generated after a long jet (jetting mode). Moreover, the frequency of generated droplet was also dependent on velocity ratio and the con- figuration size. for the same geometry, droplets breakup frequency in- creases while the droplet size decreases as the velocity ratio increases.

The numerical results allowed to plot a global phase diagram for Ca< 0.1,Re< 300, it revealed that droplet breakup regime and size are strongly dependent on both theflow velocity ratio and the config- uration size. Finally, droplets generation frequency and size in the trapped channel device can be controlled by adjusting theflow velocity ratio, the viscosity ratio of both phase and outer collection tube.

Furthermore, it is shown that Volume of Fluid Method is very useful way to simulate the droplets breakup in a trapped channel devices.

Acknowledgements

Fig. 6.Droplet generation frequency as function offlow velocity ratio of bothfluids and collection channel diameter; D*=Dout/ Dwater,μ*=μo/μw.

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