Chemical Engineering Science 64 (2009) 4515 -- 4524

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Chemical Engineering Science

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Effect of viscoelasticity on drop dynamics in 5:1:5 contraction/expansionmicrochannel flow

Changkwon Chunga, Ju Min Kimb, Martien A. Hulsenc, Kyung Hyun Ahna, Seung Jong Leea,∗aSchool of Chemical and Biological Engineering, Seoul National University, Seoul 151-744, Republic of KoreabDepartment of Chemical Engineering, Ajou University, Suwon 443-749, Republic of KoreacDepartment of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands

A R T I C L E I N F O A B S T R A C T

Article history:Received 16 October 2008Received in revised form 12 May 2009Accepted 22 May 2009Available online 10 June 2009

Keywords:MicrofluidicsDrop deformationCirculation in dropletContraction and expansion flowViscoelastic fluidFinite element methodFront tracking methodOldroyd-B model

Recently, we reported how viscoelasticity affects drop dynamics in a microchannel flow using the finiteelement-front tracking method (FE-FTM). In this work, we investigate drop dynamics for a wider rangeof parameters: viscosity ratio between droplet and medium (�), capillary number (Ca), droplet size, andfluid elasticity. The Oldroyd-B model is adopted as the constitutive equation for the viscoelastic fluid.We observe that the drop deformation in a microfluidic channel is dependent on Ca, which is morepronounced for smaller � values. The present work shows that viscoelasticity plays an important rolein drop dynamics with increasing � values for Newtonian droplet in viscoelastic medium, which can beattributed to high normal stress developed in narrow film thickness between droplet and channel forhigher � values. We also study circulation problem inside droplets, which is important in practice, suchas in droplet reactor application. The present work shows that circulation intensity is enhanced withdecreasing � values. We find that the relevance of viscoelastic effects on internal circulation is dependenton � values, and the circulation intensity is distinctively decreased with increasing elasticity for high �values for Newtonian droplet in viscoelastic medium. We expect that the present work be helpful not onlyin controlling droplets but also to improve our physical insight on drop dynamics in microchannel flows.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Droplet-basedmicrofluidics is one of the rapidly developing fieldsdue to its inherent high throughput manner, high process reliabilityand low sample consumption (Teh et al., 2008). Droplet-based mi-crofluidics has much potential to be applied to the high-tech indus-try like drug delivery (Champion et al., 2007), DNA assay (Srisa-Artet al., 2007), and photonic devices (Seo et al., 2005). Accordingly, itis essential to understand relevant physics on droplet microfluidicsfor future applications.

Numerous experimental studies have been proposed on dropletmanipulations at small length scale (Christopher and Anna, 2007;Squires and Quake, 2005; Cristini and Tan, 2004). The droplet ma-nipulation has been conducted in simple geometries as T-junction(Thorsen et al., 2001; Nisisako et al., 2002; Priest et al., 2006) andflow focusing device (Anna et al., 2003; Tan et al., 2006; Woodwardet al., 2007; Yobas et al., 2006). Recently, channel designs become

∗ Corresponding author. Tel.: +8228807410; fax: +8228801580.E-mail address: sjlee@snu.ac.kr (S.J. Lee).

0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2009.05.049

improved and complicated for better performance with novel ideas(Lee et al., 2009; Niu et al., 2008; Su and Lin, 2006). As a basic unit,contraction/expansion geometry has been frequently used in manip-ulating droplets in microfluidic channels. For instance, Cabral andHudson presented a microfluidic tensiometer utilizing drop defor-mation in contraction/expansion flows (Cabral and Hudson, 2006).Dendukuri et al. (2005) devised an efficient microfluidic platform tofabricate non-spherical particles, which utilizes the shape changesof droplets that pass through microfluidic expansion geometry. Theyshowed that ellipsoidal or disk-like particles can be freely fabri-cated by controlling the location where UV light is shed (Dendukuriet al., 2005). To understand drop dynamics in microfluidic contrac-tion/expansion geometries, several studies have been performedon axisymmetric (Harvie et al., 2007; Zhou et al., 2007; Queguinerand Barthes-Biesel, 1997) and planar drop deformation (Harvieet al., 2008; Chung et al., 2008). Harvie et al. (2008) reported thata two-dimensional approach provides qualitative agreements withexperiments of droplet microfluidics. We recently showed that finiteelement-front tracking method (FE-FTM) can be successfully appliedto drop dynamics in mixed flows such as microfluidic contrac-tion/expansion geometries (Chung et al., 2008). We also showed thatviscoelastic effect can be important in drop dynamics. In the present

4516 C. Chung et al. / Chemical Engineering Science 64 (2009) 4515 -- 4524

Fig. 1. Schematic diagram on front tracking method.

Fig. 2. Schematic diagram of drop deformation in 5:1:5 contraction/expansion chan-nel.

work, we present further results on drop deformation in microfluidic5:1:5 contraction/expansion geometry for the wider range of param-eters. We consider the effect of various conditions on drop dynamicsin microfluidic contraction/expansion geometry: viscosity ratio be-tween droplet and medium (�), capillary number (Ca), droplet size,and fluid elasticity.

A droplet in microfluidics has been utilized as a microreactorwith homogenous reaction conditions (Bringer et al., 2004; Songet al., 2003a,b). In order to mix chemical reagents in the droplet,several methods have been suggested such as drop coalescence andside feeding in microfluidics (Li et al., 2007; Song et al., 2006a,b;Tice et al., 2003). Recently, a re-circulating flow pattern inside thedroplet was numerically predicted (Galusinski and Vigneaux, 2008),where the effects of inlet velocity, viscosity ratio and confinement inthe straight channel were reported, from the viewpoint of internalcirculation problem. However, the effect of viscoelasticity on circu-lation inside the droplets has not yet been well understood. In the

Fig. 3. Mesh configurations of M2 for whole domain and zoomed view of contractionregion. M2 is reproduced from Chung et al. (2008).

Table 1Detailed information of meshes used in this study.

Name Elements Nodes DOF �xmin/w

M1 16,561 67,045 369,952 0.0625M2 25,010 101,005 533,419 0.05M3 35,158 141,687 747,292 0.04

present work, we report the effect of viscoelasticity on circulation inthe droplet as well as the effects of viscosity ratio, droplet size andcapillary number.

This paper is structured as follows. The governing equations, theproblem definition, flow geometry and computational details arepresented in the next section. In Section 3, we successively presentresults for drop deformation and internal circulation in the droplets.We will deal with the effects of viscosity ratio, capillary number andfluid elasticity on drop dynamics. Finally, we draw useful conclusionsfor some applications with droplet microfluidics.

2. Numerical method

2.1. Governing equations

We consider isothermal, incompressible, creeping and immiscibletwo-phase fluid flow in a two-dimensional planar case. The Oldroyd-B model is adopted as the constitutive equation for the viscoelasticfluid. Momentum, continuity, and constitutive equations can be de-noted as follows:

∇p − �s∇ · (∇u + (∇u)T ) − ∇ · s− ��nl�(x − xp) = 0, (1a)

∇ · u = 0, (1b)

�C�t

+ u · ∇C − (∇u)T · C − C · ∇u = −1�(C − I), (1c)

C. Chung et al. / Chemical Engineering Science 64 (2009) 4515 -- 4524 4517

Ca =0.1

Ca =0.2

Ca =0.5Ca =1.0

moving distance of drop (yd/w)

D

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

M3M2M1

Ca = 0.1

Ca = 0.2

Ca = 0.5Ca = 1.0

20151050

5

10

15

y d /w

Fig. 4. Drop deformation in contraction/expansion flows with increasing Ca (NN,� = 1,d= 50): (a) drop deformation parameter, D with mesh refinements, (b) droplet shapedepending on the position at Ca = 0.1 with M2, and (c) droplet shape depending on the position at Ca = 1 with M2.

moving distance of drop (yd/w)

D

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Ca = 0.1Ca = 0.2Ca = 0.5Ca = 1.0

Ca increases

2015105

5

10

150

y d /

w

Fig. 5. Drop deformation in contraction/expansion flows with increasing Ca (M2, NN, (M2,NN,�=1,d=70): (a) drop deformation parameter, D with increasing Ca, (b) dropletshape depending on the position at Ca = 0.1, and (c) droplet shape depending on the position at Ca = 1.

Fig. 6. Droplet shapes depending on droplet size, and contours of velocity fields at the contraction flow region (a)–(d) and in the expansion flow region (e)–(h)(M2,NN,� = 1,Ca = 1). (a) d = 30; (b) d = 50; (c) d = 70; (d) velocity magnitude; (e) d = 30; (f) d = 50; (g) d = 70; (h) ux .

where p is the pressure, �s the solvent viscosity, u the velocity vector,s the extra-stress tensor, � the interfacial tension coefficient, � twicethe local mean curvature of the surface, nl the outward unit normal

vector from the interface, and �(x−xp) the Dirac delta function whichis non-zero only at x=xp. Here, x is the position vector in the domainand xp the position vector which designates the interface location.

4518 C. Chung et al. / Chemical Engineering Science 64 (2009) 4515 -- 4524

In the constitutive equation (1c), C is the conformation tensor, tthe time, � the relaxation time of the polymer and I the identitytensor, respectively. Superscript T means the transpose of a tensor.

Fig. 7. Droplet shapes depending on Ca (M2,NN,� = 0.01, d = 50): (a) Ca = 0.5, (b)Ca = 1, and (c) Ca = 2.

moving distance of drop (yd/w)

D

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Ded =0 (NN)Ded =0.4 (VN)Ded =1 (VN)

D

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Dem=0 (NN)Dem=0.2 (NV)Dem=0.4 (NV)

D

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Ded =0 (NN)Ded =1 (VN)Ded =2 (VN)

D

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Dem=0 (NN)Dem=0.2 (NV)Dem=0.4 (NV)

20151050

moving distance of drop (yd/w)

20151050

moving distance of drop (yd/w)

20151050

moving distance of drop (yd/w)

20151050

Fig. 8. Effect of fluid elasticity on deformation parameter along the position of moving droplet (M2, d = 50, � = 0.5). (c) and (d) are reproduced from Chung et al. (2008).(a) VN (� = 1, Ca = 0.1); (b) NV (� = 1, Ca = 0.1); (c) VN (� = 100, Ca = 0.01); (d) NV (� = 100, Ca = 0.01).

The relationship between the extra stress tensor s and conformationtensor C is given by

s= �p

�(C − I), (2)

where �p is the polymer viscosity. The relative solvent contributionto the solution viscosity � is defined as

� = �s

�p + �s= �s

�, (3)

where �s is the solvent viscosity. In the present work, � is assumedto be 0.5 for viscoelastic fluids, following our previous work (Chunget al., 2008). We discretize the above Eqs. (1a)–(1c) with a finiteelement formulation. DEVSS-G (Liu et al., 1998), SUPG (Brooks andHughes, 1982), and matrix logarithm (Hulsen et al., 2005) algorithmsare adopted as stabilization techniques for viscoelastic fluid flowsimulation.

2.2. Front tracking method

For interface tracking and calculation of interfacial tension, a fronttracking method (Tryggvason et al., 2001) is used. As shown in Fig. 1,the interface is comprised of successive front elements where eachfront element has two front particles. The interface is advected withfluid velocity at each particle position using the following equation:

dxpdt

= up, (4)

where up is the particle velocity at the position xp. The temporalintegration is conducted using the Runge–Kutta second-order (RK2)method. We keep the size of front element (�sl) in the range of

C. Chung et al. / Chemical Engineering Science 64 (2009) 4515 -- 4524 4519

Fig. 9. Contours of shear stress Nxy = �((�ux/�y) + (�uy/�x)) near droplets in thenarrow channel (M2, d = 50): (a) � = 0.01, Ca = 0.5 for NN, (b) � = 0.01, Ca = 0.5,Dei = 0.4 for VN, (c) � = 0.01, Ca = 0.5, Dei = 0.4 for NV, (d) � = 1, Ca = 0.1 for NN,(e) �=1, Ca = 0.1, Dei = 0.4 for VN, (f) �=1, Ca = 0.1, Dei = 0.4 for NV, (g) �=100,Ca = 0.01 for NN, (h) � = 100, Ca = 0.01, Dei = 1 for VN, and (i) � = 100, Ca = 0.01,Dei = 0.4 for VN.

5−20% with respect to the diagonal size (h) of the characteristicmesh in finite element domain. For the calculation of interface ten-sion, unit tangent vectors tlp should be defined at the position xp ofeach front element l. Consequently, the resultant forces indicate in-ward direction of the droplet as Fig. 1. The forces calculated on theinterface are transferred to the nodes of the finite elements using theimmersed boundary method (Mittal and Iaccarino, 2005). The read-ers who are interested in more details are referred to the literature(Chung et al., 2008; Chung, 2009).

Fig. 10. Contours of normal stress difference pyy − pxx near droplets in the narrowchannel (M2,D = 50,�i = 0.5): (a) � = 0.01, Ca = 0.5, Dei = 0.4 for VN, (b) � = 0.01,Ca = 0.5, Dei = 0.4 for NV, (c) � = 1, Ca = 0.1, Dei = 0.4 for VN, (d) � = 1, Ca = 0.1,Dei = 0.4 for NV, (e) � = 100, Ca = 0.01, Dei = 1 for VN, and (f) � = 100, Ca = 0.01,Dei = 0.4 for VN. (e) and (f) are reproduced from Chung et al. (2008).

2.3. Problem definition

The schematic diagram for the present study is represented inFig. 2. The length of the narrow channel is 10 times the channelwidth w, and the length of both upstream and downstream is also10w. In the present study, the length scale is non-dimensionalizedwith 1

40 of w (characteristic length scale). Motivated by a pre-vious work (Dendukuri et al., 2005) in microfluidics, we keep200�m:40�m:200�m as 5:1:5 contraction/expansion channel inmind. Therefore, the characteristic length scale can be consideredas 1�m. It is also reasonable to interpret the present system asmicroscale platform since the droplet confinement occurs in themicroscale. On both inlet and outlet, a fully developed condition isimposed. The corners at the entrance and exit of the narrow chan-nel are rounded with r = w/2 to prevent the interface from movingout of the corner and to avoid stress singularity at the sharp corner.

4520 C. Chung et al. / Chemical Engineering Science 64 (2009) 4515 -- 4524

Fig. 11. Streamlines and velocity field in the drop frame of reference in the narrow channel (M2,NN, � = 1): (a) d = 50, Ca = 0.1, (b) d = 50, Ca = 0.5, (c) d = 70, Ca = 0.1,and (d) d = 70, Ca = 0.5.

Viscosity and relaxation time of the fluids are designated as �i and�i, where subscript i represents droplet (d) or medium (m). An initialdroplet with diameter d is positioned at 5w upwards from the en-trance of the narrow channel, and is deformed passing through thecontraction, straight channel and expansion regions, respectively.The drop deformation parameter D is measured using two lengthsL and B along principal directions as D = (L − B)/(L + B), which isshown in Fig. 2. To avoid confusion, only single droplet dynamics isconsidered in the current study. We present multiple snapshots forcomparison purpose only in some figures.

In this problem, three important dimensionless numbers are con-sidered: the viscosity ratio between droplet and medium (�), cap-illary number (Ca) and Deborah number (Dei) of fluid i, which aredefined as follows:

� = �d

�m, (5)

Ca = �mV̄�

, (6)

Dei =�iV̄w/2

, (7)

where V̄ is the mean velocity in the narrow channel. Dei representstwo cases with �i, i.e., Ded is defined for a viscoelastic droplet in aNewtonian medium (VN) and Dem is for a Newtonian droplet in aviscoelastic medium (NV). �i is also defined for viscoelastic fluids inthe same way.

We prepare three meshes: M1, M2, and M3. Mesh configurationsof M2 are shown in Fig. 3 with a zoomed view near the contraction

flow region. The mesh configurations of M1 and M3 are almost thesame as M2 except for size of the elements. We notice that M2 isthe same as in our previous study (Chung et al., 2008). All resultsgiven in the current study are produced with M2 except in the meshrefinement test, in which the results with M1 and M3 are presentedfor comparison purpose. More details on meshes used in the studyare presented in Table 1.

3. Results and discussion

3.1. Drop deformation

3.1.1. Newtonian droplet in Newtonian medium (NN)We begin with equiviscous droplet (� = 1) deformation in 5:1:5

contraction/expansion flow. First, we investigate the effect of Caon drop dynamics for the NN case (Newtonian droplet/Newtonianmedium) with a slightly larger droplet (d = 50) than channel width(w = 40). The drop deformation parameter D and the correspond-ing droplet shapes are presented in Figs. 4(a)–(c), respectively. InFig. 4(a), we show the result of mesh refinement as well, where thedifference among the three meshes is negligible for Ca in the rangeof 0.1–1. As Ca increases, we observe that the droplet becomes moreelongated in the narrow channel, i.e., L becomes larger and B smaller.At the exit of the narrow channel, the droplet is more swollen to-wards the cross-stream direction (x-direction in Fig. 2) at high Cavalues as shown in Fig. 4(c), which is the reason for negative D val-ues in Fig. 4(a).

We also investigate drop deformation with a larger droplet(d = 70) as shown in Fig. 5. By comparing Figs. 4 and 5, we find

C. Chung et al. / Chemical Engineering Science 64 (2009) 4515 -- 4524 4521

Ca

Vc,

max

/ V

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

χ = 0.01

χ = 1

χ = 100

V d /

V

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

χ = 0.01

χ = 1

χ = 100

2.01.51.00.50.0

Ca

2.01.51.00.50.0

Fig. 12. Circulation dynamics of confined droplets depending on viscosity ratio andCa (M2, NN, d = 50). (a) Circulation intensity and (b) mobility of droplet.

that the D value becomes larger with increasing droplet size at thesame Ca in the narrow channel, and the minimum D value (Dmin)is slightly shifted to downstream in the exit region. The drop de-formation towards the cross-stream direction is alleviated sincethe extensional flow field is restricted only to the exit region asshown in Fig. 6(h), where the x-component of fluid velocity (ux)is developed near the exit region. Under the extensional flow inthe x-direction, we expect that the viscous stress is developed inthe restricted region independent of the droplet size. Therefore, thedrop deformation is mainly affected by the capillary stress due tothe droplet interface in the exit region. For a droplet smaller thanthe channel width (w) as shown in Fig. 6(e), the droplet is suffi-ciently deformed (Dmin = −0.31) in the restricted extensional flowregion. As the droplet size increases, the drop deformation to thecross-stream direction is slightly reduced in Figs. 6(f) and (g), whereDmin = −0.22 and −0.16, respectively. Meanwhile, in the contrac-tion flow region, the droplet is more extended with increasing sizetowards the stream direction (y-direction). We measure the dropextension parameter towards the stream direction (2L/d) when themid-point of the droplet passes the entrance of the narrow chan-nel. As shown in Figs. 6(a)–(c), 2L/d increases with increasing size:2L/d = 1.69, 1.90, and 2.06 for d = 30, 50, and 70, respectively.

We also study drop deformation in case of a less viscous dropletwith viscosity ratio of�=0.01. The droplet shows moderate deforma-tion at low Ca as in Fig. 7(a). At higher Ca cases, the shape of dropletis similar to a tadpole in the narrow channel as shown in Figs. 7(b)and (c). We observe that the tail part becomes longer with increas-ing Ca. We can observe the tadpole-like shape only for small � value.The tadpole-like shape is not observed up to Ca = 100 in case of �=1.Recently, in the study of axisymmetric drop deformation in contrac-tion/expansion flow (Harvie et al., 2006), the droplet was severelydeformed at � = 0.0001. The shapes were like `string of sausages'and a `packet of smaller droplets' in the narrow pipe. Therefore, weconclude that the droplet is more deformable for small � values.

3.1.2. Effect of fluid elasticityIn our previous work (Chung et al., 2008), it was observed that

fluid elasticity plays an important role in drop deformation due tothe development of extra stress between the droplet and the chan-nel wall for � = 100. In the present study, however, we observe thatthe drop deformation parameter D is almost identical for � = 1 irre-spective of elasticity of droplet or medium (VN or NV case) as shownin Figs. 8(a) and (b). We also check that the same tendency is repro-duced for � = 0.01 as well. These results mean that fluid elasticityon the drop deformation becomes dominant as � increases. For ref-erence, we present the drop deformation data of the previous work(Chung et al., 2008) for � = 100 in Figs. 8(c) and (d).

To elucidate the difference in the effect of fluid elasticity ac-cording to the viscosity ratio, we investigate the development ofstress in the film between the droplet and the channel wall. Inmost experiments concerned with droplet microfluidics, a thin filmis formed between droplet and wall since the medium wets thechannel wall preferentially (Bico and Quere, 2002a,b). In the filmregion, we observe that the higher shear stress is developed asthe film thickness decreases as shown in Fig. 9. In the figures, wepresent the contours of Newtonian shear stress for three cases: lessviscous droplet (� = 0.01), equiviscous droplet (� = 1), and moreviscous droplet (� = 100). We observe that the film thicknessdecreases as �increases and the shear stress at � = 100 is thehighest among three cases. The dimensionless film thicknesses fordroplets of � = 100 (Figs. 9(g)–(i)) are smaller than those of � = 0.01(Figs. 9(a)–(c)). For example, �f /(w/2) = 0.111, 0.115 and 0.186 inFigs. 9(g)–(i), and �f /(w/2)= 0.333, 0.334 and 0.327 in Figs. 9(a)–(c),where the thickness �f is estimated in the middle of the droplet asdenoted in Fig. 9(a).

The high shear flow in the film region results in the developmentof extra stress as well, i.e., a high level of normal stress difference.That is, the drop deformation in the narrow channel is strongly af-fected by the normal stress difference as shown in Fig. 10. We ob-serve that the highest normal stress difference is developed in �=100and NV case and is two orders of magnitude larger than other cases.In Fig. 10(f), the normal stress is forcing inward direction; as a resulta `symmetric ellipsoidal shape' is obtained while a `bullet shape' isobserved in other cases. Therefore, in order to get more symmetricshape between front and rear curvature in experiments, it is sug-gested to manipulate more viscous Newtonian droplet in the vis-coelastic medium.

3.2. Circulation dynamics in droplet

In the straight channel, bilateral symmetric vortices are formedin a droplet (Tice et al., 2004), and major circulation occurs in eachvortex. In the current study, we define the mean velocity of thedroplet (Vd) as follows:

Vd =∫v dAAd

, (8)

4522 C. Chung et al. / Chemical Engineering Science 64 (2009) 4515 -- 4524

Fig. 13. Velocity field in the drop frame of reference at the contraction and expansion flow region (M2,NN,�=1,d=50): (a) Ca = 0.1, (b) Ca = 0.5, (c) Ca = 0.1, and (d) Ca = 0.5.

where v is the velocity vector of fluid and Ad the area of the droplet.The velocity fields in the drop frame of reference (v′ = v − Vd) arepresented in the narrow channel in Fig. 11. Recently, less viscousdroplets (�=0.1 and 0.5) were investigated (Galusinski and Vigneaux,2008), in which three circulation zones were clearly predicted in caseof � = 0.1 and Ca∼0.133 in the confined channel: a major vortex inthe center of droplet and smaller vortices in the front and rear zonesof the major vortex, which was also observed in the experimentalstudy (Sarrazin et al., 2006) for �=0.05. For �=1 we checked that thethree vortices are developed in the Ca range of O(10−2) although therelevant figures are not shown. For higher Ca, we report two trendson the internal circulation concerned with the major vortex and thefront vortex in Fig. 11. First, the front circulation zone is enhanced asCa decreases as shown in Figs. 11(a) and (b) for � = 1 as well, whichwas reported in the previous study (Galusinski and Vigneaux, 2008)for less viscous droplets as �= 0.1. This tendency is also reproducedfor more viscous droplets (� = 100) though the relevant figures arenot shown. The second is that the vortex pattern is not significantlychanged by the droplet size in case of same Ca as Figs. 11(c) and(d). Comparing to Figs. 11(a) and (b), almost similar patterns areobserved even in approximately three times longer droplets.

We compare the maximum velocity (Vc,max) along the center-line in the drop frame of reference in the major circulation zone. Inthe current study, we define the circulation intensity and mobilityof the droplet as Vc,max/V̄ and Vd/V̄ , respectively. In Fig. 12(a), weshow the circulation intensity for three different values of �, wherecirculation intensity is enhanced in case of � = 0.01 compared with

other cases, e.g., Vc,max of � = 0.01 is approximately six times largerthan for � = 1 at Ca = 0.5, while the mobility of the less viscousdroplet (�=0.01) is increased only about 10% as shown in Fig. 12(b).Therefore, we expect that the enhancement of circulation with de-creasing � is clear whereas drop mobility is almost constant.

In earlier works Song et al. (2003a,b), enhanced internal circula-tion was induced from chaotic advection in the droplet by passingthe droplet through winding channels. In the current study, we ex-pect similar effect by passing a droplet through the contraction andthe expansion flow regions since chaotic flow is observed in time-dependent two dimensional flow as well (Bringer et al., 2004). Toconfirm this effect, we investigate velocity fields in the drop frameof reference at the contraction and the expansion flow regions asshown in Fig. 13. We observe that different vortex pattern is devel-oped at the contraction and expansion flow regions, while only oneis developed in the narrow channel. Therefore, it will be possibleto enhance the internal circulation if we make repeated patterns ofcontraction/expansion channels.

As for the effect of fluid elasticity on circulation dynamics, re-markable difference is observed in case of � = 100, while no signif-icant effect is observed in other cases as � = 1 and 0.01. Basically,in NN case, circulation intensity is weaker in case of � = 100 thanother viscosity ratios as shown in Fig. 12(a). In case of viscoelasticfluids, as shown in Fig. 14(a), circulation intensity dramatically de-creases with increasing Dem while drop mobility is nearly constantas in Fig. 14(b). For example, circulation intensity at Dem = 0.4 inNV case decreases to 26% of the intensity of NN case for � = 100.

C. Chung et al. / Chemical Engineering Science 64 (2009) 4515 -- 4524 4523

Dei

Vc,

max

/ V

0.00

0.05

0.10

0.15

0.20

VN

NV

V d /

V

1.00

1.05

1.10

1.15

1.20

VN

NV

2.01.51.00.50.0

Dei

2.01.51.00.50.0

Fig. 14. Effect of fluid elasticity on circulation dynamics of confined droplets(M2,� = 100,d = 50, ca = 0.01,�i = 0.5). (a) Circulation intensity and (b) mobility ofdroplet.

This suggests that using a more viscous droplet in a less viscous vis-coelastic medium might be useful in some application (e.g., to avoidrapid chemical reaction) where the droplet is used as a transportingdevice preferentially.

4. Conclusions

We investigated drop deformation and circulation in the dropletpassing through 5:1:5 contraction/expansion flows focusing on theeffects of viscosity ratio, droplet size and fluid elasticity as well asCa. As for a Newtonian drop deformation in a Newtonian medium,we investigated the effect of Ca and droplet size on drop deforma-tion for equiviscous droplets. We observe that the droplet deformstowards the stream direction in the contraction flow region, and to-wards the cross-stream direction in the expansion flow region. Itis predicted that higher deformation occurs for smaller �. We alsoobserve that the effect of fluid elasticity on drop deformation is re-markably enhanced for � = 100, which is attributed to the highlydeveloped normal stress difference due to thinner film thicknesscompared to the cases with smaller viscosity ratios. Therefore, theeffect of fluid elasticity becomes more important in drop dynamicswhen more viscous droplet is manipulated in microfluidics. As forinternal circulation in the narrow channel flow, the circulation in-tensity in a less viscous droplet is more enhanced compared to an

equiviscous droplet or more viscous droplet. Furthermore, it wasfound that the circulation intensity decreases significantly due tothe effect of fluid elasticity in a more viscous droplet of � = 100 forNV case. We hope this numerical analysis provides not only physicalinsight on drop dynamics in confined geometries but also usefulguidelines to manipulate droplets in microfluidic channel flows.

Acknowledgments

This work was supported by the Korea Research FoundationGrant funded by the Korean Government (MOEHRD) (KRF-2005-213-D00033). The authors wish to acknowledge the National Re-search Laboratory Fund (M10300000159) of the Ministry of Scienceand Technology in Korea. The authors would like to acknowledge thesupport from KISTI Supercomputing Center (KSC-2007-S00-1021).

References

Anna, S.L., Bontoux, N., Stone, H.A., 2003. Formation of dispersions using “flowfocusing” in microchannels. Appl. Phys. Lett. 82, 364–366.

Bico, J., Quere, D., 2002a. Rise of liquids and bubbles in angular capillary tubes.J. Colloid Interface Sci. 247, 162–166.

Bico, J., Quere, D., 2002b. Self-propelling slugs. J. Fluid Mech. 467, 101–127.Bringer, M.R., Gerdts, C.J., Song, H., Tice, J.D., Ismagilov, R.F., 2004. Microfluidic

systems for chemical kinetics that rely on chaotic mixing in droplets. Philos.Trans. R. Soc. A 362, 1087–1104.

Brooks, A.N., Hughes, T.J.R., 1982. Streamline upwind Petrov–Galerkin formulationsfor convection dominated flows with particular emphasis on the incompressibleNavier–Stokes equations. Comput. Method Appl. Mech. Eng. 32, 199–259.

Cabral, J.T., Hudson, S.D., 2006. Microfluidic approach for rapid multicomponentinterfacial tensiometry. Lab Chip 6, 427–436.

Champion, J.A., Katare, Y.K., Mitragotri, S., 2007. Particle shape: a new designparameter for micro- and nanoscale drug delivery carriers. J. Control Release121, 3–9.

Christopher, G.F., Anna, S.L., 2007. Microfluidic methods for generating continuousdroplet streams. J. Phys. D Appl. Phys. 40, R319–R336.

Chung, C., 2009. Finite element-front tracking method for two-phase flows ofviscoelastic fluids. Ph.D. Thesis, Seoul National University.

Chung, C., Hulsen, M.A., Kim, J.M., Ahn, K.H., Lee, S.J., 2008. Numerical study on theeffect of viscoelasticity on drop deformation in simple shear and 5:1:5 planarcontraction/expansion microchannel. J. Non-Newtonian Fluid Mech. 155, 80–93.

Cristini, V., Tan, Y.C., 2004. Theory and numerical simulation of droplet dynamicsin complex flows—a review. Lab Chip 4, 257–264.

Dendukuri, D., Tsoi, K., Hatton, T.A., Doyle, P.S., 2005. Controlled synthesis ofnonspherical microparticles using microfluidics. Langmuir 21, 2113–2116.

Galusinski, C., Vigneaux, P., 2008. On stability condition for bifluid flows with surfacetension: application to microfluidics. J. Comput. Phys. 227, 6140–6164.

Harvie, D.J.E., Davidson, M.R., Cooper-White, J.J., Rudman, M., 2006. A parametricstudy of droplet deformation through a microfluidic contraction: low viscosityNewtonian droplets. Chem. Eng. Sci. 61, 5149–5158.

Harvie, D.J.E., Davidson, M.R., Cooper-White, J.J., Rudman, M., 2007. A parametricstudy of droplet deformation through a microfluidic contraction: shear thinningliquids. Int. J. Multiphase Flow 33, 545–556.

Harvie, D.J.E., Cooper-White, J.J., Davidson, M.R., 2008. Deformation of a viscoelasticdroplet passing through a microfluidic contraction. J. Non-Newtonian Fluid Mech.155, 67–79.

Hulsen, M.A., Fattal, R., Kupferman, R., 2005. Flow of viscoelastic fluids past a cylinderat high Weissenberg number: stabilized simulations using matrix logarithms.J. Non-Newtonian Fluid Mech. 127, 27–39.

Lee, C.-Y., Lin, Y.-H., Lee, G.-B., 2009. A droplet-based microfludic system capable ofdroplet formation and manipulation. Microfluid. Nanofluid. doi 10.1007/s10404-10008-10340-10402.

Li, L., Boedicker, J.Q., Ismagilov, R., 2007. Using a multijunction microfluidic device toinject substrate into an array of preformed plugs without cross-contamination:comparing theory and experiments. Anal. Chem. 79, 2756–2761.

Liu, A.W., Bornside, D.E., Armstrong, R.C., Brown, R.A., 1998. Viscoelastic flow ofpolymer solutions around a periodic, linear array of cylinders: comparisons ofpredictions for microstructure and flow fields. J. Non-Newtonian Fluid Mech.77, 153–190.

Mittal, R., Iaccarino, G., 2005. Immersed boundary methods. Annu. Rev. Fluid Mech.37, 239–261.

Nisisako, T., Torii, T., Higuchi, T., 2002. Droplet formation in a microchannel network.Lab Chip 2, 24–26.

Niu, X., Gulati, S., Edel, J.B., deMello, A.J., 2008. Pillar-induced droplet merging inmicrofluidic circuits. Lab Chip 8, 1837–1841.

Priest, C., Herminghaus, S., Seemann, R., 2006. Generation of monodisperse gelemulsions in a microfluidic device. Appl. Phys. Lett. 88, 024106.

4524 C. Chung et al. / Chemical Engineering Science 64 (2009) 4515 -- 4524

Queguiner, C., Barthes-Biesel, D., 1997. Axisymmetric motion of capsules throughcylindrical channels. J. Fluid Mech. 348, 349–376.

Sarrazin, F., Loubiere, K., Prat, L., Gourdon, C., Bonometti, T., Magnaudet, J., 2006.Experimental and numerical study of droplets hydrodynamics in microchannels.A.I.Ch.E. J. 52, 4061–4070.

Seo, M., Nie, Z.H., Xu, S.Q., Lewis, P.C., Kumacheva, E., 2005. Microfluidics:from dynamic lattices to periodic arrays of polymer disks. Langmuir 21,4773–4775.

Song, H., Tice, J.D., Ismagilov, R.F., 2003a. A microfluidic system for controllingreaction networks in time. Angew. Chem. Int. Ed. 42, 768–772.

Song, H., Bringer, M.R., Tice, J.D., Gerdts, C.J., Ismagilov, R.F., 2003b. Experimentaltest of scaling of mixing by chaotic advection in droplets moving throughmicrofluidic channels. Appl. Phys. Lett. 83, 4664–4666.

Song, H., Li, H.W., Munson, M.S., Van Ha, T.G., Ismagilov, R.F., 2006a. On-chip titrationof an anticoagulant argatroban and determination of the clotting time withinwhole blood or plasma using a plug-based microfluidic system. Anal. Chem. 78,4839–4849.

Song, H., Chen, D.L., Ismagilov, R.F., 2006b. Reactions in droplets in microflulidicchannels. Angew. Chem. Int. Ed. 45, 7336–7356.

Squires, T.M., Quake, S.R., 2005. Microfluidics: fluid physics at the nanoliter scale.Rev. Mod. Phys. 77, 977–1026.

Srisa-Art, M., deMello, A.J., Edel, J.B., 2007. High-throughput DNA droplet assaysusing picoliter reactor volumes. Anal. Chem. 79, 6682–6689.

Su, Y.C., Lin, L.W., 2006. Geometry and surface-assisted micro flow discretization.J. Micromech. Microeng. 16, 1884–1890.

Tan, Y.C., Cristini, V., Lee, A.P., 2006. Monodispersed microfluidic droplet generationby shear focusing microfluidic device. Sens. Actuators B 114, 350–356.

Teh, S.Y., Lin, R., Hung, L.H., Lee, A.P., 2008. Droplet microfluidics. Lab Chip 8,198–220.

Thorsen, T., Roberts, R.W., Arnold, F.H., Quake, S.R., 2001. Dynamic pattern formationin a vesicle-generating microfluidic device. Phys. Rev. Lett. 86, 4163–4166.

Tice, J.D., Song, H., Lyon, A.D., Ismagilov, R.F., 2003. Formation of droplets and mixingin multiphase microfluidics at low values of the Reynolds and the capillarynumbers. Langmuir 19, 9127–9133.

Tice, J.D., Lyon, A.D., Ismagilov, R.F., 2004. Effects of viscosity on droplet formationand mixing in microfluidic channels. Anal. Chim. Acta 507, 73–77.

Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han,J., Nas, S., Jan, Y.J., 2001. A front-tracking method for the computations ofmultiphase flow. J. Comput. Phys. 169, 708–759.

Woodward, A., Cosgrove, T., Espidel, J., Jenkins, P., Shaw, N., 2007. Monodisperseemulsions from a microfluidic device, characterised by diffusion NMR. SoftMatter 3, 627–633.

Yobas, L., Martens, S., Ong, W.L., Ranganathan, N., 2006. High-performance flow-focusing geometry for spontaneous generation of monodispersed droplets. LabChip 6, 1073–1079.

Zhou, C.F., Yue, P.T., Feng, J.J., 2007. Simulation of neutrophil deformation andtransport in capillaries using Newtonian and viscoelastic drop models. Ann.Biomed. Eng. 35, 766–780 43 Zhou, C.F., Yue, P.T., Feng, J.J., 2007. Simulationof neutrophil deformation and transport in capillaries using Newtonian andviscoelastic drop models. Ann. Biomed. Eng. 35, 766–780.