Numerical Study on the Effect of Viscoelasticity on Drop Deformation

8/6/2019 Numerical Study on the Effect of Viscoelasticity on Drop Deformation

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J. Non-Newtonian Fluid Mech. 155 (2008) 8093

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j n n f m

Numerical study on the effect of viscoelasticity on drop deformation in simple

shear and 5:1:5 planar contraction/expansion microchannel

Changkwon Chung a, Martien A. Hulsen b, Ju Min Kim c, Kyung Hyun Ahn a,, Seung Jong Lee a

a School of Chemical and Biological Engineering, Seoul National University, Seoul 151-744, Republic of Koreab Department of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlandsc Department of Chemical Engineering, Ajou University, Suwon 443-749, Republic of Korea

a r t i c l e i n f o

Article history:

Received 6 March 2008

Received in revised form 21 May 2008

Accepted 4 June 2008

Keywords:

Drop deformation

Shear flow

Contraction/expansion flow

Finite element method

Front tracking method

Oldroyd-B model

DEVSS-G

SUPG

Matrix logarithm

a b s t r a c t

We implemented a finite element-front tracking method (FE-FTM) to understand the drop dynamics in

microfluidic applications. We investigated the effect of viscoelasticity of both drop and medium. The

Oldroyd-B model was used for viscoelastic fluid, and DEVSS-G/SUPG/matrix logarithm algorithms were

applied to improve numericalstability.We first verified thereliabilityof thealgorithm by comparingwith

previous results under simple shear flow. The results were in good agreement with previous reports in a

wide range of parameters such as capillary number ( Ca) and Deborah number (De). Then we applied the

algorithm to a 5:1:5 planar contraction narrow channel expansion flow which is typical microfluidic

flows. One of the goal of this study is to explore the effect of viscoelasticity of drop and medium on

drop deformation, and to propose the strategy to control the drop shape. When a Newtonian drop was

suspendedin a viscoelastic medium, in thenarrow channel region, we observed an ellipsoid-like drop in

contrast to a bullet-like drop which is the typical shape for the case of a Newtonian drop in Newtonian

medium. When a viscoelastic drop is suspended in Newtonian medium, the extent of drop swell to the

cross-stream directionwas enhanced at the exitof the narrow channel.We explainedthese phenomena in

terms of thenormal stressdifference developedin theviscoelastic fluid. Thepresent study shows that the

viscoelasticityplays a significant role ondrop dynamicsand drop shape inparticular.We expect this studywill be helpful to understand the drop dynamics in microchannel flow, and provide useful information in

manipulating drops in complicated geometries such as microfluidic channels.

2008 Elsevier B.V. All rights reserved.

1. Introduction

Recently, microparticle fabrication attracts much attention due

to the increasing demand for potential applications such as drug

delivery [1], photonic crystal [2] and DNA multi-plexed analysis

[3]. Microfluidic channels have been extensively used due to their

inherent high-throughput way of operation [4,5]. In microparti-

cle fabrication, one important issue is to control the shape of the

microparticles, which is typically realized through two-phase fluid

systems [57] or continuous flow lithography [3,4,8]. Thus, it is

essential to understand drop dynamics in two-phase systems to

generate a targeted shape. In the conventional industries, drop

deformation problem is also quite important in polymer blends

[912] and polymer processing [13] to control the microstructure

and rheology and thus to enhance the product quality.

Corresponding author. Tel.: +82 2 880 8322; fax: +82 2 880 1580.

E-mail address: [email protected] (K.H. Ahn).

In these problems, drop dynamics is governed by diverse forces

such as surface tension and viscous forces, and the rheology of the

fluid can be important when non-Newtonian fluids are involved.

Several phenomenological models [1418] have been proposed to

explain drop dynamics in Newtonian and viscoelastic fluids which

show good agreement with experiments. However, these theories

are limited to small deformations in simple viscometric flows since

thedropshape is assumed tobe ellipsoidal.Thus,thereis a growing

demand for numerical analysis of drop deformation to gain phys-

ical insights for academic interest as well as practical purposes.

A number of numerical studies that focused on non-Newtonian

effects have been conducted for drop deformation in viscometric

flows, e.g., shear flow [1925] and extensional flow [2628]. Also,

non-Newtonian effects in complicated flow problems have been

investigated for various applications such as gas injection in a cap-

illary tube [29], bubble rising [30], pendant drop formation [31,32]

and drop deformation in contraction/expansion flows [33,34]. As

for drop deformation in contraction/expansion flows, only a few

numerical studies have been reported on axisymmetric drop defor-

mation in contraction/expansion flow [33,35] and on planar drop

0377-0257/$ see front matter 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.jnnfm.2008.06.002
http://www.sciencedirect.com/science/journal/03770257mailto:[email protected]://dx.doi.org/10.1016/j.jnnfm.2008.06.002http://dx.doi.org/10.1016/j.jnnfm.2008.06.002mailto:[email protected]://www.sciencedirect.com/science/journal/03770257


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C. Chung et al. / J. Non-Newtonian Fluid Mech. 155 (2008) 8093 81

Table 1

Previous works for viscoelastic effects on drop deformation in shear flow

Authors Algorithm Dimension VE model Fluids Ca De

Pillapakkam and Singh [25] Level-set 2D, 3D Oldroyd-B NV, VN 0.24 0.4

0.6 8

60 8

Yue et al. [24] Diffuse interface 2D Oldroyd-B NV 0.24 0.4

Yue et al. [22] Diffuse interface 2D Oldroyd-B NV, VN 0.1, 0.2 2Yue et al. [23] Diffuse interface 2D Oldroyd-B NV 0.1 10

VN 0.1 2

Chinyoka et al. [21] Volume of fluid 2D Oldroyd-B NV, VN, VV 0.24, 0.6 0.4

NV, VN 60 8

Khismatullin et al. [20] Volume of fluid 3D Oldroyd-B, Giesekus NV 0.24, 0.6 0.4

Aggarwal and Sarkar [41] Front tracking 3D Oldroyd-B VN 0.35 3

Fig. 1. Schematic diagram for remeshing algorithm of front mesh: (a) element addition; (b) element deletion.

deformation in convergentdivergent flow [34,36]. Furthermore,

these studies were limited to Newtonian fluids [35,36] or shear-

thinning fluids modelled by the Carreau model [33], and the effect

of viscoelasticity has rarely been reported.

There are many studies on the effect of viscoelasticity on drop

deformation in simple shear flow. Boger fluids have been consid-

ered as a model fluid to investigate the effect of viscoelasticity

in shear flow [3740]. The elasticity of medium fluid helps to

align a drop along the flow direction and to enhance deforma-

tion in shear flow, whereas the elasticity of a drop contributes

to resist drop deformation [3840]. The adverse tendency of vis-

coelastic effects on drop dynamics was qualitatively reproduced in

two-dimensional simulations for small deformation (Ca = 0.1, 0.2)[22] and for large deformation (Ca = 0.6) [21,25]. Recently, three-

dimensional numerical simulations were performed [20,41]. How-

ever, a discussion for accurate viscoelastic solutions is still under

way even in two-dimensional approaches [2125]. Detailed infor-

mation for related works is presented in Table 1. Comparing our

solutions with previous works [2125] is quite meaningful since

the problem is regarded as a good model problem to validate the

numerical algorithm. We also study drop deformation in 5:1:5 pla-

nar contraction/expansion flow which is frequently encountered in

microfluidics. Though numerous experimental studies have been

conducted on drop manipulation at small length scale [42,43], the

numerical study on the effect of viscoelaticity for drop dynamics

has rarely been presented. Here, we present how viscoelasticity of

the fluid affects drop dynamics in microchannel flows.

We implemented a front tracking method [44] to dealwithtwo-

phase flow problems for various applications. The advantage of a

front tracking method is that it provides robust solutions since a

large number of front particles on the interface are advected, i.e.,

tracking interfaces into the subgrid level is available [43,45,46].

As a sharp interface method, the front tracking method is also

applicable to viscoelastic fluid problems.A front tracking algorithm

based on finite difference method (FD-FTM) has been presented

Fig. 2. Definition of unit tangent vector and unit normal vector in front elements.


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82 C. Chung et al. / J. Non-Newtonian Fluid Mech. 155 (2008) 8093

Fig. 3. Schematic diagram of the drop deformation under shear flow: (a) initial drop and computational domain; (b) characterization of deformed drop.

for two-dimensional dropdeformation in periodic extensional flow

[28,47] and for three-dimensional viscoelastic drop deformation in

shear flow [41]. However, the finite difference based approaches

[28,41,44,47,48] are subjected to geometrically simple problems

with structured meshes. Here, we propose a finite element-front

tracking method (FE-FTM) to investigate the effect of viscoelas-

ticity on two-dimensional drop deformation. We believe that this

method is more adaptable to microfluidic applications since finite

element method has no geometrical constraints.

In the next section, we will explain details of the numerical

formulation of the FE-FTM algorithm. Then we apply the present

algorithm to drop deformation problems under simple shear flow

to validate the reliability of the algorithm for both Newtonian and

viscoelastic fluids. Then we investigate the dropdynamicsin a 5:1:5

Fig. 4. Mesh configurations: (a) UM1; (b) UM2; (c) UM3; (d) UC2.


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planar contraction/expansion flow to show possible applications

to microfluidics. We will investigate the effect of viscoelasticity

on drop dynamics. As for effect of fluid elasticity, we will discuss

the role of the normal stress difference on drop deformation in

microchannel flow, which would be useful in controlling the drop

shape using viscoelastic fluids in microfluidic applications.

2. Numerical method

2.1. Governing equations

We consider incompressible and immiscible two-phase fluid

flow. Inertia is assumed to be negligible. The Oldroyd-B model

is adopted as a constitutive equation for the viscoelastic fluid.

Momentum, continuity and constitutive equations are denoted as

follows:

p s (u + (u)T) nl(x xp) = 0, (1a)

u = 0, (1b)

C

t+ u C (u)T C C u =

1

(C I), (1c)

wherep denotes the pressure,s the solvent viscosity, u the velocityvector, the extra stress tensor, C the conformation tensor, t the

time, the relaxation time of polymer and I the identity tensor,respectively. In Eq. (1a), the last term nl(x xp) correspondsto capillary force, whereis the surface tension coefficient, twice

the local mean curvature of the surface, nl an outward unit normal

vector fromthe surface, and(x xp) theDirac deltafunctionwhichisnon-zeroonlyatx =xp.Here,x is theposition vector in thedomain

andxp is thepositionvector to designate theinterface location.The

relationship between the extra stress tensor and conformation

tensor C is given by

=(1 )

(C I), (2)

where I is theidentitytensor. represents the solvent contributionto solution viscosity as follows:

=s

p + s=

s, (3)

where p is the polymer contribution to the solution viscosity.

2.2. Numerical formulation

Thefinite element method is employed to discretize thegovern-

ing equations Eqs. (1a)(1c). The DEVSS-G scheme [49] and SUPG

scheme [50] are adopted as stabilization schemes. We reformulate

the momentum Eq. (1a), the continuity Eq. (1b) with the DEVSS-G

scheme [49] into following weak form:

;pI + ;(u + (u)T) (1 )(GT + G)

+; {;nl(x xp)} ; t

= 0, (4a)

; u = 0, (4b)

; G (u)T = 0, (4c)

where and are bilinear andbiquadratic shape functions in two-dimension, respectively. and { } denote domain integral and line

integral along the finite element, and { }r means the line integral

along the interface, respectively. is a one-dimensional quadraticshape function, G the velocity-gradient tensor. Variables such as

p, G and are approximated in terms of bilinear shape functions,

while u is discretized with biquadratic shape function. Traction on

the boundary is t =Tnd, whereT = pI ++s(u + (u)T) and nd is

an outward normalvector from the boundary in the computational

domain. With Eq. (4c), the momentum balance Eq. (4a) is stabilized

by introducing an elliptic term [49].

In this study, we employ the matrix logarithm scheme [51] to

enhance numerical stability, such that the positive definiteness

of the conformation tensor C be preserved during computation.

The conformation tensor can be diagonalized with the relation-ship C = RcRT, where R is a matrix composed of the eigenvectors

of C and the diagonal tensor c is constructed from the corre-

sponding eigenvalues [52,53]. We replace the C-based constitutive

equation with the logarithm tensor based formulation. Thus, we

deal with the evolution equation of s = log c =dim

i=1log(ci)nini =dim

i=1sinini , where s is the logarithm tensor in the principal frame.

ci denotes the eigenvalue of the conformation tensor, si is the

eigenvalue of the logarithmtensor, ni is the principaldirection con-

jugated with ci and dim is set to 2 as the dimension number. The

evolution equation makes the problem more stable by taking the

logarithm in order to change the growth of the stress tensor form

exponetial to linear, where the exponential growth is regarded as

the source of the numerical breakdown. The time derivative ofs for

Oldroyd-B model can be written as follows [51]:

s=

dimi=1

2Gii

1

(ci Iii)

ci

nini+

dimi=1

dimj=1

si sjci cj

i /= j

(cjGij + ciGji)ninj,

(5)

Fig. 5. Steady drop deformation with increasing Ca. Viscosity of drop fluid is same

with the medium (= 1). (a) Deformation parameter D; (b) orientation angle .


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Fig. 6. Contour of extensional rate N based on drop orientation (UM3). N = m1 GT

m1, where m1 is the unit vector directing major principal direction of deformed drop.

(a) Ca = 0.1; (b) Ca = 0.3; (c) Ca = 0.5; (d) Ca = 1.0.

where Gii is the velocity-gradienttensor in the principalframe. Con-

sequently, the equation described by the logarithm tensor S in the

global frame is obtained as follows:

S

t+ u S = S, (6)

where S is the transformed tensor of s through the matrix diago-

nalization as S = R s RT. Since C and S are coaxial, it is possible

to build R using the eigenvectors of C. The discrete form of Eq. (6)

with SUPG [50] scheme can be written as

+ s; Sn+1

Sn

t

+ un+1 Sn+1 = + s; Sn, (7)where s is the element-wise upwinding shape function,s =(|uchx| + |uchy|)/(2ucuc). uc denotes the velocity vector atthe center node of an element, hx and uy are the size vectors of the

element which are projected on x-axis and y-axis, respectively. In

this study, a streamline upwinding coefficient of 2 is used, fol-

Table 2

Detailed information of meshes used in this study

Name Elements Nodes DOF xmin/0.5lw

UM1 2,300 9,333 49,437 0.1

UM2 7,935 31,909 168,078 0.05

UM3 26,868 107,689 566,079 0.025

UC2 14,494 58,353 307,585 0.05

lowing some literatures [5456], though it is usually assumed as 1

in other viscoelastic fluid studies [50,57]. It is generally conceded

that proper value of is adopted depending on the problem sincesomewhat large value may introduce a serious error [57]. Super-

script n and n + 1 denote each time step. With the solution S of Eq.

(7), we need to transform it into the principal frame by s = RTSR to

get the conformation tensor C using C = RcRT= RexpsRT. Finally,

the extra stress tensor is calculated by Eq. (2). Then, the solu-

tion vector [G, u, p]T can be obtained after solving the coupled Eqs.

(4a)(4c) at every time step.

2.3. Front tracking method

2.3.1. Temporal integration of interfaces

In the front tracking method, successive front elements com-

prise the interface as in Fig. 1(a), where each front element has two

front particles. The position of front particle, xp on the interface is

Table 3

Comparison of dropdeformationat steady state (t= 3 1, Ca = 0.24,De =0.4, =0.5)

Our data (UC2 mesh) Chinyoka [68]

D D

NN 0.2674 28.90 0.2878 32.26

VN 0.2582 30.32 0.2799 32.53

NV 0.2550 25.38 0.2656 28.12

VV 0.2455 27.00 0.2598 28.22


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Fig. 7. Steady drop deformation with increasing De (Ca = 0.1, = 0.5, UC2). N means

Newtonian fluid and V means viscoelastic fluid. The first character is drop fluid

and the latter one is medium fluid. (a) Deformation parameter D; (b) orientation

angle .

convected with fluid velocity. The interface moves according to the

following equation:

dxpdt

= up, (8)

Fig. 9. Steady drop deformation with increasing De (Ca =0.5, = 0.5, UC2).

where up is the particle velocity of the coordinate xp, which is

interpolated with a biquadratic shape function. New position of

the particle xn+1p is updated from the previous step xnp using the

Runge-Kutta second-order method (RK2) as follows:

xp = tup(xnp, t), (9a)

Fig.8. Shear component ofpolymerconformation tensor C basedon droporientation (Dei =0.5,i =0.5, Ca = 0.1,UC2).The shear component ofC is representedas C = m1Cm2,

where m1 and m2 are the unit vectors directing major and minor principal directions of deformed drop. (a) NV; (b) VN.


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xp = tup(xnp + 0.5xp, t), (9b)

xn+1p = xnp + xp, (9c)

where xp and xp denote the intermediate position vectors of front

particles.

2.3.2. Remeshing of front mesh

The distances between two successive front nodes are initiallysetto a singleuniform value. However, these distancesmay become

nonuniform as the front moves along the flow field. Thus, we need

to keep the distance in a proper range to conserve the accuracy

of interface tracking during calculation. In this study, we conduct

front element addition/deletion as shown in Fig. 1 to keep the size

of front element, sl in the range of 2050% with respect to h, thatis the diagonal size of the smallest mesh in the domain. If a front

element longer than 0.5h is detected as in Fig. 1(a), the element is

divided into two equally sized elements by inserting a new front

particle. If a front element shorter than 0.2h is detected, two neigh-

boring elements are connected at the center of the element after

the element is removed as in Fig. 1(b).

2.3.3. Treatment of surface tension

Surface tension is treated as a force term nl(x xp) in Eq.4(a). The surface tension at finite element node, Fi is computed

using the force at a front element l,fl, with the immersed boundary

method [58]:

Fi(x) =l

fl(xp)2h(x xp). (10)

In the two-dimensional case, the smoothed delta function 2h witha range of two times mesh size h is approximated as follows:

2h(x xp) = d2h(x xp)d2h(y yp), (11)

where d2h(r) is defined as follows [59]:

d2h(r) = 1 + cos(r/2h)

4h |r| 2h0 |r| > 2h

. (12)

Surface tension at a front element fl is calculated based on the

interface topology. Here, we calculate the surface tension using

the curvature of front elements directly [46] instead of polynomial

fitting of interfaces [44,48]. Surface tension fl is calculated using

Frenet-Serret theorem [60] at xp as follows:

fl =

ds

nlds =

ds

dtl

dsds =

pbpa

dtl = (tlpb tlpa

), (13)

where denotes the surface tension coefficient, is the Dirac deltafunction which has non-zero only at the interface and ds is the

length of a front element. tl is a unit tangent vector at front par-

ticles pa and pb of the front element l as shown in Fig. 2. Thisfront element-wise integration results in inward force at the front

particles as shown in Fig. 2.

In this study, we consider two-phase system of purely immis-

cible fluids. However, we deal with this system in a way that the

material properties continuously change across the interface. We

utilize the Heaviside function H(x) to interpolate material prop-

erties of two-phase fluids. The Heaviside function is computed by

solving a Poisson equation with essential boundary condition and

a simple filtering algorithm [48]:

2H(x) = N =

ds

nc2h(x xc)ds, (14)

where N is the outward unit normal vector in the finite element

mesh, which is distributed from the front mesh (interfaces) and

nc is the outward unit normal vector at the center of a front ele-

ment as designated in Fig. 2. In this case, the immersed boundary

method [58] is also used to calculate N in the finite element mesh.

xc is the position of the center of a front element. In the present

study, fluid viscosity (), relaxation time of polymer () andsolventcontribution to solution viscosity () are interpolated as follows:

(x) = m + (d m)H(x), (15a)

(x) = m + (d m)H(x), (15b)

(x) = m + (d m)H(x), (15c)

wherethe subscriptd and m denote drop andmedium, respectively.

Fig. 10. Schematic diagram of drop deformation in contraction/expansion flow.


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3. Results and discussion

3.1. Drop deformation in simple shear flow

3.1.1. Problem description

We first consider drop deformation under simple shear flow to

verifyour finite element-front tracking algorithmand to gain phys-

ical insights on dropdynamics. Thedropwiththe radius a is initially

positioned at the center of the simulation domain. We assume that

two side walls move in reverse directions with a constant velocity,

which results in a constant shear rate . The computational domainhas a width oflw and a length of 24a, as shown in Fig. 3(a). In this

study, four different cases areconsidered: Newtonian drop in New-

tonian medium (NN),Newtonian dropin viscoelastic medium (NV),

viscoelastic drop in Newtonian medium (VN) and viscoelastic drop

in viscoelastic medium (VV). The drop deformation parameter D is

defined as D = (L B)/L + B as shown in Fig. 3(b), where L and B are

the length along two principal directions m1, m2 of the deformed

drop, respectively. We define L and B as the longest and the short-

est distance from the drop center to the interface. The orientation

angle is defined as an included angle between the principal axisof the deformed drop (m1) and the flow direction as shown in

Fig. 3(b).

3.1.2. Parameters

The system is governed by the following dimensionless num-

bers: capillary number (Ca), Deborah number (De) and viscosity

ratio of drop to medium () defined as follows:

Ca =ma

, (16)

Dei=i, (17)

=dm

, (18)

where Dei is defined for drop and medium (i = d, m), respectively.

In our simulation, drop and medium are assumed to have the same

viscosity, i.e.,= 1 forconvenience, andthe relaxation times of dropand medium for the VV system are assumed to be the same. Mesh

configurations are shown in Fig. 4 with wall confinement ratio

(a/lw) of 0.25. We refined the center region with structured ele-

ments to more accurately resolve the drop deformation. Detailed

information of each mesh is given in Table 2.

3.1.3. Code verification for NN

3.1.3.1. Effect of wall confinement. The drop deformation is knownto increase as wall confinement ratio (a/lw) increases [6163]. In

Fig. 11. Mesh configurations (25,010 elements, 101,005 nodes,xmin/w = 0.05): (a) whole domain; (b) zoomed view of contraction region.


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this section, we investigated how the wallconfinementaffects drop

deformation in two-dimensional flow as a preliminary study. With

an additional mesh,we confirmed thatthe dropdeformationshows

a very weak dependency on the wall confinement: e.g., 6.1% of D

deviation for Ca = 0.1 between a/lw = 0.25 and a/lw = 0.125, and 7.9%

ofD deviation for Ca = 0.5. In this study, we set a/lw as 0.25 to com-

pare our solutions with previous studies [21,24,64], following the

convention that the wall confinement effect can be neglected for

a/lw = 0.25 so that the drops do not interact with the wall [21].

3.1.3.2. Effect of Ca. We first investigated the drop deformation

problem for the NN system with unstructured meshes UM1, UM2

and UM3. A fully developed velocity profile was imposed at the

inlet. The results for D and at steady state are presented in Fig. 5.As Ca increases, D increases and decreases, i.e., the drop is morealigned with the flow direction. In Fig. 5(a), our results are con-

sistent with previous reports [19,21,24,64] in a wide range of Ca.

Hence, it can be concluded that the present FE-FTM is reliable. For

small deformation, Ca 0.3 in Fig. 5(b), coincides with the pre-diction from the perturbation theory [65,66]. Though the theory is

based on three-dimensional drop deformation, it coincides with

the present and existing results, which implies that the viscous

stress to the vorticity direction gives little effect on the orienta-

tion angle. In the previous study [64], the drop did not show the

steady-state shape over critical Ca (Cac) 0.875, i.e., the drop was

continuously extended for Ca > Cac. However, in our simulations,

the steady-state shape is maintainedup to Ca 1. Major differences

between the study in Ref. [64] and the present study are the size

of the computational domain and the inlet boundary condition. In

Ref. [64], the domain wassmaller and periodic boundaryconditions

were imposed, therefore, closely packed drops were considered,

while a steady velocity profile was imposed in the present study,

i.e., we investigate singledrop dynamicsexcluding dropdrop inter-

action. In addition, it is noteworthy that there were no differences

between fully developed condition and periodic condition, which

implies thatour domain is large enoughalong the streamwisedirec-

tion.Tounderstand the kinematic characteristics of the sheared drop,

we decoupled the extensional and rotational component of the

velocity-gradient tensor since shear flow is kinematically a mix-

ture of stretching and rotation [67]. Following Yue et al. [22], the

extensional rate N was estimated as N = m1 GT

m1, where G

is the velocity-gradient tensor and m1 is the unit vector in the

major principal direction as shown in Fig. 3(b). The drop shape

is related to the extensional rate N since the extensional stressexerted in the orientation direction is considered as a primary

factor on drop deformation. The contours of the extensional rate

are shown for various Ca in Fig. 6. For weakly deformed drops

(Ca = 0.10.5; Fig. 6(a)(c)), a compressive rate (equivalent to nega-

tive N) is observed inside the drop, while a high extensional rate is

found near the tip; therefore the ellipsoidal shape is developed atmoderate Ca. For a highly extended drop (Ca = 1; Fig. 6(d)), a com-

pressive rate is seen atbothends insidethe drop,and an extensional

rate is observed in the drop center for Ca = 1, i.e., the extensional

stress at the center tends toseparatethe dropin two drops, and the

compressive stress at both ends tends to develop bulbous shapes.

Therefore, we can conclude that the drop shape is determined by

the distribution of the extensional stress.

3.1.4. Code verification for effect of viscoelasticity

3.1.4.1. Effect of viscoelasticity (Ca = 0.24). The effect of viscoelas-

ticity on drop deformation was first studied with numerical

simulation by Pillapakkam and Singh [25] using the level-set

method. They employed the Oldroyd-B model with = 0.5 and

obtained D =0.48 for Ca = 0.24 and De = 0.4 when both drop and

medium are viscoelastic. Later, Yue et al. [24] considered the same

problem using the diffuse-interface method and showed a D value

similar to Pillapakkam and Singh [25]. Recently, Chinyoka et al. [21]

revisited the same problem using the volume of fluid method and

obtained D = 0.26. In the present study, the UC2 mesh (Fig. 4(d))

was used to compare steady D values for Ca =0.24 and De = 0.4.

The number of degrees of freedom (DOF) is provided in Table 2.

Periodicboundaryconditions for u, G and S were imposed, connect-

ing the inflow and outflow boundary. With the above conditions,

our result is similar to that of Chinyoka et al. [68]. Other solu-

tions for NN, NV and VN cases are also compared with Ref. [68]

in Table 3.

3.1.4.2. Effect of viscoelasticity (Ca= 0.1 and Ca = 0.5). In order to

investigate the effect of fluid elasticity on drop deformation,

the simulations were conducted at Ca = 0.1 with the UC2 mesh

(Fig. 4(d)). Calculated D values for NV and VN agree with Yue et al.

[22] as shown in Fig. 7(a). For instance, D was 0.122 (NV) and 0.103

(VN) for Dei = 2 in Ref. [22], while it was 0.122 (NV) and 0.109 (VN),

respectively in our simulation. Comparing NV with VN in Fig. 7,

an increase of medium elasticity (Dem) results in an increase of D

and decrease of. Comparing VV with NV, D was predicted some-what lower than NV at all De, while was almost constant, which

implies that the medium elasticity plays a dominant role in drop

orientation and the drop elasticity tends to resist drop deformation

at small Ca.

In order to understand the drop dynamics in shear flow more

quantitatively, the extensional componentof the conformation ten-

sor was separated from drop orientation through C = m1Cm1. We

Fig. 12. Drop deformation along the position of moving drop (=100, Ca = 0.01,

d = 50): (a) drop shapes and D; (b) drop size (L, B).


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also separated the shear component of the conformation tensor

from through C = m1Cm2, where m2 is the unit vector towardsthe minor principal direction of the deformed drop as in Fig. 3(b).

C is closely related to the drop orientation. For instance, a positive

shear stress in the upward flow region means the stress towards

right (positive x-direction), while a negative shear stress in the

upward flow region means the stress towards left (negative x-

direction). With these concepts, we denote the direction of the

stresswith respect tothe flow directionas an arrow in Fig.8. Forthe

NV case as in Fig. 8(a), the contribution ofC near the drop makesthe drop align to the shear flow, i.e., decreases. On the contrary,for VNcase asin Fig. 8(b), the stress exerts to the opposite direction

(denoted as white arrows) against the flow direction since the fluid

inside the drop flows clockwise. Therefore, the medium elasticity

plays a role to rotate the drop more to the flow direction, whereas

the drop elasticity resists the orientation to the flow direction. For

NN case, it was found thatthe viscous shear stress of both drop and

medium tended to reduce the orientation angle, andconsequently decreases as Ca increases as in Fig. 5(b).

In case of Ca =0.5 (Fig. 9), a different pattern of D and was

observed compared to Ca =0.1 (Fig. 7). For all cases (NV, VN, VV),

D was less than that of NN case. The only similarity between

Ca = 0.5 and Ca = 0.1 is the VN case, i.e., slight decrease of D and

slight increase of as De increases, which is qualitatively inagreement with Ref. [25] for Ca =0.6 even though Ca is not the

same.

3.2. Drop deformation in 5:1:5 planar contraction/expansion flow

3.2.1. Problem descriptionHere, we consider drop deformation in contraction/expansion

flow as a typical microfluidicproblem. Schematic diagram for5:1:5

contraction/expansion channel is presented in Fig. 10. The width of

the narrow channel isw, and the radius of curvature at the contrac-tion/expansioncorner was set as r= w/2. There are two reasons forrounding the corner: first to prevent the interface from going out

of the corner during temporal integration of the interface and sec-

ond to avoid the ambiguity due to stress singularity at the abrupt

corner. The length of the narrow and expansion channel was long

enoughcompared tothe dropsize. Initiallya dropwith the diameter

d was positioned at the center of the upstream region, namely 5wupward from the entrance of the narrow channel. Fully developed

profiles of velocity and extra stress were imposed at the inlet and

outlet.

Fig.13. Dropshapes in theentrance andexitregion dependingon drop size (= 100, Ca =0.01). (a) d =50,yd/w = 3.50, 5.15, 7.08, (b)d =60,yd/w = 2.73,4.85,7.30, (c)d =70,

yd/w = 2.45, 5.10, 8.15, (d) d =50, yd/w = 13.71, 15.47, 17.14, (e) d =60, yd/w = 13.33, 15.57, 17.40, (f) d =70, yd/w = 12.77, 15.60, 17.73.


11/14


3.2.2. Parameters

In this problem, we defined the dimensionless numbers with

the characteristic variables in the narrow channel as follows:

Ca =mV

, (19)

Dei = iV

w/2, (20)

=dm

, (21)

wherem denotes the medium viscosity,the surface tension coef-ficient,i the relaxation timeof viscoelastic fluidi (i = d, m)and Vthemean velocity of thefluid in thenarrow channel. Here, theviscosity

ratio of drop to medium was set as = 100 to consider the dynam-ics of more viscous drops [5]. The mesh configuration is provided

in Fig. 11. Center regions in the upstream and downstream and the

narrow channel were refined with structured elements, where the

characteristic mesh size was set xmin/w = 0.05 to resolve dropdynamics as accurately as possible with affordable computation

time. The mesh was composed of 25,010 elements with 101,005

nodes.

In this study, we re-defined the smoothed delta function h(x xp) in place of Eq. (12) to treat the surface tension using the

immersed boundary method [58]:

h(x xp) = dh(x xp)d

h(y yp)

=1

h2

1

|x xp|

h

1

|y yp|

h

(22)

where h =xmin =ymin. In this application, we should considerthe case that the drop interface exists near the wall within a mesh

size, especially in the narrow channel. In order to calculate the sur-

face tension along the wall where the normal direction to the wall

is the x-direction, a half-ranged delta function dh/2

(x xp) should

be substituted for dh

(x xp) in Eq. (22) to avoid a loss of surface

tension transferred out of the domain [69], i.e., using d

h/2(x xp)means that the smoothed region becomes half in the x-direction

and the weighting factor is doubled in the region. For the same

reason, we also applied a modified smoothed delta function as

h/2

(x xp) = dh/2(x xp)dh/2

(y yp) along the rounded wall at

both the contraction and expansion region.

3.2.3. Newtonian drop/Newtonian medium (NN)

We investigated the dynamics of a viscous drop (= 100) forCa = 0.01 to reproduce the experimental conditions as far as pos-

sible [5]. Drop shape and deformation parameter D are presented

in Fig. 12(a) for the drop with d = 50, which is larger than the nar-

row channel width (w = 40). Dimensionless position of themovingdrop, yd/w is designated on the x-coordinate in the graph, where

the region of the narrow channel corresponds to from 5 to 15.The position of the drop is the mean value of both leading and

trailing meniscus ends. The drop exhibits an oblong shape at the

contraction region and a bullet shape in the narrow channel at

steady state. The drop is swelled in the cross-stream direction (x-

direction in Fig. 10) just after a drop escapes from the exit region,

where D is negative since L < B. Surface tension tends to relax the

deformed dropbackto theinitialshapeof thecircle atthe expanded

channel. In Fig. 12(b), L exhibits an overshoot at the contraction

region where the extensional rate shows a sharp peak along the

center line, whereas B is maintained as a constant due to the

confinement effect by channel walls. At the exit region, a reverse

phenomenon is observed such that B is larger than L, which shows

the drop swell to the cross-stream direction. It is to be noted

that L and B decrease below the initial length due to numerical

Fig. 14. Effect of fluid elasticity on deformation parameter along the position of

moving drop (= 100, Ca = 0.01, =0.5, d = 50): (a) effect of drop elasticity (VN); (b)

effect of medium elasticity (NV).

error. We observed a 2.5% reduction of the radius at the final state

in Fig. 12(b). It was confirmed that this error reduces with mesh

refinement.

The shapes of larger drops of d =50, 60, 70 are presented in

Fig. 13 to investigate the effect of drop size. Drop deformation

seems to be insignificant at the entrance region and in the nar-

row channel, i.e., the oblong shape is commonly observed due to

the geometric constraint as in Fig. 13(a)(c). A slight alleviation of

drop swell is observed at exit region as d increases as evidenced

in Fig. 13(d)(f). It might originate from a slight difference of cap-

illary force (ds nds) since a larger drop at exit region shows arelatively smaller interface curvature of the trailing meniscus.

3.2.4. Effect of viscoelasticity

The drop deformation parameter was also studied for a vis-

coelastic drop in a Newtonian medium (VN) and for a Newtonian

drop in a viscoelastic medium (NV) as in Fig. 14. Increase of D in

the narrow channel is clearly observed as the elasticity of medium

(Dem) increases in Fig. 14(b), whereas the effect of drop elastic-

ity (Ded) is relatively small on drop deformation in Fig. 14(a). D

increases at lower Dem = 0.2 and 0.4, and the drop swells to the

cross-stream direction at the exit region (at around 15 on the x-

coordinate in Fig. 14(a)). For the VN case, the drop-swell is clearly

observed and the recovery process to the initial shape is faster than

for the Newtonian drop, while the drop-swell is remarkably alle-

viated for the NV case. The drop shape and velocity vectors in the


12/14


Fig. 15. Drop shape with velocity vectors in the narrow channel: (a) NN, f/0.5w = 0.103; (b) VN (Ded =2), f/0.5w = 0.113; (c) NV (Dem = 0.4),f/0.5w = 0.186.

narrow channel are provided in Fig. 15. For the NV case, the drop

shows an ellipsoid-like shape with symmetric curvatures of lead-

ing and trailing menisci as shown in Fig. 15(c) which leads to a

D increase, whereas a bullet shape with a relatively flat trailing

meniscus is developed for the NN and VN cases. It was reported

that there is a film of liquid of uniform thickness between drop

interface and tube wall, which is in a region of uniform pressure

with no tangential stress on the interface [70]. Here, we estimated

the dimensionless film thickness between dropinterfaceand chan-

nel wall as f/0.5w = 0.103 (NN), 0.113 (VN), 0.186 (NV), where

Fig. 16. Contour of normal stress difference near drop in the narrow channel (=100, Ca = 0.01, =0.5, d = 50). (a) yy xx (VN, Ded = 2); (b) yy xx (NV, Dem =0.4).


13/14


the thickest one is the NV case. To understand the differences in

drop shape and film thickness, we investigated the contours of

normal stress difference for the VN and NV cases in Fig. 16. For

VN in Fig. 16(a), the normal stress difference develops inside the

drop near the wall, which is considered to help drop swell to the

cross-stream direction at the exit region since the normal stress

difference is exerting to the wall direction during the confinement

in the narrow channel. For the NV case, the normal stress differ-

ence is highly developed in the film region in Fig. 16(b), which is

regarded as a primary factor for the increase of drop deformation

in the narrow channel and for alleviation of drop swell at the exit

region. Therefore, it might be concluded that the normal stress dif-

ference strongly affects drop dynamics including drop shape, film

thickness in the narrow channel and drop swell at the exit region.

4. Concluding remarks

In this paper, we investigate viscoelastic effects on drop dynam-

ics utilizing a finite element-front tracking method (FE-FTM) with

stabilizing schemes as DEVSS-G/SUPG/ML. First, drop deformation

in shear flow is considered. We compare our steady solutions with

previous results [19,21,24,6466] for Newtonian system (NN) to

verify our algorithm, where we present steady solutions up to

Ca = 1. We also validate our viscoelastic solutions in comparison

with previous results [22,25,68]. We explain drop dynamics with

extensional components of Newtonian viscous fluid (N) and ofpolymer conformation tensor (C), and shear component of poly-mer conformation tensor (C) based on drop orientation. Since the

shear component is considered to relate to the drop rotation, a vis-

coelastic mediumtends to rotate a drop to the flow direction, while

a viscoelastic drop tends to resist the rotation to the shear flow.

We also treat drop deformation in 5:1:5 planar contrac-

tion/expansion flow which is a typically confronted problem in

microfluidics. We confirmthat FE-FTM describes physical phenom-

ena qualitatively by presenting the effect of initial drop size in a

Newtonian system (= 100, Ca = 0.01). In the NN and VN cases, the

drop shows two distinctive features: a bullet shape in the nar-row channel and a drop swell to the cross-stream direction at

the exit region. Especially, for the VN case, the normal stress dif-

ference developed inside the drop contributes to the drop swell

at the exit region. Meanwhile, in the NV case, the drop is more

deformed by the normal stress difference of the medium in the

narrow channel. Normal stress differences developed in the film

region between drop interface and channel wall seems to make a

drop into an ellipsoid-like shape instead of a bullet shape, which

induces D enhancement in the narrow channel. Alleviation of the

drop swell to the cross-stream direction at the exit region is also

attributed to the normal stress difference of the medium. These

results tell us that viscoelastic fluids canbe usefulto control shapes

of microparticles in microfluidics.

It is expected that knowledge as stated above on the dropdynamics in shear flow or in mixed flow would be helpful to

manipulate drops in microfluidics. In both cases, the normal stress

difference near the drop plays a significant role for drop shapes

and deformation. We infer that viscoelastic drop is unlikely to be

deformed easily while viscoelastic medium enhances drop defor-

mation since drop is more deformed in the NV case than the VN

case. Wehopethatthis studywould give physicalinsightsto control

the drop shapes for various applications.

Acknowledgements

This work was supported by the Korea Research Foundation

Grant funded by the Korean Government (MOEHRD) (KRF-2005-

213-D00033). The authors wish to acknowledge the National

Research Laboratory Fund (M1030000 0159) of the Ministry of

Science and Technology in Korea. The authors would like to

acknowledge the support from KISTI Supercomputing Center (KSC-

2007-S00-1021).

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Numerical Study on the Effect of Viscoelasticity on Drop Deformation

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