ILASS Americas 27th Annual Conference on Liquid Atomization and Spray Systems, Raleigh, NC, May 2015

Viscoelastic Droplet Formation in a Microfluidic T-junction

O. Y. Shonibare∗, K. Feigl, F. X. TannerDepartment of Mathematical SciencesMichigan Technological University

Houghton, MI49931-1295, USA

AbstractThe production of uniformly-sized droplets has numerous applications in various fields including thebiotechnology and chemical industries. For example, in the separation of mixtures based on their relativeabsorbency, an optimal arrangement of monodispersed droplets in columns is desired for an effectiveseparation. However, very few numerical studies on the formation of viscoelastic droplets via cross-flowshear are available, none of which have considered the case when the flow of the continuous phase isCouette. In this work, the effect of flow type and fluid elasticity on drop size and droplet formationdynamics was investigated in a viscoelastic-Newtonian system. Two-dimensional numerical simulations,using the Volume of Fluid (VOF) method within OpenFOAM, have been performed to predict the sizeand detachment behavior of a viscoelastic droplet in a Newtonian matrix. The results obtained show goodqualitative agreement with experimental work. In both cases where the flow of the continuous phase ispressure-driven (P-flow) and plane Couette (C-flow), there was a decrease in drop size as the cross-flow shearrate increased. However, for a fixed average shear rate, the drop sizes generated in C-flow were found to besmaller than that in P-flow. It was also found that the influence of elasticity on drop size became accen-tuated as the cross-flow shear increased. An increase in elasticity was accompanied by a decrease in drop size.

Keywords: microfluidics, T-junction, viscoelasticity, Giesekus model, non-Newtonian fluid, satellitedroplets.

∗Corresponding Author: oyshonib@mtu.edu

Introduction

The dispersions of two or more immiscible liq-uids is referred to as an emulsion. Some examplesof emulsions include mayonnaise, vinaigrettes andbutter. Emulsions have found great uses in manyindustries. For example, in the agriculture industry,emulsion technology aids dilution and provides bet-ter sprayability of insecticides and pesticides; in thepharmaceutical industry, they are applied to makedrugs more edible and fine-tune dosage of active in-gredients while in the food industry, emulsions influ-ence the physical appearance and mouthfeel of foodproducts. Major emulsification methods used toproduce uniformly-sized microdroplets include mi-crofluidic processes, microchannel emulsification andmembrane emulsification. A very good understand-ing of droplet formation mechanism enables the de-termination of the feasibility and boundary of theuse of membrane emulsification in different kinds ofapplications.

Typical apparatus for the production of emul-sions include agitators, rotor-stator systems, high-pressure valve homogenizers and ultrasound sys-tems. These apparatus rely on turbulence for thedisintegration of large drops in a pre-emulsion. Amajor drawback with this technique is that in mostcases it results in droplets that are highly polydis-perse [1]. The drop size distribution has a stronginfluence on the physical and chemical propertiesof emulsions. For example, in the food industry,monodispersed emulsions improve the qualities ofa product such as mouthfeel, physical appearance,flavor and shelf-life [11]. Also, the generation ofdroplets in these devices are accompanied with en-ergy consumption and shear stresses that are veryhigh. This is not only expensive but has a damag-ing effect on food and pharmaceutical products [9].Improvements on these techniques were made overthe years, some of which include membrane emul-sification, microchannel emulsification and microflu-idic processes. In membrane emulsification, dropletsare generated either as a result of the decomposi-tion of a coarse emulsion after being forced througha membrane channel or shearing of the pure injec-tion source by the continuous phase. The membraneused in membrane emulsification devices could beeither fixed or dynamic, where the rotation/ vibra-tion of the membrane also aids in the pinch-off ofdroplets from the membrane surface [16, 18]. Ingeneral, microfluidic devices are categorized as ei-ther flow focusing or microfluidic junctions. Of allmicrofluidic junctions, the T-junction is easiest toconstruct [5, 11]. In the T-junction, the dispersedphase is injected at normal direction into a flow-

ing stream of the continuous phase, droplets thendetach from the tip of the injection source as a re-sult of several factors including the accumulation ofpressure upstream of the growing droplet and dragfrom the continuous phase. Although productivityof droplets via membrane emulsification is higherwhen compared to either microchannel emulsifica-tion or microfluidic processes, they are highly poly-disperse [17]. On the other hand, with microflu-idic processes, the user gains precision over the size,homogeneity and even the inner composition of thedroplets.

In a membrane emulsification apparatus, whenthe ratio between the pore distance and size is small,it can result in droplet coalescence. In an attemptto resolve this problem, Schadler and Windhab [10]devised a rotating membrane emulsification appa-ratus with an adjustable distance between pores.They conducted a study to determine the effects ofrotational speed and the volume ratio of the dropand matrix phase on drop detachment characteris-tics. They found that the rotational speed of themembrane has a direct relationship on the size ofdroplets formed and also claimed that the width be-tween gaps has a strong impact on the formationmechanism. Similar results were found in other ex-periments [19] . The presence of surfactant in emul-sions reduces the interfacial tension between differ-ent pairs of phases which helps to lower the emulsi-fication pressure and promote stability of droplets.Several authors have investigated the role of surfac-tants in the droplet formation process of membranedevices [6, 7, 9, 12]. Their results showed that in-terfacial tension has an inverse relationship with thesize and formation time of droplets. Van der Graafet al. [11] studied droplet formation in a T-shapedmodel system for a cross-flow membrane emulsifica-tion device. He observed a direct relationship be-tween droplet size and flow rate. It has also beenshown in some studies that size of droplets decreasesas the wall shear stress is increased [7, 8].

Most experimental studies on droplet formationusing elastic fluids have utilized flow focusing devices[13, 14, 15] while only a few were performed with aT-shaped microchannel [20, 22]. Hong and Cooper-White [13] investigated the formation of carbopoldispersions that shear thins and possess yield stressvia a flow-focusing micro geometry. They claimedthat below a critical value of the continuous phaseflow rate, Qc, there is a direct relationship betweenthe size of droplets formed and the viscosity ratio butbeyond this critical value, non-Newtonian propertiesof the fluid begin to surface and results in a decreasein droplet size when viscosity ratio increases. They

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also argued that in the absence of satellite droplets,the shear thinning and elastic property of the fluidresults in the formation of drops with smaller sizesthan the case when elasticity is neglected. Theyattributed this result to the formation time beingshorter. Steinhaus et al. [15] studied the effect ofchannel dimension and fluid elasticity on the gen-eration of polymeric drops within a Newtonian ma-trix in a flow-focusing micro channel. Their resultsshowed that increasing elasticity produced longerthread lengths and longer detachment times. Sim-ilar results were also found in [13, 14]. Husny andCooper-White [20] studied the detachment dynam-ics of droplets formed in a T-junction. A Boger fluidwas used as the dispersed phase and silicone oil forthe continuous phase. It was determined that thepresence of elasticity in the drop phase precipitatedelongated filaments, which also resulted in the pro-duction of satellite droplets between drops producedat regular intervals. The form of these filaments wasshown to depend on the viscosity ratio and molecularweight of the polymers. In addition, the characteri-zation of secondary drops formed was investigated ingreat detail and it was concluded that the monodis-persity of these satellite droplets depended predomi-nantly on the viscosity ratio and the flow rate of thecontinuous phase.

A few numerical studies have been conductedon the characterization of viscoelastic droplet for-mation in a Newtonian stream. For example, vis-coelastic drop formation at an aperture [3, 4] andin a flow-focusing channel [2]. To the best of theauthors knowledge, no numerical study on the for-mation of viscoelastic droplets in a T-junction hasbeen considered.

The goal of this work is to study the formationand detachment of viscoelastic drops in a Newto-nian matrix. We focus on the T-shaped microchan-nel. This work extends the work of Feigl et al. [28]in which Newtonian/Newtonian fluid systems wasstudied using numerical simulations. The role of im-posed flow type, wall shear rate, interfacial tensionand elasticity on formation process of viscoelasticdroplets in a Newtonian fluid is critically examined.Emulsification process with a fixed membrane is cap-tured in the T-cell as the case when the flow of thecontinuous phase is Poiseuille (P-flow) and for a ro-tating membrane, the flow of the continuous phase isplane Couette (C-flow) (see Fig. ). Throughout thiswork, the dimensions H = 500µm and D = 50µmwere fixed.

Computational methodology

Governing equations

The immiscible fluid flow of a Newtonian andviscoelastic fluid in a microchannel is considered inthis study. The flow is assumed to be incompressibleand isothermal.

To track the interface between both phases, theVolume of Fluid methododolgy (VOF) was used. Ascalar-valued function, φ that can take on only val-ues in the range [0, 1] is used to identify the twofluids, with φ ∈ (0, 1) signaling the interface whileφ = 0 and φ = 1 connotes a control volume filledonly with the continuous and dispersed phase re-spectively. The evolution of the scalar field, φ, isthen typically governed by

∂φ

∂t+∇ · (vφ) = 0. (1)

To avoid numerical issues associated with thediscretization of the advection term in Eq. (1), theInter-gamma compressive scheme [26] is employed.With this scheme, the sharpness of the interface andmonotonicity of the volume fraction field is main-tained by introducing an artificial compression terminto Eq. (1) to obtain

∂φ

∂t+∇ · (vφ) +∇ · (φ(1− φ)vc) = 0 (2)

where vc is the difference between the velocity ofthe dispersed phase and the continuous phase. As isevident in Eq. (2), the artifical term is only activewithin the interface region. The physical propertiesof fluids used in the remaining set of equations arethen obtained as

ζ = φζd + (1− φ)ζc, (3)

where ζd and ζc represents a generic property of thedispersed and continuous phase respectively.

The conservation equations of the fluid systemare given by the continuity equation:

∇ · v = 0 (4)

and the momentum equation:

∂ρv

∂t+∇·(ρvv) = −∇p+∇·τs+∇·τp+ρg+Fs (5)

In Eq. (5), τs represents the solvent contribution tostress and satisfies the Newtonian constitutive law:

τs = 2ηsD (6)

3

Figure 1. (i) Droplet formation in a membrane emulsification device with a fixed membrane (ii) Transversesection of a rotating membrane device (iii) Schematic representation of droplet formation in a T-cell. H andD denotes the width of the main channel and the dispersed phase channel respectively.

ηs is the viscosity of the solvent, D denotes the de-formation rate tensor, defined by

D =1

2

(∇v + (∇v)T

)(7)

and the polymer stress, τp, is governed by theGiesekus constitutive law [29]:

τp + λOτp + α

λ

ηp(τp · τp) = 2ηpD (8)

where λ is the relaxation time, ηp denotes the zero-shear-rate polymeric viscosity, α is a parameter thataccounts for the anisotropy of the drag on polymer

molecules in fluid flow, andOτp represents the upper

convected time derivative of τp defined as

Oτp ≡

∂τp∂t

+ v · ∇τp − (∇v)T .τp − τp · ∇v. (9)

In Eqs. (2) to (9), v represents the velocity, p isthe pressure, ρ is the density of the mixture and isobtained as ρ = ρdφ+ ρc(1−φ) (see Eq. (3)), whereρd is the density of the dispersed phase (Giesekusfluid) and ρc is the density of the continuous phase(Newtonian fluid), and g is the gravitational accel-eration. To account for the interfacial tension forceFs in Eq. (5), the Continuum Surface Force model(CSF) [24] is employed, and is obtained as

Fs = −∇ ·(∇φ|∇φ|

)(σ∇φ) (10)

where σ denotes the constant interfacial tension.

Numerical method

A detailed analysis and validation of the numer-ical method employed in this study has been care-fully outlined in a paper by Favero [27], so only abrief discussion would be presented here.

The governing equations and the computationaldomain were discretized using the finite volumemethod. The first-order implicit backward Eu-ler method was used for the discretization of timederivatives. Second order schemes were used to dis-cretize all spatial derivatives except the stress diver-gence term in Eq. (5) which was discretized using afirst-order upwind scheme [27].

At time, t = 0, the velocity, v, pressure, p, andstress, τp field was set to zero. The volume fraction,φ was initialized to 0 in the continuous phase channeland 1 in the channel of the dispersed phase.

For P-flow, the no-slip condition (v = 0) holdsat all walls of the computational domain. A zeronormal gradient was specified at both inlets for pand τp, at the walls for p, τp and φ, and at theoutlet for v and φ. A value of zero was assumed atthe outlet for the polymeric stress, τp, and pressure.The boundary conditions for the C-flow case differed

4

from the aforementioned only at the upper wall andinlet of the continuous phase channel where a non-zero fixed value was set and zero normal gradientwas specified for v respectively.

The sets of linear equations obtained were thensolved using BiCGstab with diagonal incomplete LU(DILU) preconditioning for stress and velocity, andconjugated gradient method (CG), using PCG withdiagonal incomplete-Cholesky (DIC) precondition-ing for pressure.

Results and discussion

In steady shear flow, the exact solution for theshear viscosity, η, and first normal stress coefficient,Ψ1, of the Giesekus model is obtained, respectively,as [25]

η = ηo

(λ2

λ+

(1− λ2

λ

)(1− f)2

1 + (1− 2α)f

)(11)

and

Ψ1 = 2ηo(λ− λ2)f(1− αf)

(λγ)2α(1− f), (12)

whereηo = ηs + ηp, (13a)

λ2 = ληs

ηs + ηp, (13b)

f =1− χ

1 + (1− 2α)χ, (13c)

χ2 =(1 + 16α(1− α)(λγ)2)

12 − 1

8α(1− α)(λγ)2, (13d)

In Eqs. (11) to (13), λ2 represents the retarda-tion time and γ denotes the shear rate.

The graphs of viscosity and stress ratio (Ψ1γη )

against shear rate, obtained using Eqs. (11) and(12), are shown in Fig. . In (i), the effect of therelaxation time on the shear-thinning property of afluid is delineated. As the relaxation time increases,the shear rate at which the fluid begins to shear thindrops. As expected, when λ = 0, the viscosity is in-dependent of shear rate. The stress ratio providesa measure of elasticity in simple shear flow. It canbe seen in (ii) that the elasticity of the fluid flowincreases as the relaxation time, λ, increases.

Grid Independence Study

To confirm grid independence of the results ob-tained in this study, numerical simulations were per-formed on three different meshes [23]. For descrip-tive purposes, the meshes are identified as Mesh 1,

Mesh 2 and Mesh 3 in order of fineness with Mesh 1being the coarsest. A summary of the characteristicsof all meshes are shown in Table 1.

Mesh 4xmin/D 4ymin/D Number of cells

1 0.091 0.058 55,577

2 0.063 0.038 124,960

3 0.042 0.026 281,160

Table 1. Grid properties

Mesh 1 is a two dimensional mesh that com-prises non-uniform hexahedral cells. The cellsaround the mouth of the dispersed phase channelare more refined to accurately predict droplet de-tachment. Cell sizes in this area were4x/D = 0.091and 4y/D = 0.058, where D is the width of the dis-persed phase channel (see Fig. ). On refining Mesh 1by a factor of 1.5 in both x- and y- directions, Mesh2 was obtained. In the same way, Mesh 3 was ob-tained from Mesh 2. The effect of wall shear rate ondroplet sizes are shown in Fig. 3 for all three meshes.

To give a quantitative account of the discretiza-tion error, the Normalized Percent Error (NPE) de-fined as

NPE =n

maxk=1

(|xik − x

refk |

max(xref )

)× 100 (14)

was computed for the result obtained in P-flow andC-flow. In Eq. (14), n denotes the number of wallshear rates considered for each mesh (See Fig. 3),i = 1, 2, 3 identifies the mesh and ref = 3 (i.e. Mesh3 was chosen as the reference mesh). In P-flow, theNPE for the normalized drop size with Mesh 2 wasobtained as 4% while in C-flow, it was obtained as5% . Hence Mesh 2 was chosen as the grid of choice.

Influence of flow type

To investigate the effect of imposed channelflow type, two sets of experiments were conducted.The first set was performed in a pressure-drivenflow (P-flow) while the second was conducted in aplane Couette flow (C-flow). The interfacial ten-sion, σ = 0.0415N/m, the density of the continuousphase, ρc = 960 kg/m3, the continuous phase viscos-ity, ηc = 0.106Pa.s, and the density of the dispersedphase, ρd = 803.87 kg/m3 were fixed for all simula-tions in this section. The relaxation parameter wasset to λ = 5ms. For comparison purposes, the casewith these parameter values shall be identified as thebase case. The range of values for the important pa-rameters in this study and the wall shear rate (γw)

5

101

102

103

104

10-2

10-1

100

λ=0sλ=0.005sλ=0.01s

101

102

103

10410

-2

10-1

100

101

102

λ=0.005sλ=0.01s

Figure 2. (i) Viscosity as a function of shear rate. (ii) Stress ratio as a function of shear rate. The followingparameter values have been used: ηs = 0.002, ηp = 1.2 and α = 0.05. In (ii), the stress ratio when λ = 0(Newtonian) is zero.

is given in Table 2.

Type γw(1/s) Rec Cac Dec Ded

P-flow 600-3000

0.23-1.13

0.13-1.33

0 – 12 0-0.11

C-flow 600-3000

0.68-3.4

0.38-4.0

0 – 15 0-0.11

Table 2. Range of parameter values for P-flow andC-flow. Rec = ρcvcH/ηc, Cac = ηcvc/σ, Dec =λγac, Ded = λγad, where γac denotes the averageshear rate in the continuous phase channel and γad isthe average shear rate in the dispersed phase chan-nel.

The influence of the imposed flow type on thesize of droplet is depicted in Fig. 4. The casewhen both flow types have the same velocity ratio,vc/vd = 136.4, have also been included for compari-son. As shown, the droplet size formed in C-flow islarger. Also, as expected for each flow, the dropletsize decreases as Cac increases. A similar result wasreported by Husny and Cooper-White [20].

Different behavior was observed concerning therole of the continuous phase flow rate, Qc, on dropletsize. In Fig. 5, the leftmost point on each curveconnotes the threshold flow rate for which dropletdetachment occurs and the right end-point indi-cates the flow rate above which iterative conver-

gence could not be attained. All droplet detachmentshowed a dripping behavior. As shown in Fig. 5,which is also evident in Fig. 4, a decrease in dropletsize is seen asQc increases for a fixed dispersed phaseflow rate. Also noticed is that for each continuousphase flow rate, the size of droplets generated inthe C-flow is greater than that in the P-flow. Thisis expected since the applied shear rates are largerin P-flow than in C-flow for a given vc. For addi-tional understanding of the differences seen in thedroplet size between the flow types, the shear ratejust above the mouth of the dispersed phase channelwas monitored from time, t = 0s to t = 1s. Theaverage velocity of the continuous phase and dis-persed phase was kept fixed at vc = 0.15m/s andvd = 0.0011m/s respectively. As shown in Fig. 6,just after the initial time, the shear rate in the P-flow had risen more than twice that in the C-flow.This behavior is seen almost throughout the simula-tion. Consequently, droplets detached in P-flow aresmaller due to the presence of higher shear stress.This is also evident by comparing the number of de-tachments for both cases in Fig. 6 represented byopen symbols.

The relationship between the drop size and themain channel’s Reynolds number, capillary numberand wall shear rate is depicted in Fig. 7. For C-flow,the drop size decreases approximately exponentiallywith Re, Ca and γw at the same rate, the index of

6

500 1000 1500 2000 2500 30000.5

1

1.5

2

2.5

3

3.5

4

Mesh 1Mesh 2Mesh 3

500 1000 1500 2000 2500 3000

1

2

3

4

5

Mesh 1Mesh 2Mesh 3

Figure 3. Drop size as a function of wall shear rate(i) C-flow (ii) P-flow. For C-flow, the wall shearrate was computed as γw = 2vc

H while for P-flow,

γw = 6vc

H

the power law being about −0.744. Also, it could beinferred from Fig. 7(iii) that for a given wall shearrate, the drop size produced in P-flow is larger thanthat in C-flow.

Influence of interfacial tension

Interfacial tension plays a significant role inchemical processes that involves the mixture of twoor more immiscible fluids. For example, it influencesthe likelihood of phases to detach in the productionof emulsions. It is also invaluable in the case of flood-ing during oil production. The use of emulsifiers toreduce interfacial tension aids in assembling of theorganic phase after being inundated with water. Toaccount for the effect of interfacial tension on dropletsize and formation mechanism, different cases wereset up with the parameters kept fixed as the basecase while the interfacial tension coefficient was var-ied from 0.02N/m to 0.0415N/m for both P-flow

Figure 4. Effect of flow type on droplet size. (i)P-flow (ii) C-flow.

and C-flow.The effect of interfacial tension on the size of

droplet is depicted in Fig. 8. For a fixed flow rateof the continuous phase, Qc, a direct relationshipis seen between the drop size and interfacial ten-sion. Droplet sizes smaller than the width of thedispersed phase channel were obtained at high Qcfor σ = 0.02N/m and 0.03N/m. In Fig. 8, the sud-den change in droplet size for σ = 0.02N/m and0.03N/m coincides with the transition from jettingto dripping regime. For example, when σ = 0.03 inthe P-flow, this occurs when Cac increases from 0.35to 0.53 for a fixed Wed (= ρdv

2dh/σ, where ρd and vd

are the density and average velocity of the dispersedphase respectively).

7

0 100 200 300 400 500 600 7001

1.5

2

2.5

3

3.5

4

P-flowC-flow

Figure 5. Influence of flow type on droplet sizes.Vc is the average velocity for both flow types; Theaverage velocity of the dispersed phase,Vd = 0.0011,ηc = 0.106 and ηd = 1.202. On the horizontal axis,the droplet size, d, is normalized by the width ofthe dispersed phase channel, D. The error bars in-dicate the standard deviation from the mean size,computed from n = 5 droplets formed in the chan-nel.

The interfacial tension had a significant effecton droplet detachment behaviour. Two pinch-offregimes were observed as the interfacial tension wasvaried; dripping regime, where drops were generatedat the tip of the pore channel and jetting regime,where there is an incomplete draw back of the neckafter droplet pinch off from the tip of the filament.For the case with σ = 0.0415N/m (base case), alldroplets were formed in the dripping regime. Witha reduced interfacial tension i.e. σ = 0.03N/m and0.02N/m, a transition from jetting to dripping wasseen as the cross-flow shear increased. The tran-sition was earlier for σ = 0.03N/m. Fig. 9 com-pares the snapshot of the alpha field when vc =0.07m/s for σ = 0.0415N/m and σ = 0.02N/mimmediately after droplet detaches from the parentsource. As shown, dripping and jetting is seen inσ = 0.0415N/m and σ = 0.02N/m respectively.

The impact of interfacial tension on themonodispersity of droplets formed was also consid-ered in this study. Fig. 8 shows the standard errorbased on the size of the first five droplets formed inthe microchannel. As discussed in the previous para-graph, the droplet formation mechanism changes asσ is varied. When σ = 0.0415N/m, all droplets

0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

C-flowP-flow

Figure 6. Evolution of shear rate at thepoint,(−1.84 × 10−5m, 4.75 × 10−5m, 0) , just up-stream the t-junction. Open symbols indicate thetime when droplet detachment occurs.

formed showed dripping behavior for the ranges ofQc considered and monodispersity was high. How-ever, as σ reduced to 0.03N/m and 0.02N/m, bothdripping and jetting behavior was seen. Althoughthe monodispersity is high in the dripping regimefor all cases, this worsened as σ is reduced in the jet-ting regime. In P-flow, the maximum coefficient ofvariation, CVmax, for σ = 0.0415, 0.03 and 0.02 was2.5%, 3.1% and 7.2% respectively while in C-flow,CVmax for σ = 0.0415, 0.03 and 0.02 were obtainedas 0.93%, 2.83% and 4.67% respectively. At a fixedwall shear rate, the droplets generated in C-flow werethus found to be more monodisperse.

Satellite droplets are drops formed along withthe primary drop as a result of the non-linear behav-ior of the fluid motion near the pinch-off point [21],hence also known as secondary droplets. In manyapplications, the occurrence of satellite droplets isan undesirable phenomena. With the same testcases, the formation of satellite drops was seen athigh velocity (vc = 0.25m/s) for σ = 0.02N/m and0.03N/m only in the P-flow but not in C-flow. Thiscorresponds to a fixed Rec = 1.13, and Cac = 1.33and 0.71 respectively. For σ = 0.02N/m, satellitedroplets were formed immediately after every pri-mary drop generated at the T-junction. On theother hand, for σ = 0.03N/m, the first secondarydroplet was formed after several primary drop hadbeen formed and this was at t = 0.19s. Fig. 10 com-

8

103

100

101

P-flowC-flow

10-1

100

101

100

101

P-flowC-flow

10-2

10-1

100

100

101

P-flowC-flow

Figure 7. Drop size as a function of (i) Capillarynumber (ii) Reynolds number (iii) Wall shear rate.The dotted lines in each figure represent the line ofbest fit for each flow. The slopes of the blue andred dotted lines in (i), (ii) and (iii) are −0.827 and−0.744, −0.834 and −0.744, and −0.826 and −0.744respectively.

0 50 100 150 200 250 3000.5

1

1.5

2

2.5

3

3.5

4

4.5

σ=0.0415N/m

σ=0.02N/mσ=0.03N/m

0 100 200 300 400 500 600 7000

1

2

3

4

σ=0.0415N/m

σ=0.02N/mσ=0.03N/m

Figure 8. Effect of interfacial tension on drop size(i) P-flow (ii) C-flow. The symbols (�, ) and(�, #) connotes dripping and jetting regime respec-tively.

9

Figure 9. Transition from jetting to dripping asinterfacial tension increases in P-flow. (i) σ =0.02N/m (ii) σ = 0.0415N/m

pares the droplet generation process for three caseswith the same parameters but only differed in inter-facial tension. For the top, σ = 0.02N/m, the mid-dle, σ = 0.03N/m and the bottom, σ = 0.0415N/m.The weber number of the dispersed phase, Wed, forthe cases shown in Fig. 10 were O(10−6). Hence theinertia force from the dispersed phase could be ne-glected. It can be inferred that reducing interfacialtension precipitates satellite droplet formation andthis occurs above a critical Cac.

Influence of elasticity

In this section, the effect of elasticity on dropletsize, detachment and filament dynamics is investi-gated. The elasticity, which was measured via theDeborah number, was raised by increasing the relax-ation time, λ, while keeping the average inlet veloc-ity of the dispersed phase, vd, fixed. The relaxationtime was varied between 0s and 0.01s for each fixedvc as in the base case.

The snapshot shown in Fig. 11 depicts thedroplet formation process in a P-flow for two caseswith similar parameters but differed only in theamount of elasticity - a Newtonian dispersed phase(N) and a viscoelastic dispersed phase (V). As bothfluids enter the continuous phase stream, dropletpinch-off is seen to occur further downstream in (N)and thus has a longer filament. When the dropletdetaches, the front of the Newtonian fluid retracts toa position further downstream than the viscoelasticfluid. Also seen in Fig. 11 is that the interface atthe pinch-off point has higher curvature in (V) than(N).

Figure 10. Satellite droplet formation in P-flow.Top: σ = 0.02N/m, Middle: σ = 0.03N/m, Bottom:σ = 0.0415N/m.

At low shear rates, the viscoelastic fluid had sim-ilar behavior to the Newtonian; increasing λ appearsto have no effect on the drop size. On increasing vc,the effect of elasticity heightened. Fig. 12 is a plotof drop size as a function of relaxation time for acase with low velocity (vc = 0.07m/s) and anotherwith high velocity (vc = 0.2m/s). A slight decreasein drop size is seen for the case with high vc as λ in-creases. Husny and Cooper-White [20] studied theinfluence of elasticity on the pinch-off dynamics andsize of droplet formed within a T-junction geometry,using silicone oil as the continuous phase and botha Newtonian and Boger fluid as the dispersed phase.They reported that the presence of elasticity had noeffect on the droplet formation time and concludedthat although elasticity had a strong impact on thenecking behavior of the injection source, its effect onthe resultant droplet size is minimal.

Finally, we investigate the growth pattern of thefilament. All calculations were taken after the firstdroplet had detached from the injection source. Theevolution of the dispersed phase front (or filamentlength) for different relaxation times at a low andhigh vc is delineated in Fig. 13. The increase insparsity of the symbols as time proceeds indicatesthe non-linearity of fluid motion close to time ofbreakup. The growth rate was found to be almost

10

Figure 11. The effect of elasticity in the dropletgeneration process at a fixed dispersed phase flowrate (vd = 0.0011), viscosity ratio, β = 20 andvc/vd = 181.8. (i) Newtonian (Dec = 0) (ii) Vis-coelastic (Dec = 12).

the same for all cases. In particular, when vc is low,elasticity does not appear to have any effect on thegrowth rate of the filament. Although, the drop for-mation time reduces as elasticity increases.

Conclusions

In this work, the influence of imposed channelflow, interfacial tension and elasticity on drop de-tachment and size was studied in a microfluidic T-junction. The rheology of the viscoelastic fluid wasmodeled using the Giesekus model.

A direct relationship was found between the cap-illary number and drop size in both P-flow and C-flow. In particular, for a given wall shear rate, thesize of droplets generated in P-flow was found tobe larger than that in C-flow. The interfacial ten-sion was found to have a strong effect on the dropletformation mechanisms. Within the range of parame-ters used in this study, both jetting and dripping wasseen. Reducing interfacial tension resulted in a de-crease in drop size which is expected since reducinginterfacial tension results in a higher Ca. However,when it is reduced beyond a critical value satellitedroplets are formed at high shear rate. The existenceof satellite droplets results in increasing the polydis-persity of droplets and in cases where they mergewith primary droplets reduce the mixing precisionin applications. The monodispersity of droplets was

0 0.002 0.004 0.006 0.008 0.011

2

3

4

5

vc=0.2m/s

vc=0.07m/s

Figure 12. Droplet size as a function of relaxationtime (elasticity). The effect of elasticity becomesmore pronounced as Qc increases.

found to be strongly influenced by interfacial ten-sion. Also, its effect on the two types of flows con-sidered in this study - P-flow and C-flow - were differ-ent. For a given constant interfacial tension, C-flowwas found to produce more uniformly-sized dropletsthan P-flow with coefficients of variation less than4.7%. The effect of elasticity was insignificant untilabove a critical continuous phase flow rate where aminimal reduction in drop size was seen as elasticityincreased.

Acknowledgments

The authors acknowledge the computing re-sources of the Superior cluster at Michigan Tech-nological University. This work has been funded bythe Department of Mathematical Sciences, MichiganTechnological University.

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0.5 1 1.5 2 2.5 30

200

400

600

800

1000

λ = 0sλ = 0.01sλ = 0.005s

0 0.1 0.2 0.3 0.4 0.5 0.60

100

200

300

400

500

λ = 0sλ = 0.01sλ = 0.005s

Figure 13. Effect of elasticity on droplet growthdynamics. (i) vc = 0.07m/s (low shear) (ii) vc =0.2m/s (high shear).

13