FEUP - Faculdade de Engenharia da Universidade do Porto ...fpinho/pdfs/CI5-AERC2010_EOF...Numerical...

Numerical Studies of Electro-Osmotic �ows of

Viscoelastic �uids

A.M. Afonso1, F.T. Pinho2 and M.A. Alves1

1 Departamento de Engenharia Química, CEFT, Faculdade de Engenharia daUniversidade do Porto, Portugal, {aafonso, mmalves}@fe.up.pt

2 Departamento de Engenharia Mecânica e Gestão Industrial, CEFT, Faculdadede Engenharia da Universidade do Porto, Portugal, [email protected]

A.M. Afonso, F.T. Pinho and M.A. Alves AERC 2010, Gothenburg, Sweden 1 / 18

Outline

1 Introduction

Electro-Osmotic Flow (EOF): Theory

2 Governing Equations

EOF of Viscoelastic Fluid

Electrokinetics

3 Numerical Solutions

Channel �ows

Complex geometries

4 Conclusions

Outline

1 Introduction

Electrokinetics

Channel �ows

Complex geometries

4 Conclusions

Outline

1 Introduction

Electrokinetics

Channel �ows

Complex geometries

4 Conclusions

Outline

1 Introduction

Electrokinetics

Channel �ows

Complex geometries

4 Conclusions

Electro-Osmotic Flow (EOF)Surface charge

Surface charge:Solution of ions

Overall charge neutrality

Electric Double Layer (EDL):

Mobile di�usive layer

Immobile layer (Stern Model)

Debye layer: λD = 1κ =√

�kBT2noe2z2

Electro-Osmotic Velocity:

Apply an external potentialElectric force F = ρeEViscous forces drag the solution.

Negatively charged suface

Negatively charged surface

Neutral fluidPositively charged layers

Electro-Osmotic Flow (EOF)Electric Double Layer (EDL)

�kBT2noe2z2

- - - -

NeutralZone

distance

ψ DiffusiveLayer

Stern Immobile Layer

λD=1/κ σ

ζ

Electro-Osmotic Flow (EOF)Electro-Osmotic Velocity

�kBT2noe2z2

Negatively charged suface

Negatively charged surface

Induced Velocity

λD=1/κ

+ _

Electro-Osmotic Flows (EOF)Aplications

Aplications:micro �ow injection analysis, micro�uidic chromatography, microreactors,microenergy, microelectronic cooling systems and micro-mixing.

Interesting Flow Instabilities:

Newtonian �uids[1].

Viscoelastic �uids[2].

[1]Park, Shin,Huh and Kang, Physics of Fluids. (2005).

[2]Bryce and Freeman, Lab Chip. (2010).

Electro-Osmotic Flows (EOF)Electrokinetic Instabilities (Newtonian)

ages were analyzed by the image analysis software, Meta-Morph 6.1, of Universal Image.

An interfacial wave is generated and convected down-stream of the T channel in the experimental visualization ofFig. 1. The later stage of growth involves nonlinear deforma-tion of the interface. The flow speed is estimated as about1 mm/s, which is comparable to that of electro-osmotic flow.The zeta potential is measured as �l=−55.2 mV and �h=−34.2 mV for the channel width of 150 �m. The appliedpotential difference is between 0.9 and 3.3 kV and the chan-nel length is 4.5 cm from one inlet to the outlet. The down-stream electric field is estimated to be between 2.4�104 V/m and 8.8�104 V/m. The Helmholtz–Smoluchowski slip velocities are estimated as 0.92 mm/s forthe high concentration stream and 0.56 mm/s for the lowconcentration stream in the case of a threshold potential dif-ference of 0.9 kV.

Figure 2 shows fluorescent microscopic images of fourdifferent channels: A, B, C, and D. Type A is a straight chan-nel and types B, C, and D are channels with square cavitieson one side, square cavities on both sides, and herringbone-shape cavities on both sides, respectively. Square cavities areon the side of a lower ionic concentration in channel B. Thedepth and the interval between cavities are both 150 �m forthe channels B, C, and D. The applied potential difference is1.3 kV which corresponds to an average electric field of3.5�104 V/m. Unstable spiky waves are continuously gen-erated and convected downstream in Fig. 2�a� while thelower concentration stream penetrates deeper into the cavi-ties on the other side in Figs. 2�c� and 2�d�. Channels C andD show a peculiar development pattern from one cavity tothe next while the wave simply grows and gets convected inchannes A and B. Animated images show more irregularwaves as compared with the straight channel at a lower elec-tric field in Fig. 1. This is due to the formation of a strongcounterclockwise vortex in the entrance region to the hori-zontal section as the electric field increases.

Figure 3 is numerical result for channel B. Although

details of the numerical procedure are not presented here, theconservation equations7,8 including momentum, species, andPoisson equation are solved numerically to reproduce netcharge generation near the cavity corner. The result confirmsthat simulated interfacial waves closely resemble those ob-served in the experiment. Note strong counterclockwise vor-tices in the velocity vectors of Fig. 3�a�. The equipotentiallines and electric field vectors are presented in Fig. 3�b�.There is a strong potential gradient at the cavity corners dueto the no-flux boundary condition of the electric field. Thenet charge proportional to −�� ·E�� /��8 has a negative peakvalue at every other corner of the square cavities: � is con-ductivity, E is the electric field, and � is permittivity. As theinterfacial wave in the flow approaches the cavity corner, thepeak of the wave is pulled to the corner by Coulombic force.It remains attached to the cavity corner while the interface isstretched by downstream convective motion in Fig. 3�c�.Successive attractive forces against the mean flow result in amore unstable and fluctuating flow pattern than for channelA.

Figure 4 shows successive fluorescent images betweenthe second and the third cavity in channel D. An initiallygenerated wave is amplified with the lower concentrationstream penetrating into the higher concentration region. Al-though not clear in the images, there is a rapid thin jetejected from the wave front of the higher concentrationstream to the following cavity corner in Figs. 4�b�–4�d�. InFigs. 4�d� and 4�e�, the interface is rolled up clockwise toform a wave front of the same shape as in Fig. 4�a�. The netcharge generated at the cavity corners on the higher ionicconcentration side is responsible for this phenomenon. How-ever the mechanism leading to the complicated flow patternsis not fully resolved numerically at the current stage. The

FIG. 1. Instability of an electro-osmotically driven two-layer flow with dif-ferent NaCl concentrations, dc potential difference of 1 kV, and averageelectric field of 2.7�104 V/m in the channel.

FIG. 2. Experimental concentration images for different channel types at theaverage electric field of 3.5�104 V/m.

118101-2 Park et al. Phys. Fluids 17, 118101 �2005�

Downloaded 05 Feb 2010 to 193.136.33.132. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

Electro-Osmotic Flows (EOF)Electrokinetic Instabilities (Viscoelastic)

b925391b-ga.gif (Imagem GIF, 378x149 pixéis) http://www.rsc.org/ejga/LC/2010/b925391b-ga.gif

1 de 1 05-04-2010 19:12

Governing EquationsMass & Momentum Conservation

Mass Conservation:

∇ · u = 0

Momentum Conservation:

ρDuDt = −∇p+∇ · τ−ρe∇ (φ+ ψ)

Constitutive Equation:

f(τkk)τ + λ∇τ = 2ηD

∇τ = Dτ

Dt − τ .5 u−5uT .τ

Phan-Thien & Tanner (PTT)

f(τkk) = 1 + ελη τkkUpper Convected Maxwell (UCM)

ε = 0 ⇒ f(τkk) = 1

Governing EquationsConstitutive Equation

Mass Conservation:

∇ · u = 0

ρDuDt = −∇p+∇ · τ−ρe∇ (φ+ ψ)

∇τ = Dτ

Dt − τ .5 u−5uT .τ

ε = 0 ⇒ f(τkk) = 1

Governing EquationsConstitutive Equation

Mass Conservation:

∇ · u = 0

ρDuDt = −∇p+∇ · τ−ρe∇ (φ+ ψ)

∇τ = Dτ

Dt − τ .5 u−5uT .τ

ε = 0 ⇒ f(τkk) = 1

Governing EquationsElectric Body Force

Mass Conservation:

∇ · u = 0

ρDuDt = −∇p+∇ · τ−ρe∇ (φ+ ψ)

Nernts-Plankt Equations

∇2φ = 0∇2ψ = − ez� (n

+ − n−)ρe = ez (n+ − n−)∂n±

∂t + u · ∇n± = ∇ · (D±∇n±)±∇ ·

[D±n± ezkBT∇Φ

]

Governing EquationsNernts-Plankt Equations (NP)

Mass Conservation:

∇ · u = 0

ρDuDt = −∇p+∇ · τ−ρe∇ (φ+ ψ)

Nernts-Plankt Equations

∇2φ = 0∇2ψ = − ez� (n

+ − n−)ρe = ez (n+ − n−)∂n±

∂t + u · ∇n± = ∇ · (D±∇n±)±∇ ·

[D±n± ezkBT∇Φ

]

Governing EquationsPoisson-Boltzmann Equations (PB)

Mass Conservation:

∇ · u = 0

ρDuDt = −∇p+∇ · τ−ρe∇ (φ+ ψ)

Poisson-Boltzmann Equations (PB)

∇2φ = 0

∇2ψ = −2noez� sinh(

ezkBT

ψ)

ρe = 2noez sinh(

ezkBT

ψ)

Numerical SolutionsNumerical Method

Finite Volume Method[1]

Structured, collocated and non-orthogonal meshes.

Discretization (formally 2nd order)

Di�usive terms: central di�erences (CDS)Advective terms, high resolution scheme: CUBISTA[2]

Dependent variables evaluated at cell centers;

Special formulations for cell-face velocities and stresses;

Log-conformation for the extra-stress tensor[3].

sinh linearization[4]: sinh(X)=sinh(X)n+(Xn−1-Xn)cosh(X)n

[1]Oliveira, Pinho and Pinto, JNNFM (1998); [2]Alves, Pinho and Oliveira, IJNMF (2003)

[3]Afonso, Pinho and Alves, JNNFM (2009); [4]Chun, Lee and Lee, KARJ (2005)

Numerical SolutionsChannel Flow - Computational Domain

x/2H

y/H

0 2 4 6 8 10-1

-0.5

0

0.5

1

Computational meshes

nº cells 4xmin 4yminx10−4

M1 1800 0.2 8

M2 3600 0.2 4

M3 7200 0.2 2

Numerical SolutionsChannel Flow - Mesh convergence

x/2H

y/H

0 0.5 1 1.5 2

0.96

0.98

M1 1800 0.2 8

M2 3600 0.2 4

M3 7200 0.2 2

x/2H

y/H

0 0.5 1 1.5 2

0.96

0.98

M1 1800 0.2 8

M2 3600 0.2 4

M3 7200 0.2 2

y/H0.5 0.75 1

-1

-0.5

0

0.5

1

AnalyticalM1M2M3

ψ/ζ1

κ=200

κ=20

y/H0.5 0.75 10

0.5

1

1.5

AnalyticalM1M2M3

ε=0.25Deκ=1κ=200

u/ush

x/2H

y/H

0 0.5 1 1.5 2

0.96

0.98

M1 1800 0.2 8

M2 3600 0.2 4

M3 7200 0.2 2

y/H-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

AnalyticalM1M2M3

ψ/ζ1

κ=200

κ=50

y/H-1 -0.5 0 0.5 1

-1

0

1

2

AnalyticalM1M2M3

ε=0.25Deκ=1κ=50Rζ=-1

u/ush

Numerical SolutionsChannel Flow - NP/PB equations

y/H0 0.5 1

0

0.5

1

PB (mesh M2)NP (mesh M2)

u/ush0 β=0.2kH=20εDe2κ0=1

y/H0 0.5 1

0

0.5

1

ψ/ζ1β=0.2kH=20εDe2κ0=1

Analytical Solution

u(y)ush0

= 1β

(1− sinh(κ̄y)cosh(κ̄) − (1− β)

[Ω (1)− sinh(κ̄y)cosh(κ̄) Ω (y)

])

Ω(y) =∞∑n=0

(

13

)n

(12

)n

(23

)n(

32

)n

(32

)n

(− 272 βεDe

2κ0

(sinh(κ̄y)cosh(κ̄)

)2)nn!

β = ηsη0 =ηs

ηp+ηs

Deκ0 = ush0κλ

Numerical SolutionsChannel Flow - NP/PB equations

y/H0 0.5 1

0

0.5

1

y/H0 0.5 1

0.997

1

1.003

n/n0β=0.2kH=20εDe2κ0=1

n+

n-

Analytical Solution

u(y)ush0

= 1β

])

Ω(y) =∞∑n=0

(

13

)n

(12

)n

(23

)n(

32

)n

(32

)n

(− 272 βεDe

2κ0

)2)nn!

β = ηsη0 =ηs

ηp+ηs

Deκ0 = ush0κλ

Numerical SolutionsChannel Flow - NP/PB equations and Analytical solution (PTT model)

y/H0 0.5 1

0

0.5

1

PB (mesh M2)NP (mesh M2)analytical

y/H0 0.5 1

0.997

1

1.003

PB (mesh M2)NP (mesh M2)analytical

n/n0β=0.2kH=20εDe2κ0=1

n+

n-

Analytical Solution

u(y)ush0

= 1β

])

Ω(y) =∞∑n=0

(

13

)n

(12

)n

(23

)n(

32

)n

(32

)n

(− 272 βεDe

2κ0

)2)nn!

β = ηsη0 =ηs

ηp+ηs

Deκ0 = ush0κλ

Numerical SolutionsCross Slot: geometry

Computational mesh (same re�nement of M1)

nº cells 4xminx10−4 4yminx10−4

MCS 12801 4 4

Numerical SolutionsCross Slot: meshes

MCS 12801 4 4

Numerical SolutionsPure newtonian EOF: external potential

κH0 20 40 60 80 100

1.6

1.8

2

2.2

Δφ[kV]

Numerical SolutionsPressure pro�les in the Cross-slot

Pout

x,y/H

P/(

ηU/H

)-40 -20 0 20 40

-1.5

-1

-0.5

0

0.5

1

1.5κH=5κH=10κH=20κH=50κH=100

Numerical SolutionsE�ect of Debye Layer size (κH = 100)

Numerical SolutionsCreeping �ow of pure Viscoelastic EOF using UCM (ε = 0)

Results κH=100

Deκ = λUκ DeH =λUH

0 0

1 0.01

2 0.02

3 0.03

4 0.04

4.88 0.0488

4.9 0.049

5 0.05

10 0.1

x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=100

Results κH=100

0 0

1 0.01

2 0.02

3 0.03

4 0.04

4.88 0.0488

4.9 0.049

5 0.05

10 0.1

x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=100

Results κH=100

0 0

1 0.01

2 0.02

3 0.03

4 0.04

4.88 0.0488

4.9 0.049

5 0.05

10 0.1

x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=100

Results κH=100

0 0

1 0.01

2 0.02

3 0.03

4 0.04

4.88 0.0488

4.9 0.049

5 0.05

10 0.1

x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=100

Results κH=100

0 0

1 0.01

2 0.02

3 0.03

4 0.04

4.88 0.0488

4.9 0.049

5 0.05

10 0.1

x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=100

Results κH=100

0 0

1 0.01

2 0.02

3 0.03

4 0.04

4.88 0.0488

4.9 0.049

5 0.05

10 0.1

x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=10

Results κH=100

0 0

1 0.01

2 0.02

3 0.03

4 0.04

4.88 0.0488

4.9 0.049

5 0.05

10 0.1

x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=10

Results κH=100

0 0

1 0.01

2 0.02

3 0.03

4 0.04

4.88 0.0488

4.9 0.049

5 0.05

10 0.1

x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=100

Results κH=100

0 0

1 0.01

2 0.02

3 0.03

4 0.04

4.88 0.0488

4.9 0.049

5 0.05

10 0.1

Results κH=20

0 0

0.8 0.04

2. 0.1

4 0.2

4.6 0.23

4.7 0.235

4.8 0.24 x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=2

Results κH=20

0 0

0.8 0.04

2. 0.1

4 0.2

4.6 0.23

4.7 0.235

4.8 0.24 x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=20

Results κH=20

0 0

0.8 0.04

2. 0.1

4 0.2

4.6 0.23

4.7 0.235

4.8 0.24 x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=20

Results κH=20

0 0

0.8 0.04

2. 0.1

4 0.2

4.6 0.23

4.7 0.235

4.8 0.24 x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=20

Results κH=20

0 0

0.8 0.04

2. 0.1

4 0.2

4.6 0.23

4.7 0.235

4.8 0.24 x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=20

Results κH=20

0 0

0.8 0.04

2. 0.1

4 0.2

4.6 0.23

4.7 0.235

4.8 0.24 x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=20

Results κH=20

0 0

0.8 0.04

2. 0.1

4 0.2

4.6 0.23

4.7 0.235

4.8 0.24 x/H

y/H

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

10.80.60.40.20

-0.2-0.4-0.6-0.8-1Rotation

Shear

Extension

κH=2

Numerical SolutionsStability Maps: De.vs.κH

κH0 20 40 60 80 100

0

0.1

0.2

0.3

0.4

0.5

DeH

Unstable Flow

Stable Flow

κH0 20 40 60 80 100

2

3

4

5

6

DeκUnstable Flow

Stable Flow

Conclusions

Numerical solutions

Excellent agreement with analitical solutions;

Sharp re�nement near the EDL (Alternative: Viscoelastic

Helmholtz-Smoluchowski Velocity at the wall (Slip velocity)[1]);

Elastic instabilities

Elastic instabilities present in the viscoelastic EOF �ow in

Cross-Slot geometry;

The critical Deborah number increased with Debye layer

relative size (κH);

No steady assymetric �ow [2] were obtained, rounded corners

are needed;

[1]Park and Lee, JCIS (2009)[2]Poole, Alves and Oliveira, PRL (2007)

Conclusions

Numerical solutions

are needed;

Conclusions

Numerical solutions

are needed;

Conclusions

Numerical solutions

are needed;

Acknowledgements

Fundação para a Ciência e a Tecnologia (FCT), Portugal:

Projects PTDC/EQU-FTT/70727/2006 andPTDC/EQU-FTT/71800/2006;

Scholarship SFRH/BD/28828/2006 (A.M. Afonso).

Thanks!

Questions?

IntroductionElectro-Osmotic Flow (EOF): Theory

Governing EquationsEOF of Viscoelastic FluidElectrokinetics

Numerical SolutionsChannel flowsComplex geometries

Conclusions

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