Investigation of Three Immiscible Fluids in a Microchannel

Accounting for the Pressure Gradient and the Electroosmotic Flow

Nicolas La Roche-Carrier, Guyh Dituba Ngoma, Fouad Erchiqui and Ibrahim Hamani

School of Engineering’s Department, University of Quebec in Abitibi-Témiscamingue,

445, Boulevard de l’Université, Rouyn-Noranda, Quebec, J9X 5E4, Canada

Keywords: Immiscible Fluids, Microchannel, Modeling and Simulation.

Abstract: This study deals with the investigation of three immiscible fluids in a microchannel consisting of two parallel

plates. These fluids were composed of two electric conducting fluids and one electric nonconducting fluid.

The concept of pumping a nonconducting fluid using interfacial viscous shear stress was applied accounting

for the effect of the electroosmosis and pressure gradient. The electric potential and the flow parameters were

found resolving the Poisson-Boltzmann equation and the modified Navier-Stokes equations for a hydraulic

steady fully-developed laminar flow of an incompressible fluid. The results achieved revealed the influence

of the wall and interfacial zeta potentials, the pressure difference, and the dynamic fluid viscosity ratio on the

flow characteristics of the three immiscible fluids. The developed approach was compared with a model of

two immiscible flows to highlight the relevance of this work.

1 INTRODUCTION

Microfluidic transport is widely used in the fields of

micropumps, micropower generation, chemical

processes, biomechanical processes and heat transfer,

where surface effects dominate the flow behavior

within microdevices (Dituba Ngoma G. et al., 2005).

The precise knowledge of the immiscible fluids flow

behavior in microchannels is essential to develop

high-performance microfluidic devices to pump a

nonconducting fluid by means of conducting fluids.

This can be achieved taking relevant fluid parameters

and microchannel configurations into consideration

in the planning, design and optimization phases. Most

previous investigations of pressure gradient and

electroosmotic flow in microchannels were

performed using a single conducting fluid with the

zeta potentials at the microchannel walls (Dhinakaran

S. et al., 2010; Vainshtein P. et al., 2002; and Brask

A. et al., 2005). There, the effects of surface potential,

electric field, ionic concentration and channel size on

the velocity distribution and the effect of friction on

flow characteristics. Furthermore, Yong et al., 2011

numerically analyzed the immiscible kerosene-water

two-phase flows in microchannels connected by a T-

junction using lattice Boltzmann method. In addition,

Dituba Ngoma G. et al., 2005; and Gao Y. et al., 2005,

conducted a study on two immiscible fluids

consisting of a conducting fluid and a nonconducting

fluid in a microchannel. The electric field and the

pressure gradient were applied. Moreover, an

analytical model of mixed electroosmotic/pressure

driven three immiscible fluids in a rectangular

microchannel was developed by Li H. et al., 2009.

They analyzed the effects of viscosity ratio,

electroosmosis and pressure gradient on velocity

profile and flow rate. Thorough analysis of previous

works clearly demonstrated that the research results

obtained are specific to the microchannel

configuration depending on considered key

parameters of fluids and microchannels. Therefore,

in this work, to enhance the fluid flow of

nonconducting fluids and performances of the flow in

microchannels, an investigation was conducted

considering the fluid flow of the three immiscible

fluids in a two parallel plates to deeply analyze the

impacts of the zeta potential, the pressure difference

and the dynamic viscosity on the flow characteristics

of the three fluids.

428

Roche-Carrier, N., Ngoma, G., Erchiqui, F. and Hamani, I.

Investigation of Three Immiscible Fluids in a Microchannel Accounting for the Pressure Gradient and the Electroosmotic Flow.

DOI: 10.5220/0006481004280433

In Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2017), pages 428-433

ISBN: 978-989-758-265-3

Copyright © 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

2 MATHEMATICAL

FORMULATION

The model of three immiscible fluids in two parallel

plates considered in this study is shown in Fig. 1. The

two plates separated by a distance h. The

microchannel is filled of two electric conducting

fluids, the fluid 1 and the fluid 3, and one non-

conducting fluid, the fluid 2. These fluids have

different dynamic viscosities which are specified by

µ

1

, µ

2

, and µ

3

, respectively. The interface positions

are specified using the heights h

1

and h

2

. The forces

acting on the conducting fluids include the pressure

force and the electric body force generated by the

double layer electric field. For the non-conducting

fluid, only the pressure force acts on this.

Figure 1: Model of three immiscible fluids.

To develop the governing equations for the liquid

flow of the three immiscible fluids in a microchannel

accounting for the electroosmosis, the following

assumptions were made: (i) A steady state, one-

dimensional and laminar flow was assumed; (ii) no-

slip boundary conditions were assumed; (iii) a planar

interface between the immiscible fluids was assumed;

(iv) the fluid shear stress and the flow velocity were

the same at the fluid interface; (v) the fluids were

assumed to be incompressible; and (vi) the gravity

effect was negligible.

2.1 Electric Potential Field

According to the electrokinetic (Dituba Ngoma G. et

al., 2005; and Li H. et al., 2009), the equation of the

electric potential of ions, , in y direction for the

conducting fluids is given as:

0

e

2

2

dy

d

. (1)

where is the dielectric constant of the solution,

0

the permittivity of vacuum, and

e

the net charge

density.

The net charge density can be written assuming a

symmetric electrolyte as:

,

Tk

ez

sinhezn2

b

0

0e

(2)

where e, k

b

, n

, T and z

0

are the elementary charge,

Boltzmann constant, bulk concentration of ions,

absolute temperature and valence of ions,

respectively.

Combining the Eqs. 1 and 2, and using the Debye-

Hückel approximation, Eq. 3 is found:

,

dy

d

2

2

2

(3)

where

Tk

n2

ez

b0

0

is the Debye-Hückel

parameter and

1

is the Debye length.

The net charge density Eq. 2 can be rewritten as:

Tk

ezn2

b

22

0

e

. (4)

Introducing the Debye-Hückel parameter in Eq. 4, the

last can be expressed by:

2

0e

. (5)

To solve Eq. 3, the following boundary conditions for

the conducting fluids are used:

0 y

1

hy

2

hy

3

hy

1

for the bottom wall,

2

for the interface of the fluids

1 and 2,

3

for the interface of the

fluids 2 and 3,

4

for the upper wall.

(6)

Using the dimensionless parameters and variables,

Eq. 3 can be formulated as:

*2

2*

*2

K

dy

d

. (7)

where

,

Tk

ez

b

0

*

,

h

y

y

*

,hK

h is the distance

between the plates, it equal to h

3

.

The boundary conditions of Eq. (7) in dimensionless

form can be written as for the conducting fluid “fluid

1”:

0 *y

*

1

h *y

*

1

*

,

*

2

*

.

(8)

For the conducting fluid “fluid 3”. They are expressed

by:

*

2

*

hy

1hy

*

3

*

*

3

*

,

*

4

*

.

(9)

Investigation of Three Immiscible Fluids in a Microchannel Accounting for the Pressure Gradient and the Electroosmotic Flow

429

Where

,

h

h

h

1

*

1

,

h

h

h

2

*

2

,1

h

h

h

3

*

3

,

Tk

ez

b

10

*

1

,

Tk

ez

b

30

*

3

.

Tk

ez

b

40

*

4

In addition, the net charge density in dimensionless

form is given by:

ezn

0

e

*

e

.

(10)

When considering the expression of

*

, Eq. 10

becomes:

*

*

e

2

.

(11)

The solutions of Eqs. 7 and 11 can be written as:

,BeAe*

*Ky*Ky

.BeAe2

*Ky*ky*

e

(12)

where A and B are determined accounting for the

boundary conditions. In general case, these boundary

conditions can be expressed as follows:

*

i

*

hy

*

j

*

hy

*

i

*

,

*

j

*

.

(13)

Substituting Eq. 13 for * in Eq. 12, Eq. 14 is found

*

i

*

i

KhKh

*

i

BeAe

,

*

j

*

j

KhKh

*

j

BeAe

.

(14)

From Eq. 14, A and B are determined:

,

ee

ee

A

)hh(K)hh(K

Kh

j

Kh

i

*

j

*

i

*

j

*

i

*

i

*

j

.

ee

ee

B

)hh(K)hh(K

Kh

j

Kh

i

*

j

*

i

*

j

*

i

*

i

*

j

(15)

Furthermore, for the conducting fluids “fluid 1”

and “fluid 2”, A and B are calculated using Eq. 16:

0h

*

i

*

1

*

j

hh

*

2

*

i

hh

1h

*

j

*

1

*

i

,

,

*

2

*

j

,

*

3

*

i

.

*

4

*

j

(16)

2.2 Hydrodynamic Field

The modified Navier-Stokes equations for the fluids

1, 2, and 3 can be expressed by:

,E

y

u

P0

e

2

3

2

3x

,E

y

u

P0

e

2

1

2

1x

.

y

u

P0

2

2

2

2x

(17)

where

dx

dp

P

x

assuming that the pressure gradient

in x direction is constant. E and E

e

are the electric

field and the electric body force, respectively.

Eq. 17 represents three second-order differential

equations. Thus, six boundary conditions are required

in order to solve them. The no-slip boundary

conditions can be written as based on Fig.1:

0)0y(u

1

,

0)hy(u

33

.

(18)

Moreover, the boundary conditions for the same

velocity at the interfaces of the fluids 1 and 2, and the

fluids 2 and 3 can be formulated as follows:

)hy(u)hy(u

1211

,

)hy(u)hy(u

2322

.

(19)

In addition, the boundary conditions for the

interfacial shear stresses are described as:

,

y

u

y

u

11

hy

2

2

hy

1

1

.

y

u

y

u

22

hy

3

3

hy

2

2

(20)

Solving Eq. 17 accounting for the boundary

conditions, the dimensionless velocities are

determined:

,ayaC

2

yC

u

2

*

1

*

1

2*

0

*

1

,byb

2

yC

u

2

*

1

2*

4

*

2

.ayaC

2

yC

u

4

*

3

*

3

2*

2

*

3

(21)

where

0

1

*

1

u

u

u

,

0

2

*

2

u

u

u

,

0

3

*

3

u

u

u

, u

0

is an arbitrary

reference velocity,

,

u

Ph

C

01

x

2

0

,

u

u

C

0

1h

1

,

u

Ph

C

03

x

2

2

,

u

u

C

0

3h

3

and

.

u

Ph

C

02

x

2

4

u

h1

and u

h3

are the Helmholtz-

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

430

Smoluchowski electroosmotic velocities for the

conducting fluids “fluid 1” and “fluid 3”,

respectively. They are given by:

10

b0

1

h

ez

TKE

u

and

30

b0

3h

ez

TKE

u

.

(22)

The dimensionless boundary conditions can be

formulated as:

,00*yu

*

1

,huhu

*

1

*

2

*

1

*

1

,huhu

*

2

*

3

*

2

*

2

,01hu

*

3

*

3

,

y

u

y

u

*

1

**

1

*

hy

*

*

2

hy

*

*

1

,

y

u

y

u

*

2

*

*

2

*

hy

*

*

3

hy

*

*

2

(23)

where

1

2

and

1

3

.

Accounting for the boundary conditions Eq. 23, the

system of six equations in matrix form are found in

order to determine the constants a

1

,a

2

,a

3

,a

4

,b

1

and

b

2

:

6x1 6x1 6x6

,

G

G

G

G

G

G

b

b

a

a

a

a

0 - 0 0 0

0 - 0 0 0 1

1- h- 1 h 0 0

1- h- 0 0 1 h

0 0 1 1 0 0

0 0 0 0 1 0

6

5

4

3

2

1

2

1

4

3

2

1

*

2

*

2

*

1

*

1

(24)

where:

,C G

*

111

,C

2

C

G

*

43

2

2

,

2

hC

C

2

hC

G

2*

14

*

21

2*

10

3

,

2

hC

C

2

hC

G

2*

24

*

33

2*

22

4

,eBeAKC

hChCG

*

1

*

1

Kh

1

Kh

11

*

10

*

145

.eBeAKC

hChCG

*

2

*

2

Kh

3

Kh

33

*

24

*

226

(25)

2.3 Dimensionless Flow Rates

The flow rate between the parallel plates is

determined by integrating the flow velocity

distribution over the cross-sectional area. For the

fluids 1, 2, and 3, the dimensionless flow rates are

expressed respectively by:

,1eB1eA

K

C

hah

2

a

h

6

C

Q

*

1

*

1

Kh

1

Kh

1

1

*

12

2

*

1

1

3

*

1

0

*

1

,hhb

2

hh

b

6

hhC

Q

*

1

*

22

2*

1

2*

2

1

3*

1

3*

24

*

2

.eeBeeA

K

C

h1ah1

2

a

h1

6

C

Q

*

2

*

2

Kh

K

3

Kh

K

3

3

*

24

2*

2

3

3*

2

2

*

3

(26)

3 NUMERICAL RESULTS AND

DISCUSSION

Numerical simulations were done using the

MATLAB software to investigate and analyze the

effects of the wall and interfacial zeta potentials, the

pressure difference, the interface position, the

dynamic viscosity ratio on flow characteristics of the

three immiscible fluids in a microchannel between

two plates. The main reference data for all simulation

runs in this study are given as:

0

= 8.854 x 10

-12

C/(m

V), = 80, n

∞

= 6.022 x 10

20

1 m³, z

0 = 1

, E =15000 V/m ,

T = 298 K, k

b

= 1.381 x 10

-23

J/K,

1

= 0.001 Pa s, = 1 ,

= 1, L = 0.02 m, and u

0

=1 m/s.

3.1 Impact of the Zeta Potential

To analyze the impact of the zeta potential on the

electric potential for the conducting fluids “fluid 1”

and “fluid 3”, all parameters were kept constant

except the wall, and interfacial zeta potentials. Fig. 2

shows the dimensionless electric potential as a

function of the dimensionless height of the fluid with

the dimensionless wall and the interfacial zeta

potentials as parameters. There, it can be seen that the

effect of the zeta potential is very pronounced on the

microchannel bottom and upper walls, and the

interface positions of the three fluids. The electric

potential is zero for the nonconducting “fluid 2”.

Investigation of Three Immiscible Fluids in a Microchannel Accounting for the Pressure Gradient and the Electroosmotic Flow

431

Figure 2: Dimensionless electric potential versus

dimensionless height.

3.2 Impact of the Pressure Difference

To investigate the impact that the pressure difference

in a microchannel has on the flow velocity, the

dynamic viscosity ratios, the wall zeta potentials, the

interfacial zeta potentials, the interface positions; the

pressure differences were varied using the

dimensionless values. Fig. 3 shows the dimensionless

flow velocity distribution in the microchannel cross-

section. From this figure, it can be observed that the

dimensionless flow velocities for the three

immiscible fluids increase when the dimensionless

pressure difference between the microchannel inlet

and outlet increases. This well explain the pumping

of the nonconducting fluid “fluid 2” by means of two

conducting fluids “fluid 1” and “fluid 3”.

Figure 3: Dimensionless flow velocity versus

dimensionless height.

Moreover, Fig. 4 represents the dimensionless

interfacial flow velocity for the three fluids as a

function of the dimensionless pressure difference,

where it can be seen that the dimensionless interfacial

flow velocity increases with the dimensionless

pressure difference.

Figure 4: Dimensionless interfacial flow velocity versus

dimensionless pressure difference.

3.3 Impact of the Dynamic Viscosity

Ratio

To examine the effect of the dynamic viscosity ratio

“” between the fluids 1 and 2 on the interfacial flow

velocity, and the flow velocity; all parameters were

kept constant, except the dynamic viscosity ratio “”.

Fig. 5 shows the distribution of the dimensionless

flow velocity for the three fluids as a function of the

dimensionless height of the fluid with the dynamic

velocity ratio ““ as parameter. From this figure, it

can be seen that the dimensionless flow velocities for

the three fluids decrease when the dynamic viscosity

ratio “ “ increases. This can be explained by the fact

that the resistance to fluid flow increases when the

dynamic viscosity of a fluid increases.

Figure 5: Dimensionless flow rate versus dimensionless

height.

3.4 Model Comparison

The developed model of the three immiscible fluids

in microchannel consisting of two parallel plates was

compared with the model of two immiscible fluids in

two parallel plates (Dituba Ngoma G. et al., 2005).

The results obtained shows that the maximum

dimensionless velocity was achieved for the model of

three fluids as depicted in Fig. 6. That highlights the

relevance to consider the concept of two conducting

fluids to drive a conducting fluid.

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

432

a) Model of two immiscible fluids

b) Model of three immiscible fluids

Figure 6: Model comparison.

4 CONCLUSION

In this work, a model of the flow of three immiscible

fluids in a microchannel formed by two parallel plates

was investigated. The concept of pumping an electric

nonconducting fluid using two electric conducting

fluids was applied. The combined effect of the

pressure gradient and electroosmosis was accounted

for to identify the flow parameters that improve the

flow of the nonconducting fluid. Based on the

modified Navier-Stokes and the Poisson-Boltzmann

equations, numerical simulations were accomplished.

The results obtained demonstrate that, among other

things, the dynamic viscosity ratios, the zeta

potentials and the pressure difference affects the flow

behavior in a microchannel in a strong yet different

manner. A comparison between the developed

approach of three fluids and a model of two fluids was

done to show the relevance of this study.

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