PERIODICAL PRESSURE-DRIVEN FLOWS IN MICROCHANNEL WITH WALL SLIP VELOCITY AND ELECTRO-VISCOUS EFFECTS*

WANG Lei, WU Jian-kang

Department of Mechanics, Huazhong University of Science and Technology, National Laboratory for Optoelectronics, Wuhan 430074, China, E-mail: wangleisabrina@yahoo.com.cn

PERIODICAL PRESSURE-DRIVEN FLOWS IN MICROCHANNEL WITH WALL SLIP VELOCITY AND ELECTRO-VISCOUS EFFECTS*

WANG Lei, WU Jian-kang

Department of Mechanics, Huazhong University of Science and Technology, National Laboratory for Optoelectronics, Wuhan 430074, China, E-mail: wangleisabrina@yahoo.com.cn

(Received October 27, 2009, Revised December 25, 2009)

In a microfluidic system, the flow slip velocity on a solid wall can be the same order of magnitude as the average velocity in the microchannel. The flow-electricity interaction in a complex microfluidic system subjected to a joint action of wall slip and electro-viscosity is an important topic. An analytical solution for the periodical pressure-driven flow in a two-dimensional uniform microchannel, with consideration of wall slip and electro-viscous effect is obtained based on the Poisson–Boltzmann equation for the Electric Double Layer (EDL) and the Navier-Stokes equations for the liquid flow. The analytic solutions agree well with the numerical solutions. The analytical results indicate that the periodical flow velocity and the Flow-Induced Electric Field (FIEF) strongly depend on the frequency Reynolds number (Re=ωh2/ν), that is a function of the frequency, the channel size and the kinetic viscosity of fluids. For Re＜1, the flow velocity and the FIEF behave similarly to those in a steady flow, whereas they decrease rapidly with Re as Re＞1. In addition, the electro-viscous effect greatly influences the periodical flow velocity and the FIEF, particularly, when the electrokinetic radius κH is small. Furthermore, the wall slip velocity amplifies the FIEF and enhances the electro-viscous effect on the flow.

electrokinetic flow, frequency Reynolds number, wall slip, electro-viscous effects, Flow-Induced Electric Field (FIEF)

1. Introduction

2. Problem formulation

A two-dimensional uniform microchannel with an isolated wall is schematically shown in Fig.1, where H is the half width of the channel. Because of the symmetry, only the half region of the microchannel will be considered.

2.1 Governing equations and boundary conditions

For a symmetric binary electrolyte solution, both the electrical potentialand the net charge densityeρ are described by Poisson–Boltzmann equation

where0n and z are the bulk ionic concentration and the valence of ions, respectively, e is the elementary charge, ε is the dielectric constant of the solution,bk is the Boltzmann constant, and T is the absolute temperature. Due to the symmetry, the boundary conditions related to Eq.(1) are as follows

where ζ is the zeta potential on the interface between solid wall and bulk solutions. Because the length of microchannels is usually much larger than its width, the flows in the microchannel can be well approximated by a fully developed laminar flow except at the entrance and the exit of the channel. The Navier-Stokes (N-S) equation for incompressible viscous fluids in a two-dimensional uniform microchannel is given by

in which u is the velocity along the channel direction x, ρ and μ are the density and the dynamic viscosity of the fluid, f is the body force in the x direction, E is the flow induced electric field along the channel direction x andeEρ represents the flow induced electro-viscous force.

The continuity equation for the incompressible fluid in the channel is

Without considering any other body forces except for the electrokinetic forces, the N-S Eq.(4) reduces to

The slip boundary conditions related with the N-S equation are

where β is the slip length of the channel wall. The periodical pressure p, the velocity u and the flow induced electric field E can be expressed in complex variable functions as

in which0p,0u,0E are the amplitudes of the applied pressure, the flow velocity, and the flow induced electric field, respectively, ω is the frequency.

Substituting Eqs.(8) into Eq.(6) yields:

Particularly, from Eq.(9), the electric field0E can be obtained from the balance between the electric currents from the fluid flow and the electrical conduction[5]. Specifically, the ions in the double layer are carried by the flow, which results in a streaming currentsI along the direction of the flow, which can be expressed by

The resultant electrokinetic potential, termed as the streaming potential, can induce a flow of ions, known as the electrical conduction currentcI, in the direction opposite to the flow direction, which can be expressed as

where σ is the total electrical conductivity of the bulk fluid. The flow-induced electric field can be calculated by setting the net current in the channel to be zero[5], i.e., Is+Ic=0

2.2 Non-dimensional equations

To facilitate the analysis, all the variables in the governing equations were non-dimensionalized as follows

and the characteristic thickness of the EDL can be expressed as

represents the ratio of a half channel width to the thickness of the EDL,maxu represents the maximum velocity in the channel (at=0y) in the steady flow without electro-viscous effects, and can be expressed as

E?is the flow induced electric field without considering electro-viscous effect, which is calculated from the Helmholtz–Smoluchowski equation in a steady flow[5]as

Finally, the dimensionless form of Poisson-Boltzmann Eqs.(1) and (2) are

whereα=zeζkT ,χ=(κH)2α,κH is defined

b as the electrokinetic radius, the ratio of the half channel width to the thickness of the EDL. The boundary conditions for Eq.(17) become

The Eqs.(17) and (18) are numerically solved for the charge density, which will be used for solving N-S equation. The N-S Eq.(9) can be written in the dimensionless form as

where B2=iRe, Re=ωH2/ν is the frequency Reynolds number, ν is the kinetic viscosity of fluids, γ=ε2ζ2κ2/μσ is the electro-viscous number defined to represent the ratio of the flow-induced electric resistance to the viscous force. The boundary conditions of N-S Eq.(19) become

The dimensionless electric field0E induced by flow is written as

3. Analytic solution of periodical flow in microchannel with consideration of wall slip and electro-viscous effects

The solution of Poisson-Boltzmann Eq.(17) can be expressed as[3]

where

While the solution of N-S Eq.(19) is

From Eqs.(26) and (27), we have

After substituting Eqs.(28) into Eq.(19), we have

Then,

Equations (30) can be numerically integrated by using the charge density obtained early on. So we have

The solution of N-S Eq.(24) can be expressed as

Imposing the boundary conditions (20), one obtains

Substituting Eq.(33) into Eq.(32), the final solution for the flow velocity is as follows

and

In cases without electro-viscous effect (=0γ), Eq.(34) reduces to

In cases without wall slip (β=0),Eq.(34) reduces to

In cases without both electro-viscous effect and wall slip, Eq.(34) reduces to

Substituting Eq.(34) into Eq.(21), the flow-induced electric field is found to be

If the electro-viscous effect is neglected (=0γ), Eq.(41) reduces to

4. Results and discussion

The parameters of a typical electrokinetic flow in microchannels are specified as

The amplitude and the phase angle of the periodical flow velocity in the microchannel with varying frequency Reynolds numbers are shown in Figs.2 and 3. where numerical solutions are also given for a comparison. The lines in all of the figures below represent analytrc solutions and the dots represent numerical solutions. A good agreement is found.

It can be seen that the amplitude of the electroosmotic flow decreases and the lag phase angle of the electroosmosis increases as the frequency Reynolds number increases. As the frequency increases, the response of the fluids to the pressure gradient becomes weak and lags behind the pressure gradient. The flow amplitude decreases to about 16%, and the maximum phase angle is about –70owhen=5Re in the present example. At a low frequency (0Re≈), the fluid moves with the pressure gradient in phase. The amplitude and the phase angle of the periodical electroosmosis with without the electro-viscous effects are shown in Figs.4 and 5. It can be seen that the electro-viscous effect reduces both the flow amplitude and the phase angle. The electro-viscous effect is less significant at a high frequency (say =5Re).

The transient velocity profile of the periodical electroosmotic flow in the microchannel is shown in Fig.6.

It can be seen that the flow velocity decreases with increase of the frequency Reynolds number, and the figure also shows a parabolic profile at the low frequency, and the wave profile at the high frequency. The amplitude and the phase angle of the slip velocity on the channel wall are shown in Figs.7 and 8. It can be seen that the amplitude and the phase angle vary slowly with the frequency Reynolds number when Re≤1. When Re＞1 the slip velocity amplitude rapidly decreases with the frequency Reynolds number while the phase angle rapidly increases.

It can also be seen that the slip velocity amplitude and the phase angle increase with decrease of the electrokinetic radiusHκ, which implies that the wall slip velocity and the phase angle increase when the channel is narrowed for a fixed wall slip length. The amplitude and the phase angle of the FIEF in the microchannel are shown in Figs.9 and 10. It is found that the effect of the frequency Reynolds number on the FIEF is the same as that of the wall slip velocity. Furthermore, it can be seen that the amplitude of the FIEF decreases and the phase angle of FIEF increases when the microchannel is narrowed.

The effects of the wall slip length on the FIEF are shown in Figs.11 and 12. It can be seen that thewall slip length amplifies the FIEF in both amplitude and phase angle. The effects of the wall slip length on the wall slip velocity are shown in Figs.13 and 14. A similar behavior is found as that with respect to the FIEF.

5. Concluding remark

The analytic solution of periodical pressuredriven flows in a uniform microchannel is obtained in this work with consideration of wall slip and electro-viscous effects. By utilizing a non-dimensional method, the periodical flows and the FIEF in microchannels are analyzed. The results are found to depend on four parameters: (1) the electro-viscous number γ=ε2ζ2κ2/(μσ) reflecting the ratio of the electro-viscous force to the viscous force, (2) the frequency Reynolds number Re=ωH2/ν, which is the ratio of the periodical inertial force to the viscous force, (3) the dimensionless wall slip length β/H and (4) the electrokinetic radius κH. The periodical flow behavior and the FIEF strongly depend on the frequency Reynolds number. For Re＜1, the flow velocity and the FIEF behave similarly to those of a steady flow, whereas they decrease rapidly with Re as Re＞1. For a small electrokinetic radius κH, the electro-viscosity is found greatly affecting the periodical flows and the FIEF. Furthermore, the wall slip increases the flow velocity and the FIEF , but the effects on the phase angle are less significant.

[1] LI Dong-qing. Electrokinetics in microfluidics, interfaces science and technology[M]. New York: Elsevier, 2004, 2.

[2] KARNIADAKIS G. E., BESKOK A. Micro flows fundamental and simulation[M]. New York: Springer-Verlag, 2002.

[3] PROBSTEIN R. F. Physicochemical hydrodynamics[M]. New York: Wiley and Sons Inc, 1994.

[4] RICE C. L., WHITEHEAD R. Electrokinetic flow in a narrow cylindrical capillary[J]. J. Phys. Chem., 1965, 69(11): 4017-4024.

[5] HUNTER R. J. Zeta potential in colloid science[M]. London: Academic Press, 1981.

[6] YANG Chun, LI Dong-qing and MASLIYAH J. H. Modeling forced liquid convection in rectangular microchannels with electrokinetic effects[J]. Int. J. Heat Mass Transfer, 1998, 41(24): 4229-4249.

[7] CHUN B., LADD A. J. C. The electroviscous force between charged particles: Beyond the thin-doublelayer approximation[J]. J. Colloid Interface Sci., 2004, 274(2): 687-694.

[8] REN C. L., Li Dong-qing. Improved understanding of the effect of electrical double layer on pressure-driven flow in microchannels[J]. Analytica Chimica Acta, 2005, 531(1): 15-23.

[9] GONG L., WU J. K. and WANG L. et al. Streaming potential and electroviscous effects in periodical pressure-driven microchannel flow[J]. Physics of Fluids, 2008, 20(6): 063603.

[10] BAUDRY J., CHARLAIX E. and TONCK A. et al. Experimental evidence for a large slip effect at a non-wetting fluid-solid interface[J]. Langmui, 2001, 17(17): 5232-5236.

[11] PIT R., HERVET H. and LéGER L. Direct experimental evidence of slip in hexadecane: Solid interfaces[J]. Physical Review Letters, 2000, 85(5): 980-983.

[12] PIT R., HERVET H. and LéGER L. Friction and slip of a simple liquid at a solid surface[J]. Tribology Letters, 1999, 7(2-3): 147-152.

[13] ZHU Y. X., GRANICK S. Limits of the hydrodynamic no-slip boundary condition[J]. Physical Review Letters,2002, 88(11): 106102.

[14] BARRAT J. L. Large slip effect at a non-wetting fluid-solid interface[J]. Physical Review Letters, 1999, 82 (23): 4671-4674.

[15] TRETHEWAY D. C., MEINHART C. D. Apparent fluid slip at hydrophobic microchannel walls[J]. Physics of Fluids, 2002, 14(3): L9-L12.

[16] OU Jia, PEROT B. and ROTHSTEIN J. P. Laminar drag reduction in microchannels using ultrahydrophobic surfaces[J]. Physics of Fluids, 2004, 16(12): 4635-4643.

[17] CHUN M. S., LEE T. S. and LEE K. Microflow of dilute colloidal suspension in narrow channel of microfluidic-chip under Newtonian fluid slip condition[J]. Korea-Australia Rheology Journal, 2005, 17(4): 207-215.

[18] TRETHEWAY D. C., MEINHART C. D. A generating mechanism for apparent fluid slip in hydrophobic microchannels[J]. Physics of Fluids, 2004, 16(5): 1509-1515.

[19] JOSEPH P., TABELING P. Direct measurement of the apparent slip length[J]. Physics Review E, 2005, 71(3): 035303(R).

[20] DAVIDSON C., XUAN X. C. Electrokinetic energy conversion in slip microchannels[J]. Journal of Power Sources, 2008, 179(1): 297-300.

[21] REN Y. Q., STEIN D. Slip-enhanced electrokinetic energy conversion in microfluidic channels[J]. Microtechnology, 2008, 19(19): 195707.

[22] BRUNET1 E., AJDARI A. Generalized Onsager relations for electrokinetic effects in anisotropic and heterogeneous geometries[J]. Physical Review E, 2004, 69(1): 016306.

10.1016/S1001-6058(09)60123-2

* Project supported by the National Natural Science Foundation of China (Grant No. 50805059)

Biography: WANG Lei (1983-), Female, Ph. D. Candidate

WU Jian-kang,

E-mail: wujkang@mail.hust.edu.cn