1 Introduction

With the development of scientific research and technology,microfluidics becomes one of the most rapid and vast research fields. It not only poses a series of challenge to study small-scale phenomena but also enables the development of many new and high-impact technologies ^[1] . In this field,electroosmotic flow is very attractive because of the advent of various microfluidic devices. Electroosmotic flow is the motion of liquid induced by an applied potential across porous material,capillary tube,microchannel,or any other fluid conduit. The flow is caused by the electrical forces on ions in an electrical double layer (EDL),which is a thin layer of ions near the interface of a solid surface and an electrolyte solution ^{[2, 3, 4]} .

Numerous studies on electroosmotic flow have been carried out since electroosmotic flow was first reported in 1809 by Reuss ^{[5, 6]} . Most of the existing studies regarding electrokinetics flow focus on Newtonian fluids. However,microfluidic devices are usually used to analyze biofluids, such as blood and DNA solutions,which cannot be treated as Newtonian fluids. Therefore,it is necessary to study the electroosmotic flow of non-Newtonian fluids. In view of this,the complex behaviors of electroosmosis with different non-Newtonian fluid models have been widely studied in recent years. Das and Chakraborty ^[7] and Chakraborty ^[8] obtained the analytical solutions for the electroosmotic velocity distribution of power-law fluids in a microchannel. Zhao et al. ^[9] and Zhao and Yang ^[10] carried out the analyses of the electroosmosis of power-law fluids in a slit microchannel,and discussed the non-Newtonian effects on electroosmostic flow. Tang et al. ^[11] numerically studied the electroosmotic flow in microchannels. They also used the power-law non-Newtonian model. Park and Lee ^[12] gave the Helmholtz-Smoluchowski velocity for pure electroosmotic flow of the Phan-Thien-Tanner (PTT) fluid. Berli and Olivares ^[13] studied the electrokinetic flow of different non-Newtonian fluids in slit and cylindrical microchannels. The power-law model,Bingham model,and Eyring model were used as the constitutive equations. Afonso et al. ^[14] developed the analytical solutions for hydrodynamic characteristics of com- bined electroosmotical and pressure driven flow of viscoelastic fluids with the PTT and finitely extensive nonlinear elastic-Peterlin (FENE-P) models. With the same viscoelastic constitutive equations,Sadeghi et al. ^[15] investigated the thermal transport characteristics of the steady fully developed electroosmotic flow in a slit microchannel. Hayat et al. ^[16] investigated the time periodic electroosmotic flow of a generalized Burgers fluid. Zhao et al. ^[17] studied the unsteady electroosmotic flow of the Oldroyd-B fluid in a capillary. With the method of integral trans- forms,the distributions of velocity profiles were obtained analytically and discussed. Zhao and Yang ^[18] investigated the transient electroosmotic flow in rectangular microchannels with the generalized fractional Oldroyd-B model.

As indicated in the review of Zhao and Yang ^[6] ,different constitutive relationships are used in the description of the electroomotic flow of non-Newtonian fluids. However,the question is that which fluid model can realistically describe the fluid flow in microchannels or nanotubes. Recently,by analyzing the flow behavior of an Eyring fluid in nanotubes,Yang ^[19] concluded that the Eyring fluid bridged the Newtonian fluid to the fluid of the plug-like type,and suggested that the Eyring fluid could be used as a mechanistic model to study nanoscale fluid flow. Thus,in the present investigation,we shall study the electroosmotic flow of the Eyring fluid in a slit microchannel,considering the interfacial interaction over the solid-fluid interface ^[20] and Navier’s slip boundary condition. The exact and approximate solutions of the velocity distribution are derived and discussed numerically with graphics.

2 Problem formulation and exact solutions

The problem under consideration is a steady and full developed electroosmotic flow of an incompressible Eyring fluid with the dielectric constant ǫ through a channel with the height 2H (see Fig. 1). It is assumed that the channel wall is uniformly charged with a zeta potential ψ w . When an external electric field E x is imposed along the x-direction,the fluid sets in motion due to the electroosmosis. Furthermore,for the electroosmotic flow,no pressure is applied,the gravitational force is negligible,and the temperature is uniform.

Under the above conditions,the basic governing equations are the continuity equation

and the modified Cauchy equation where V is the velocity vector,and τ is the stress tensor. ρ e E reprints a body force per unit volume,where E is the applied external electric field,and ρ e is the net electric charge density in the EDL of the channel wall. In view of the symmetry of the geometry,half of the slit channel (0 6 y 6 H) is considered. We assume that the velocity distribution is Then,the continuity equation (1) is satisfied automatically,and the relevant equation of motion (2) reduces to

According to the theory of electrostatics ^[3] ,the potential distribution ψ can be expressed by means of the Poisson equation

With the assumptions of the Boltzmann distribution and small values of the electrical potential ψ of the EDL,the Debye-Hückel approximation can be used successfully,and the electrical potential profile in the EDL can be governed by the following linearized Possion-Boltzmann equation: Here,κ² is the Debye-Hückel parameter expressed by κ² = (2n_∞e²z²)/(ǫk_bT),and κ⁻¹ means the EDL thickness,where n_∞ is the bulk concentration of the ions,e is the elementary electronic charge,z is the valence of the ions,k_b is the Boltzmann constant,and T is the absolute temperature.

The boundary conditions for the Possion-Boltzmann equation (6) are

Equation (6) can be solved by the above boundary conditions as follows: Finally,the net charge density distribution,Eq.(5),in conduction with Eq.(7) reduces to

From the above analysis,the momentum equation (4) reduces to

For this study,a non-Newtonian fluid that follows the Eyring model will be used. As we all know,Eyring in 1936 ^[21] proposed a hyperbolic sine relation between the shear rate and the shear stress,i.e.,the Eyring fluid model. The Eyring model was derived from a molecular theory of liquid,and it has been improved by additional empirical constants ^[22] . This model has the following rheological equation: where τ₀ and γ₀ are the rheological parameters. We consider Eqs.(9) and (10) with the most popular slip boundary condition,i.e.,the linear Navier relation ^{[19, 20]} ,as follows: and the symmetry boundary condition as follows: Here,L⁰_s is a constant slip length.

Now,we give the solution to (9)−(12). Integrating Eq.(9) from 0 to y with the consideration of the boundary condition given by Eq.(12),we have

From Eqs.(10) and (13),the velocity distribution can be found to satisfy the following ordinary differential equation: Finally,integrating Eq.(14) from y to H with the Navier slip boundary condition given by Eq.(11) yields the velocity distribution where

3 Approximate analytical solutions

In this section,we shall present an approximate approach to obtain the velocity distributions from Eq.(15). As we all know,the hyperbolic sine function can be approximated as

Such an approximation is mathematically amenable and successfully used in Refs.^{[9, 23]} to obtain the approximate solutions in different problems. As for the small shear stress τ_yx ＜＜τ₀ ,the Eyring fluid is very close to the Newtonian fluid,and τ₀ /γ₀ represents the viscosity of the fluid. In practical applications,such as elastohydrodynamic lubrication,τ₀ is usually very large (see Ref. ^[24] and the references therein). Thus,we first assume τ_ws ＜ τ₀ in order to analytically obtain the velocity distribution approximately with the consideration of two cases,i.e.,κH > 1 and 0 ＜ κH ≤ 1.

3.1 Case I: κH > 1

In this case,the slip velocity v H can be replaced approximately by the following expression:

If κy > 1,Eq.(15) can be computed as If 0 ≤ κy ≤ 1,Eq.(15) can be integrated piecewise,and an analytical expression can be evaluated as From Eqs.(18) and (19),the average velocity can be obtained as follows:

Specially,with no slip boundary condition (i.e.,L⁰_s= 0 in Eq.(11)) and τ_ws ＜＜ τ₀ ,we can obtain the velocity distribution for the Newtonian fluid from Eq.(15) as

where µ₀ = τ₀ /γ₀ . In the case of κH ＞＞ 1,we have the average velocity which can also be given by Eq.(20). From Eqs.(18),(19),and (20),we can see that the dependence of the velocity on the electric field strength and wall potential is linear,but the EDL thickness is nonlinear. It should be pointed out that V_s is the Smoluchowski velocity and will be used as a reference velocity to obtain the dimensionless velocity. Furthermore,the velocity expressions can be rewritten in terms of the average velocity as follows: We can find that the velocity distribution given by Eq.(23) is continuous at κy = 1 although the approximation expression in Eq.(16) leads to discontinuity at x = 1.

3.2 Case II: 0 ＜ κH ≤ 1

Likewise, from Eqs.(15) and (16), the velocity and the average velocity can be given as

4 Results and discussion

Now,the exact and approximate solutions can be derived for the electroosmotic flow of the Eyring fluid in a slit microchannel. To examine the flow,we give some numerical results. In the simulation,we choose the dielectric constant ǫ to be 6.95 × 10⁻¹⁰C²· N⁻¹ · m⁻² ,the wall zeta-potential of ψ_w to be −50mV,and the applied electric field E_x to be 500V · m⁻¹ . In order to illustrate realistic fluid,we take τ₀ = 43.4N · m⁻² for the Eyring fluid model,which is the value of the 3.5% M-3 Napalm used in Ref.^[25].

To illustrate the effectiveness of the approximate solutions presented in Section 3,we give the numerical solutions,which are calculated from Eq.(15) by using Romberg’s method,and the approximate solutions,which are given by Eq.(24) when κH ≤ 1 and by Eqs.(18) and (19) when κH > 1. The results are shown in Fig. 2. It can be seen that the approximate solutions can give a good estimation of the velocity field for large and small values of κH. The reason of the deviation of the approximation solution from the exact solution comes from Eq.(16). From the theory of Taylor series,we know that the maximum error induced by the approximate solution occurs when κH = 1. But,as the values of κH are usually very large for microfluidic applications ^{[1, 2, 3, 4, 9]} ,Eqs.(18) and (19) are more useful and will be used instead of Eq.(15) in the following numerical simulations. It can also be observed that the Eyring fluid flow exhibits a plug-like profile at high values of κH but a parabolic-like profile at small values of κH. Decreasing κH reduces the maximum velocity at the slit channel centerline.

As the thickness of the EDL on the channel wall is usually measured by the parameter κ⁻¹ , the dimensionless electrokinetic parameter κH = H/κ⁻¹ can characterize the relative importance of the half channel height to the EDL thickness ^[6] . Figure 3 describes the normalized velocity v_x(y)/V_M calculated from Eqs.(18),(19),and (24) for different values of the dimensionless slip length d = L⁰ _s /H and κH. Here,V_M is the fluid velocity at the centerline of the slit channel,i.e.,V_M = v_x(0). It is clear that the velocity profile becomes more plug-like as the electrokinetic parameter κH increases (thin EDL or large channel). With the slip boundary condition,the velocity on the channel wall is not zero. We can also see that an increase in the EDL thickness κ −1 reduces the slip effect,and an increase in the dimensionless slip length L⁰_s /H increases the slip velocity on the channel wall.

Figure 4 shows the non-dimensional average velocity versus the electrokinetic parameter κH for various slip lengths. Under the no-slip boundary condition,it can be seen that the average velocity of the Eyring fluid V tends to the Smoluchowski velocity V_s when κH → ∞. This can also be found from Eq.(20). But,when the slip length is not zero (L⁰_s≠ 0) and the electrokinetic parameter κH is larger,the average velocity can be several times larger than that for the Newtonian fluid with the no-slip boundary condition. With the same electrokinetic parameter κH,the dimensionless average velocity increases with the increase in L⁰_s/H. This shows that the effect of the slip flow becomes an important factor in determining the average velocity for large κH (thin EDL or large channel).

5 Conclusions

The flow behavior of the electroosmotic flow of the Eyring fluid in a slit microchannel with Navier’s slip boundary condition is studied. An exact solution of the velocity distribution is presented under the Debye-Hückel approximation. This solution is obtained by solving the linearized Poisson-Boltzmann equation,the modified Cauchy equation,and the constitutive equation of the Eyring fluid. The approximate solutions of the velocity distribution are also obtained by the approximation of the hyperbolic sine functions. The approximate solutions are effective when it is compared with some numerical results.

The results show that the dependence of the approximate solutions on the Smoluchowshi velocity V_s is linear,but the electrokinetic parameter κH is nonlinear. The calculations reveal that the velocity profile of the Eyring fluid becomes more plug-like at high values of κH,i.e., κH ＞＞ 1. It can also be concluded that the value of the electrokinetic parameter κH affects the slip velocity. Increasing the values of κH will remarkablly enhance the slip effect.