Abstract
The electroosmotic flow of nonNewtonian fluid–Eyring fluid in microparallel pipes under high zeta potential driven by the combination of pressure and electric force is studied. Without using the Debye–Hückel (DH) linear approximation, the numerical solutions of the fluid potential distribution and velocity distribution obtained using the finite difference method are compared with the analytical approximate solutions obtained using the DH linear approximation. The results show that the numerical method in this article is effectively reliable. In addition, the influence of various physical parameters on the electroosmotic flow is discussed in detail, and it is obtained that the velocity distribution of the Eyring fluid increases with the increase in the electric potential under the high zeta potential.
1 Introduction
With the rapid growth of microfluidic technology, related applications have been promoted, for instance chemical analysis, medical diagnosis, etc., [1,2], so a variety of microfluidic devices appear, and the electroosmotic flow (EOF) has received more and more attention. EOF is a kind of fluid flow caused by voltage applied at both ends of porous media, microchannels, and other fluid pipes. Uematsu and Araki [3,4] discussed the adsorption and hydrodynamic effects of polymers near the wall. Most of the early studies on EOF were for Newtonian fluids. But microfluidic equipment is sometimes used to resolve biological fluids, for instance lymph, colloidal suspension, etc., which are nonNewtonian fluids. So it is of great theoretical and practical significance to research EOF of various types of nonNewtonian fluids, such as powerlaw fluid [5,6], OldroydB fluid [7], Phan–Thien–Tanner fluid [8], generalized Maxwell fluid [9,10], thirdgrade fluid [11], and so on.
In the abovementioned studies, the electrostatic potential distribution is mostly achieved through solving the Poisson–Boltzmann (PB) equation and using the Debye–Hückel (DH) linear approximation, which is only applicable to the low zeta potential (i.e., ζ ≤ 25 mV) [12]. However, in actual application, zeta potentials up to 100–200 mV are often encountered, so it is of great significance to research the EOF for nonNewtonian fluids with high zeta potential without imposing DH linear approximation, many scholars have done a lot of research on these problems. For example, based on the central difference scheme, Nekoubin [13] investigated the EOF for the powerlaw fluid through a curved rectangular microchannel under high zeta potential without imposing DH linear approximation; by directly solving the nonlinear PB equation, Xie and Jian [14] discussed rotating EOF for powerlaw fluid with high zeta potential at microchannels; Jiménez et al. [15] discussed the startup of remaining EOF for the Maxwell fluid at asymmetric high zeta potentials on the wall in rectangular microchannels; and so on.
It can be obtained from the above references that different constitutive relations are used for the EOF of different nonNewtonian fluids. However, is there a nonNewtonian fluid model that can be used to describe the EOF in microchannels or nanotubes? Eyring deduced the hyperbolic sine relationship between shear rate and shear stress in 1936 [16], namely Eyring fluid, indicating that Eyring fluid is a nonNewtonian fluid. Yang [17] analyzed the flow of Eyring fluid in nanotubes by using continuum mechanics, and the results showed that Eyring fluid can be used to study nanoscale fluid flow. The problem of slip boundary conditions is also very important in the study of heat transfer and flow characteristics of micronano fluids in microchannels, and their behaviors are different in the macroscale and microscale. As we all know, noslip boundary conditions are widely used in macroscopic fluid flow problems, Zhu and Granick [18] concluded that for micronanoscale flows, the noslip boundary condition may fail, depending on the roughness of the interface and the interface interaction between the solid and the fluid. Therefore, it is necessary to consider the slip boundary conditions when analyzing the flow in the microchannel. Navier first proposed the slip boundary condition, which is given by the linear relationship between slip velocity and wall shear rate. Later, some scholars developed different slip boundary conditions [19]; however, the Navier slip boundary condition is the easiest one and the most proverbially utilized one. For instance, Tan and Liu [20] explored the characteristics of EOF for the Eyring fluid under Navier slip boundary condition in circular microtubes; by utilizing the numerical method, Song et al. [21], investigated the rotating EOF for thirdgrade fluids at parallel plate microchannels under Navier slip boundary conditions; Tan et al. [22] discussed EOF for the Eyring fluid under Navier slip boundary conditions in narrow microchannels; Jiang and Qi [23] analyzed the EOF for Eyring fluid at parallel microchannels with Navier slip boundary condition under the combined action of applied electric field force and pressure; Afonso et al. [24] proposed under the effect of the Navier slip boundary condition, an analytical solution for mixed pressuredriven electrokinetic slip flows of viscoelastic fluids in hydrophobic microchannels; Jamaati et al. [25] presented an analytical solution for pressuredriven electrokinetic flows in planar microchannels with velocity slip at the walls; Soong et al. [26] studied an analysis of pressuredriven electrokinetic flows in hydrophobic microchannels with emphasis on the slip effects under coupling of interfacial electric and fluid slippage phenomena. In previous studies, Navier slip boundary conditions have been applied to predict some flow phenomena in carbon nanotubes [27].
Motivated by the above, in this article, we will study the EOF of a nonNewtonian fluidEyring fluid in parallel microchannels under high zeta potential under Navier slip boundary conditions, derive the numerical solution of the velocity distribution, and use graphics for numerical discussion.
2 Mathematical model and its solution
We consider a steady EOF for incompressible Eyring fluid at a parallel microchannel in Figure 1, where the width, length, and height of the microchannel are W, L, and 2H, respectively, and satisfy
Neglecting the influence of gravity on the EOF, Eyring fluid satisfies the Cauchy momentum equation [22],
where
The constitutive relation of Eyring fluid is [16,28]:
where
From electrostatic theory, the electrostatic potential
Then, the net charge density distribution is
where
Combining Eqs. (4) and (3), the electrostatic potential distribution
The potential distribution meets the boundary conditions [22],
Combining Eqs. (4) and (1), Cauchy momentum Eq. (1) can be written as:
The velocity distribution satisfies the following Navier slip boundary condition [17] and symmetry boundary condition,
where
Considering the symmetry boundary condition (10) and integrating Eq. (8) from 0 to y, the shear stress
Moreover, we can deduce the following ordinary differential equation for the velocity distribution by using Eqs. (2) and (11)
For the convenience of calculation, this article cites the approximate method of ref. [22], the following approximate expression of hyperbolic sine function is used [29]:
Such an approximation is mathematically amenable and successfully used in previous studies to obtain the approximate solutions for different problems [29]. In practical applications, such as elastohydrodynamic lubrication,
Introducing dimensionless variables
here
Substituting Eq. (14) into Eqs. (5)–(7), the PB equation and the boundary conditions for the electrostatic potential distribution take the following dimensionless forms:
Applying finite difference method (step size 0.02) for the Eqs. (15)–(17), we can obtain numerical solutions of the dimensionless electrostatic potential distribution.
Furthermore, nondimensionalizing the Eqs. (9) and (12) via Eq. (14), the nondimensional forms for velocity distribution and the Navier slip boundary condition after some simplification are:
where
Integrating Eq. (18) with respect to
where
Because there are two variables
Also, integrating Eq. (21) with respect to
By applying numerical integration method (complex trapezoidal formulation, with a step size of 0.02) for Eqs. (21) and (22), we can give approximations about the velocity distribution and the average velocity.
In order to verify the reliability of solutions in this article, under low zeta potential conditions, we will compare the velocity distribution for Eyring fluid with the analytical approximate solution obtained by using the DH linear approximation [23], namely
3 Results and discussion
In this part, we will study the influence of zeta potential, ratio
Figure 2 shows the comparison of numerical solution (21) and analytical approximate solution (23) of Eyring fluid velocity distribution under low zeta potential
Figures 3 and 4 illustrate the effect of the ratio
Figure 5 displays the change in trends of the velocity distribution with the characteristic thickness
Figure 7 indicates the relationship between the dimensionless average speed and the electric parameter
Figure 8 represents the relationship between the dimensionless average velocity and the zeta potential
To compare the EOF of Eyring fluid with other fluids in the microchannel, Figure 9 shows the comparison of Eyring fluid and Newtonian fluid
4 Conclusion
The EOF for Eyring fluid under Navier slip boundary condition at high zeta potential is discussed under the combined action of applied electric field force and pressure at a parallel microchannel in this work. The electrostatic potential distribution and velocity distribution of the fluid are given by the finite difference method, and the influences of relevant physical parameters for the velocity are studied. The results show that the relationship between the dimensionless velocity distribution and the electrokinetic parameter

Funding information: The authors wish to express their sincere appreciation to the National Natural Science Foundation of China (12062018), the Natural Science Foundation of Inner Mongolia (2020MS01015) and the Colleges and Universities Youth Science and Technology Talent Support Program Funded Project of Inner Mongolia Autonomous Region (NJYT22075).

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.
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