Talk about Several Time Periodic Pulse Electroosmotic Flow of Maxwell Fluid in a Circular Microchannel

Using the method of Laplace transform, analytical expressions are derived for the time periodic pulse electroosmotic flow (EOF) velocity of the triangle and sawtooth of Maxwell fluid in circular microchannel. The solution involves analytically solving the linearized Poisson-Boltzmann (P-B) equation, togeth-er with the Cauchy momentum equation and the general Maxwell constitutive equation. By numerical computations of inverse Laplace transform, the effects of electrokinetic width K, relaxation time 1 λ and pulse width a on the above several pulse EOF velocities are investigated. In addition, we focused on the comparison and analysis of the formulas and graphs between the triangle and sawtooth pulse EOF with the rectangle pulse EOF. The study found that there are obvious differences in formulas and graphs between triangle and sawtooth pulse EOF with rectangle pulse EOF, and the difference mainly depends on the different definitions of the three kinds of time periodic pulse waves. Finally, we also studied the stability of the above three kinds of pulse EOF and the influence of relaxation time on pulse EOF velocity under different pulse widths is discussed. We find that the rectangle pulse EOF is more stable than the triangle and sawtooth pulse EOF. For any pulse, as the pulse width a increases, the influence of the relaxation time on the pulse EOF velocity will be weakened.


Introduction
Microfluidic devices have become increasingly more important because of their How to cite this paper: Li, D.S. and Li, K. sensors (such as lab-on-a-chip) [1] [2]. In general, when most substances come into contact with polar solutions, they tend to generate negative charges on the surface. The distribution of ions near the wall in the solution will be affected by this phenomenon. The opposite ions with the opposite polarity to the wall will be attracted to the wall, while the same ions will be repelled away from the wall.
In this way, an electric double layer (EDL) will be formed [3]. Further, when an external electric field is applied to both ends of the channel, the ions in the electric double layer will move under the force of the electric field. This is due to the viscosity of the fluid itself, the moving free ions will drive the movement of nearby fluid clusters, and eventually form an electro-osmotic flow (EOF).
In previous research, a large number of theoretical and experimental studies [4]- [9] on the fully developed EOF problem of Newtonian fluids in various geometric shapes of microchannels have been completed. However, this steady EOF problem requires higher voltage and larger field strength, which may bring many difficulties to the experimental conditions. Very recently, time-dependent EOF as an alternative mechanism of microfluidic transport has attracted the attention of scholars at home and abroad [10]- [15].
All of the above-mentioned studies are related to Newtonian fluids. But most of solutions of industry and biopharmaceutical are fluid that has the structural characteristic of non-Newtonian fluids, such as biological fluid and other solutions of long-chain molecules, which structural characteristics include strain force, normal shear stress, hysteresis effect, variable viscosity, memory effect and so on [16]. Therefore, the study of non-Newtonian fluids becomes very important. Additionally, the theoretical research of electroosmotic flow of non-Newtonian fluids is mainly limited to simple fluid models because of the inherent analytical difficulties introduced by more complex constitutive equation [17]. So far, some work has been done on the simple fluid models, for example Oldroyd-B fluid model [18] [27] and Jeffrey fluid model [28] [29] [30].
Although some basic characteristics of EOF of non-Newtonian fluids have been reported in the above studies, its rich properties still need to be examined. Recent study have shown that Maxwell fluid model simulation of blood in narrow conical vessels has achieved an ideal effect, and it is completely possible to analyze the blood-based microfluidics and other microbial fluid transmission systems by means of electric mechanism [30]. At the same time, the literature [31] shows that low-frequency pulses can promote local blood circulation, which has been proved in clinical medical research. Thus, the study of Maxwell fluid pulse EOF will play a very beneficial role in blood transport.
However, from the current research situation, there is almost no research on pulse EOF, and it has not attracted enough attention from the majority of researchers. Therefore, based on the rectangle pulse EOF, we have re-selected several common pulses (such as triangle pulse and sawtooth pulse) to study the time periodic pulse EOF of Maxwell fluid through a circular microchannel in Journal of Applied Mathematics and Physics present work. The target of this article is to derive the semi-analytical solutions of the above two time periodic pulse EOF for viscoelastic fluid. Meanwhile, we analyzed the effect of several parameters such as the electrokinetic width, the relaxation time and the pulse width on the pulse EOF of Maxwell fluid. Moreover, we also discussed the different effects of relaxation time on the EOF velocity of different pulses. It is mainly compared with the rectangle pulse EOF [32] completed before, and some new results have been found. In the second section, the physical model of the problem is described and the semi-analytical solutions of the governing equations of rectangle pulse EOF, triangle pulse EOF and sawtooth pulse EOF are derived. In the third section, the article discusses numerical results and the parametric which dependent on the pulse EOF velocity for different pulses. At last, in the fourth section, the article presents the conclusions.

Cauchy Momentum Equation and Constitutive Relation
The time periodic pulse EOF of an incompressible Maxwell fluid through a circular microchannel is sketched in Figure 1. The channel has a circular cross-section with a radius R and a length L, assumed to be much larger than the diameter i.e., Provided that the boundary condition of Equation (1) is no slip, and it can be given as [33] ( ) For the Maxwell fluid, the constitutive equation satisfies [34] ( ) where 1 λ is the relaxation time, 0 η is zero shear rate viscosity.

Electric Potential Field Solution
The chemical interaction between the electrolyte liquid and the solid wall produces an electric double layer (EDL), a very thin layer of charged liquid at the solid-liquid interface. A cylindrical coordinate system ( ) , , r z θ is adopted. In this theoretical model, it is assumed that the channel wall is uniformly charged, so that the electrical potential in the EDL only varies in this r direction and does not depend on θ [33]. For a symmetric binary electrolyte solution, assuming that the electrical potential ψ of the EDL is stable, and its distribution and the local volumetric net charge density ( ) e r ρ are described by the Poisson- where ε is the dielectric constant of the electrolyte liquid, ( ) r ψ is the electrical potential of the EDL, n 0 is the ion density of the bulk liquid, z ν is the valence, e 0 is the electron charge, k b is the Boltzmann constant, T is the absolute temperature, and sinh is the sine function.

D. S. Li, K. Li Journal of Applied Mathematics and Physics
Combing Equation (4) and Equation (5) gives which is subject to the following boundary conditions where 0 ψ is wall zeta potential, r is radial coordinate and R is radius of the circular microchannel.
Assuming that the electrical potential is small enough, the Debye-Hückel linearization approximation can be applied to the hyperbolic sine function appearing on the right hand side of Equation (6), which means that the electrical potential is physically small compared to the thermal energy of the charged species [28]. Thus, Equation (6) can be simplified as where κ is the Debye-Hückel parameter, which 1 κ usually represents the thickness of the EDL in physical.
By solving Equation (7) and Equation (8), the net charge density distribution for circular microchannel can be express as where 0 I is the first kind modified Bessel function of order zero.

The Analytical Solutions of the Cauchy Momentum Equation
In order to obtain the solution of the velocity field of the triangle pulse EOF and the sawtooth pulse EOF, let us first briefly review the process of solving the velocity field of the rectangle pulse EOF, and then analyze the difference among the three formulas to obtain the corresponding velocity field solution above.

Rectangle Pulse Wave
The ideal rectangle pulse can be expressed as the following form For simplicity, the following dimensionless groups are introduced: where eo U denotes steady Helmholtz-Smoluchowshi EOF velocity of Newtonian fluids, K is the ratio of the characteristic width of the microchannel to Journal of Applied Mathematics and Physics Debye length.
Using Equation (11), Equations of (1) and (3) and boundary conditions (2) are normalized as Eliminating rz τ from Equation (12) and Equation (13) yields Let us employ the method of Laplace transforms defined by (15), and the Laplace transform of ( ) f t is given by the Appendix A.
From the literature [32], the solution of the velocity field is given as where 2 1 s s β λ = + , tanh is a hyperbolic tangent function.
The inverse Laplace transform is defined by where Γ is a vertical line to the right of all singularities of ( ) , U r s in the complex s plane. The exact solution of the EOF velocity cannot be obtained analytically due to the complexity of the express of ( ) , U r s . Therefore, the numerical computation must be performed by numerical inverse Laplace transform [35].

Triangle Pulse Wave
The ideal triangle pulse can be expressed as the following form The difference from the rectangle pulse wave is in Equation (15), the others are the same, and the Laplace transform of ( ) given by the Appendix B.
If initial condition satisfies ( ) , 0 0 u r = , then the transforms of Equation (15) and Equation (14) can be written as Equation (20) can be simplified as The homogeneous solution of Equation (22) is expressed as where 0 I and 0 K are modified Bessel functions of first and second kinds of order zero, respectively.
Due to the finite of ( ) , U r s at 0 r = , the constant B equal to zero from the boundary condition Equation (21). Therefore, the homogeneous solution of Equation (22) is rewritten as here A is constant, which can be determined from boundary conditions of Equation (21).
Considering the formation of the right hand side of Equation (22) The coefficient A with boundary condition of Equation (21) can be determined as Inserting Equation (31) where 2 1 s s β λ = + , sech is a hyperbolic secant function.
As with rectangle pulse wave, the numerical computation must be performed by numerical inverse Laplace transform of Equation (32).

Sawtooth Pulse Wave
The ideal sawtooth pulse can be expressed as the following form The Laplace transform of ( ) Similar to the above several pulse waves, the numerical calculation must also be performed by the numerical inverse Laplace transform of Equation (35).

Results and Discussion
In  . We can clearly find that for different pulses, the variations of velocity are relatively significant. Therefore, it is very necessary to study different time periodic pulse EOF. At the same time, it can be seen from Figure 3 that for any pulse, the velocity amplitude increases with relaxation time 1 λ . The main reason is that the longer relaxation time, which means the greater elastic effect and weaker recovery ability of Maxwell fluid. Because of the "fading memory" phenomenon of Maxwell fluid, increasing the relaxation time leads more easily to the variation of the pulse EOF velocity profiles caused by external electric field [36]. Moreover, it is very obvious from Figure 3 that the velocity of three kinds of pulse EOF gradually attains to steady state with the elapse of time t .    EOF velocity is different. Especially for smaller pulse width is more significant (for example 1 a = ). The reason may be that for small pulse width a, the pulse force has a short duration and the flow of fluid stability is relatively weak, so it is easily affected by other forces. In addition, we can see from the above picture that as the pulse width a increases, the different change frequency of velocity profiles slows down, which means a long cycle time [37], and the time required for the velocity profiles to reach a steady state has also become longer. ). From Figure 8, we can find that a larger relaxation time 1 λ will result in larger velocity amplitude, and the velocity distribution is no longer mainly restricted to EDL. This is because that for a larger relaxation time 1 λ , the elasticity of the fluid is more obvious. And since elasticity is the physical property of the fluid as a whole, the velocity variation can extend to the entire region of the flow [38]. Additionally, by comparing the  variations in the amplitude of the three kinds of pulse EOF velocity, we can see that the relaxation time has different effects on the velocity amplitude for different pulses. In particular, it has a greater impact on triangle pulse and sawtooth pulse than rectangle pulse (see Figure 7(b) and Figure 7(c), Figure 8(b) and Figure 8(c)). The possible reason is that the velocity expressions of triangle pulse EOF and sawtooth pulse EOF have one more variable about the relaxation time 1 λ (see Equation (32) and Equation (35)) than that of rectangle pulse EOF. Thus, the rectangle pulse EOF is a relatively more stable pulse EOF among the three kinds of pulse EOF.

Conclusion
In this article, the semi-analytical solutions for both triangle and sawtooth time periodic pulse EOF of Maxwell fluid through a circular microchannel under the Debye-Hückel approximation are presented. Based on the results obtained in this work, it can be concluded that with the electrokinetic width K increases, the velocity variations are mainly limited to the narrow area close to the EDL for small relaxation time 1 λ . However, as the increase of relaxation time 1 λ , the elasticity of the fluid becomes significant and the velocity variations can be extended to the entire flow field. At the same time, the velocity amplitude will significantly larger, and the flow needs longer time to attain steady status. Moreover, the time it takes for the fluid to change from a static state to a flowing state Journal of Applied Mathematics and Physics increases with relaxation time 1 λ . For given pulse width a, the effect of relaxation time 1 λ on triangle pulse EOF and sawtooth pulse EOF is greater than rectangle pulse EOF, which implies that the rectangle pulse EOF is more stable. With the increase of pulse width a, the effect of relaxation time 1 λ on the velocity will be weakened, the change period of the velocity profiles becomes larger, and the time required for the velocity profiles to reach a steady state also becomes longer.