Analytical Solution of Electro-Osmotic Peristalsis of Fractional Jeffreys Fluid in a Micro-Channel

The electro-osmotic peristaltic flow of a viscoelastic fluid through a cylindrical micro-channel is studied in this paper. The fractional Jeffreys constitutive model, including the relaxation time and retardation time, is utilized to describe the viscoelasticity of the fluid. Under the assumptions of long wavelength, low Reynolds number, and Debye-Hückel linearization, the analytical solutions of pressure gradient, stream function and axial velocity are explored in terms of Mittag-Leffler function by Laplace transform method. The corresponding solutions of fractional Maxwell fluid and generalized second grade fluid are also obtained as special cases. The numerical analysis of the results are depicted graphically, and the effects of electro-osmotic parameter, external electric field, fractional parameters and viscoelastic parameters on the peristaltic flow are discussed.


Introduction
The electro-osmotic transport in micro-channels and nano-channels recently attract much attentions of researchers because of the development of biomedical microelectromechanical systems (Bio-MEMS) and lab-on-a-chip technologies. It has wide applications in biomedicine, biology, and biotechnology. Various experimental, theoretical, and numerical investigations of electro-kinetics have been presented in previous studies. For example, Sadr et al. [1] experimentally studied electro-osmotic flow in rectangular microchannels. Santiago et al. [2] theoretically analyzed the effects of fluid inertia and pressure on the velocity and vorticity field of electro-osmotic flows. The effects of the electrical double layer near the solid/liquid interface on liquid flow through a rectangular microchannel are discussed by Yang et al. [3]. Wang et al. [4] investigated the electro-osmotic flow through a microchannel with a semicircular cross-section and gave analytical series solutions for two basic cases. Jian et al. [5] studied the flow behavior of time periodic electro-osmosis in a cylindrical microannulus. An analytical solution of electro-osmotic flow (EOF) velocity distribution as functions of radial distance, periodic time, and relevant parameters are derived. Finite difference simulation was also applied to the problems of electro-osmosis [6]. Furthermore, taking most biofluids such as blood emerge the viscoelastic feature into consideration, more and more interests are being shown in the electro-osmotic flow of non-Newtonian fluids. Das and Chakraborty [7,8] firstly studied the non-Newtonian effects on electro-osmotic flow. Then power-law fluid, second-grade fluid, Maxwell fluid, and Oldroyd-B fluid were discussed as the model of viscoelastic fluid in electro-osmotic flow by Zhao et al. [9][10][11][12][13].
Additionally, peristaltic flow, which is generated by means of contraction and expansion of the tube and channel wall, also has wide applications in many physiological processes and industries. Peristalsis, or the mechanism of peristalsis, is used to propel the biological fluid from one organ to another-for instance, the transport of blood in vessels and the movement of the chyme in the where τ is the shear stress tensor, .
γ is the rate of strain tensor, µ is the viscosity of the fluid, λ 1 and λ 2 are constant relaxation and retardation times, D α t and D β t are the fractional calculus of order α and β with respect to t, respectively, and may be defined as [45]: This model includes the ordinary Jeffreys fluid as a special case for α = β = 1, in which λ 1 and λ 2 are relaxation and retardation time. This model also can be simplified to be the generalized second-grade fluid when α = 0, λ 1 → 0 and to be the fractional Maxwell fluid when β = 0, λ 2 → 0.
We consider the unsteady electro-osmotic peristaltic transport of a viscoelastic fluid through a cylindrical channel with an externally-applied electric field along the axial direction ( Figure 1). The geometry of the channel wall is mathematically described as follows: where a is the radius of the channel, φ, λ, c, and t are the amplitude, wavelength, wave velocity, and time, respectively. It is assumed that the channel is filled with an ionic solution, such as blood, which may be manipulated by the external electric field. According to the theory of electrostatics, the Poisson-Boltzmann equation to describe the electric potential distribution for a symmetric binary electrolyte solution is given as [31]: in which ρ e is the net charge density, ε is the permittivity. When there is no axial gradient of the ionic concentration within the micro-channel and the Debye-Hückel linearization approximation is employed [31], we have: where n 0 is the ion density of the bulk, z is the valence of ions, e is electronic charge, K B is the Boltzmann constant, T is the average temperature of the electronic solution. This model includes the ordinary Jeffreys fluid as a special case for α = β = 1, in which λ1 and λ2 are relaxation and retardation time. This model also can be simplified to be the generalized secondgrade fluid when α = 0, λ1 → 0 and to be the fractional Maxwell fluid when β = 0, λ2 → 0.
We consider the unsteady electro-osmotic peristaltic transport of a viscoelastic fluid through a cylindrical channel with an externally-applied electric field along the axial direction ( Figure 1). The geometry of the channel wall is mathematically described as follows: where a is the radius of the channel, , λ, c, and t are the amplitude, wavelength, wave velocity, and time, respectively. It is assumed that the channel is filled with an ionic solution, such as blood, which may be manipulated by the external electric field. According to the theory of electrostatics, the Poisson-Boltzmann equation to describe the electric potential distribution for a symmetric binary electrolyte solution is given as [31]: in which is the net charge density, is the permittivity. When there is no axial gradient of the ionic concentration within the micro-channel and the Debye-Hückel linearization approximation is employed [31], we have: is the ion density of the bulk, is the valence of ions, e is electronic charge, KB is the Boltzmann constant, T is the average temperature of the electronic solution. div 0 V = (6) div e d p dt where ⁄ is the material time derivative, V is the velocity vector, is the density, p is the pressure, E is the applied external electric field. In the axisymmetric cylindrical coordinate system (x, r), in which variable r is radial coordinate and x-axis along the center line of the channel, we have: where d/dt is the material time derivative, V is the velocity vector, ρ is the density, p is the pressure, E is the applied external electric field. In the axisymmetric cylindrical coordinate system (x, r), in which variable r is radial coordinate and x-axis along the center line of the channel, we have: where, u and v denote the axial velocity and radial velocity, respectively. We introduce dimensionless variables and parameters as follow: ( 1), Re, and ζ are the wave number, Reynolds number, and zeta potential of the channel wall, respectively. Then under the approximations of the long wavelength and low Reynolds number, we obtain the dimensionless equations (for simplicity, the dimensionless mark "ˆ" will be neglected from here on) from Equations (4), (5), and (8)-(10): ∂u ∂x where m = aez √ 2n 0 /εK B T = a/λ d is called the Debye-Hückel parameter, λ d is the Debye length or characteristic thickness of the electrical double layer (EDL), and U HS = −E x εζ/µc is the Helmholtz-Smoluchowski velocity.
The corresponding boundary conditions are:

Solution of the Problem
Solving Equation (11) with Equation (15), we obtain the potential distribution: where I 0 (·) is the zero-order modified Bessel function of the first kind. By integrating Equation (13) with respect to r and considering Equation (16), the solution for the axial velocity is obtained as: The volumetric flow rate in the fixed frame is defined as: The relationships of the wave frame (X, R), (U, V) moving with velocity c and the fixed frame, (x, r), (u, v) in dimensionless form, are given by: Thus, the volumetric flow rate in the wave frame is: The average of the volumetric flow rate along one time period gives: From Equations (19)- (22), we can deduce: Then, from Equation (18), we have: Let f (X, t) = ∂p/∂X. Then, in order to give the analytical solution of the pressure gradient in the wave frame, we introduce the Laplace transform: Considering the initial condition f (X, 0) = 0 and the Laplace transform formula of the fractional derivative [45], the transform of Equation (23) is given as: Finally, applying the inverse Laplace transform we obtain: where E p, q (·) is the Mittag-Leffler function [45], defined by E p, q (z) = According to the definition of fractional operator, the analytical solution of axial velocity in wave frame is also obtained: Since in wave frame the velocity U = 1 R ∂ψ ∂R , in which ψ is the stream function, we can give: The dimensionless pressure rise and friction can be obtained as follows:

Discussion and Numerical Results
In the special case, if α = 0, λ 1 → 0, corresponding to the electro-osmotic peristaltic flow of generalized second grade fluid (GSF), the pressure gradient reduces to: When β = 0 and λ 2 → 0, Equation (27) can be simplified to: which is the pressure gradient for fractional Maxwell fluid (FMF). If α =0, λ 1 → 0, β = 0 and λ 2 → 0, we obtain the solution of the electro-osmotic peristaltic flow for Newtonian fluid from (27) as follows: which is the same solution given by Tripathi et al. [31]. In addition, the influences of pertinent parameters on the flow motion are discussed through graphical illustrations. Firstly, we consider the effects of characteristic thickness of EDL, external electric field, fractional parameters, and viscoelastic parameters on the pressure gradient. Figures 2-6 present the axial pressure gradient profiles, i.e., the pressure gradient vs. the axial coordinate along the center line of the channel in one period with fixed time and flow rate. Evidently the pressure gradient profiles are uniform and exhibits periodicity due to the nature of the peristaltic flow, i.e., it is at a minimum at fully-relaxed wall locations and exhibits maximum values at fully-contracted wall locations. In Figures 2-6 we find also that, because of the contraction and relaxation of the walls of the channel, there always exists a negative pressure gradient which causes the forward propagation of the trapped bolus. Figure 2 depicts the pressure gradient for increasing Debye-Hückel parameter (which is inversely proportional to Debye length or characteristic thickness of EDL). It is noticed that the pressure gradient is elevated with the increasing Debye-Hückel parameter, i.e., with decreasing characteristic thickness of EDL, and the decreasing characteristic thickness of EDL can reduce the negative pressure gradient.
The effect of the Helmholtz-Smoluchowski velocity (which is proportional to external electric field) on the pressure gradient is shown in Figure 3. It is observed that, with an increase in the Helmholtz-Smoluchowski velocity, i.e., with an increase in the external electric field there is a consistent enhancement in the pressure gradient, and the increasing external electric field strength can restrain the negative pressure gradient.
In Figure 4a,b, we show the effects of the fractional parameters (α, β) on the pressure gradient. Figure 4a depicts the pressure gradient profiles as α is increased from 0.2 through 0.4 to 0.5 with β fixed at 0.9. The increase in parameter α is observed to reduce the pressure gradient at contracted wall locations, and to enhance the pressure gradient at relaxed wall locations, inversely. Figure 4a also depicts the pressure gradient for classical Jeffreys fluid (α = β = 1). We find that the changes in pressure gradient of the fractional Jeffreys fluid are more significant than that of classical Jeffreys fluid. Figure 4b shows that the pressure gradient decreases with increasing β at contracted wall locations and inversely at relaxed wall locations.  Figure 7a,b. Obviously the pressure rise is the linear function of the flow rate, and decreases with the greater flow rate. Figure  7a shows that increasing Debye-Hückel parameter, i.e., decreasing characteristic thickness of EDL elevates the pressure rise remarkably. As the Helmholtz-Smoluchowski velocity (i.e., external electric field) increases, Figure 7b illustrates that there is also a consistent elevation in the pressure rise. Especially when UHS = −1, the pressure is always negative for all flow rates.
Furthermore, trapping, as an important phenomenon of peristaltic transport, is considered. In Figure 8a-d, we give the stream function profiles (radial coordinates versus axial coordinates) for different values of the Debye-Hückel parameter and Helmholtz-Smoluchowski velocity (with and without an external electric field). In Figure 8a,b, it is evident that with the increasing Debye-Hückel parameter, i.e., the decreasing characteristic thickness of EDL, the range of trapped boluses is reduced. In Figure 8b-d, it is observed that when the external electric field is strong enough, the number of trapped boluses significantly reduces with a larger value of the Helmholtz-Smoluchowski velocity (which is proportional to the external electric field).    Figure 7a,b. Obviously the pressure rise is the linear function of the flow rate, and decreases with the greater flow rate. Figure  7a shows that increasing Debye-Hückel parameter, i.e., decreasing characteristic thickness of EDL elevates the pressure rise remarkably. As the Helmholtz-Smoluchowski velocity (i.e., external electric field) increases, Figure 7b illustrates that there is also a consistent elevation in the pressure rise. Especially when UHS = −1, the pressure is always negative for all flow rates.
Furthermore, trapping, as an important phenomenon of peristaltic transport, is considered. In Figure 8a-d, we give the stream function profiles (radial coordinates versus axial coordinates) for different values of the Debye-Hückel parameter and Helmholtz-Smoluchowski velocity (with and without an external electric field). In Figure 8a,b, it is evident that with the increasing Debye-Hückel parameter, i.e., the decreasing characteristic thickness of EDL, the range of trapped boluses is reduced. In Figure 8b-d, it is observed that when the external electric field is strong enough, the number of trapped boluses significantly reduces with a larger value of the Helmholtz-Smoluchowski velocity (which is proportional to the external electric field).    Figure 5a,b illustrates the effects of viscoelastic parameters (relaxation time and retardation time) on the pressure gradient distribution. In Figure 5a, the maximum value of pressure gradient decreases with increasing relaxation time, while the minimum value increases. Inversely, in Figure 5b, the maximum value of pressure gradient enhances with greater value of retardation time, while the minimum reduces. It is concluded that the elasticity of the viscoelastic fluid suppresses the negative pressure gradient because, in general, for the viscoelastic model with greater relaxation time, the fluid is more elastic.
For comparison, the pressure gradient of fractional Jeffreys fluid (FJF), generalized second fluid (GSF), fractional Maxwell fluid (FMF), and Newtonian fluid (NF) are plotted in Figure 6. It is observed that the amplitude of the pressure gradient of fractional Jeffreys fluid falls in between that of generalized second fluid and fractional Maxwell fluid in the same situation.
The profile of pressure rise vs. the time averaged volumetric flow rate for different values of Debye-Hückel parameter and Helmholtz-Smoluchowski velocity are given in Figure 7a,b. Obviously the pressure rise is the linear function of the flow rate, and decreases with the greater flow rate. Figure 7a shows that increasing Debye-Hückel parameter, i.e., decreasing characteristic thickness of EDL elevates the pressure rise remarkably. As the Helmholtz-Smoluchowski velocity (i.e., external electric field) increases, Figure 7b illustrates that there is also a consistent elevation in the pressure rise. Especially when U HS = −1, the pressure is always negative for all flow rates.   Furthermore, trapping, as an important phenomenon of peristaltic transport, is considered. In Figure 8a-d, we give the stream function profiles (radial coordinates versus axial coordinates) for different values of the Debye-Hückel parameter and Helmholtz-Smoluchowski velocity (with and without an external electric field). In Figure 8a,b, it is evident that with the increasing Debye-Hückel parameter, i.e., the decreasing characteristic thickness of EDL, the range of trapped boluses is reduced. In Figure 8b-d, it is observed that when the external electric field is strong enough, the number of trapped boluses significantly reduces with a larger value of the Helmholtz-Smoluchowski velocity (which is proportional to the external electric field).

Conclusions
In this paper, a mathematic model of the electro-osmotic peristaltic flow for fractional Jeffreys fluid through a cylindrical micro-channel is established. Considering the assumptions of long wavelength, low Reynolds number and Debye-Hückel linearization, the analytical solution of pressure gradient, velocity, and streamline function are presented by using integral transform method of fractional operater. The solutions of the corresponding flow for fractional Maxwell fluid, generalized second fluid, and Newtonian fluid are discussed as the special cases. The visualization solutions via Mathematica software have been shown to evaluate the effects of the electro-osmotic parameter (i.e., Debye-Hückel parameter), external electric field, fractional parameters (α, β), and viscoelastic parameters on the peristaltic flow and trapping phenomena. The present computations have shown that:

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The axial pressure gradient is elevated with the increasing Debye-Hückel parameter (i.e., decreasing characteristic thickness of EDL) and the increasing external electric field in all ranges of the axial coordinate. In contracted wall locations, the pressure gradient decreases with the

Conclusions
In this paper, a mathematic model of the electro-osmotic peristaltic flow for fractional Jeffreys fluid through a cylindrical micro-channel is established. Considering the assumptions of long wavelength, low Reynolds number and Debye-Hückel linearization, the analytical solution of pressure gradient, velocity, and streamline function are presented by using integral transform method of fractional operater. The solutions of the corresponding flow for fractional Maxwell fluid, generalized second fluid, and Newtonian fluid are discussed as the special cases. The visualization solutions via Mathematica software have been shown to evaluate the effects of the electro-osmotic parameter (i.e., Debye-Hückel parameter), external electric field, fractional parameters (α, β), and viscoelastic parameters on the peristaltic flow and trapping phenomena. The present computations have shown that:

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The axial pressure gradient is elevated with the increasing Debye-Hückel parameter (i.e., decreasing characteristic thickness of EDL) and the increasing external electric field in all ranges of the axial coordinate. In contracted wall locations, the pressure gradient decreases with the increasing fractional parameter (α, β) and relaxation time, and also with decreasing retardation time. In relaxed wall locations, the converse response is observed and the negative pressure gradient is generated in some cases.

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The pressure rise increases with the increasing Debye-Hückel parameter (i.e., decreasing characteristic thickness of EDL) and the increasing external electric field in all ranges of time-averaged volumetric flow rate. When the pressure rise is fixed, the flow rate is increased with a decrease in the characteristic thickness of EDL and an increase in the external electric field. The influence of the characteristic thickness of EDL on the pressure rise is more remarkable than that of the external electric field.

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The trapped boluses are suppressed with the increasing Debye-Hückel parameter, i.e., a decreasing characteristic thickness of EDL and with a larger value of the Helmholtz-Smoluchowski velocity (which is proportional to the external electric field).