# The Streaming Potential of Fluid through a Microchannel with Modulated Charged Surfaces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

**B**acts on the fluid along the y* axis, and the pressure gradient −dP*/dx* acts on the fluid along the x* axis. During the fluid flow process, there is a chemical interaction with the walls that generate the EDL. The excess ions that are generated by the flow in the electrolyte solution will gather at the downstream of the microchannel. Therefore, the streaming potential

**E**is obtained under the drive of the magnetic field and pressure gradient, where the direction is in the negative direction of the x* axis. It can be shown that the magnetic field and the pressure gradient are the basic driving mechanisms of the subsequent fluid flow. It was assumed that the flow in the microchannel was stable throughout the entire flow process. The potential based on modulation is asymmetric, and the zeta potential of the upper and lower parallel plates can be expressed as

_{1}* and ξ

_{2}* represent the amplitudes of the top and bottom surfaces respectively, α and β are constants and m* and n* are the patterning frequencies. Because the cosine term produces a vertical velocity, the fluid flow is considered to be two-dimensional. Because there are L, W >> 2H in the rectangular microchannel, it can be considered that the velocity component in the z* direction is zero.

#### 2.1. EDL Potential Distribution

_{e}is the local volumetric net charge density, ε is the permittivity of the electrolyte solution, n

_{0}is the bulk ionic concentration, z is the ion valence, e is the charge of the electron, k

_{B}is the Boltzmann constant, T

_{a}is the absolute temperature.

_{B}T

_{a}is less than unity. The hyperbolic sine function can be approximated by Debye-Hückel as follows [37]:

_{B}T

_{a}/2e

^{2}z

^{2}n

_{0})

^{1/2}is given by using the above approximations. Additionally, the Poisson–Boltzmann equation is linearized and becomes

_{1}, f

_{2}and f

_{3}can be obtained according to the corresponding boundary conditions after splitting:

#### 2.2. Velocity Distribution

**u**is the velocity field, and we only need to consider the velocities in the two directions x* and y*. ρ is the fluid density, P* is the pressure, μ is the dynamic viscosity of the fluid. In addition to the pressure gradient, the net body force

**F**also has other external forces caused by the interaction between the external magnetic field and the induced electric field:

**J**is the local ion current density satisfying Ohm’s law. σ is the electrical conductivity of the medium,

**E**is the induced electric field and

**B**is the applied magnetic field. Because the magnetic Reynolds number is small, the magnetic field is independent of the velocity. The N-S equation can be simplified to its two-dimensional component form:

_{ij}* is the stress tensor, in which τ

_{ij}* is eliminated in combination with Equations (11), (16)–(18) and then simplified to obtain the final governing equation:

_{p}is the characteristic velocity of the fluid flow driven by pressure, u

_{r}is the characteristic velocity of the electric flow, δ is the nondimensional slip length, P

_{0}is the characteristic pressure, Ha is the Hartmann number, E

_{0}is the characteristic scale of the electric field. After the dimensionless transformation, Equations (19)–(21) and boundary condition Equation (22) become

_{1}, g

_{2}and g

_{3}which are related to y can be determined as follows:

_{ij}(i = 2, 3; j = 1, 2, 3, 4) satisfies the following two equations, and we can conclude that λ

_{21}= −λ

_{22}, λ

_{23}= −λ

_{24}, λ

_{31}= −λ

_{32}, λ

_{33}= −λ

_{34}.

_{3}= 0 and γ

_{4}= 1. In addition, the rest of γ

_{k}(k = 1, 2, 5–12) can represented by the following matrix equation:

#### 2.3. Streaming Potential

_{s}which needs to be determined. Because there is no applied electric field, the streaming potential can be determined by considering the condition that the net ion current in the electrolyte solution is zero. When the fluid reaches a stable state, it satisfies the following equation.

^{±}is the velocity of cation and anion in the x* direction, and n

^{±}is the concentration of cation and anion, which satisfy the following relationship respectively.

^{2}z

^{2}µ/εk

_{B}T

_{a}f, is introduced, which is a dimensionless parameter that is equivalent to the ionic Peclet number [41]. Substituting the above parameters into Equation (36), the following equation can be obtained:

## 3. Result and Discussion

^{−3}kg/m

^{3}, the dynamic viscosity μ is about 10

^{−3}kg/(m·s), the range of conductivity σ is 2.2 × 10

^{−4}~10

^{2}S/m [42], the strength of the external magnetic field B

_{0}is 0.01~5 T. According to Ha = B

_{0}H(σ/μ)

^{1/2}, the range of the Hartmann number (Ha) can be obtained from 0 to 0.4 with Ha = 1 as the maximum permissible upper limit [43,44] theoretically. Based on the previous theoretical derivation, in order to satisfy the Debye–Hückel linearization approximation conditions, the dimensionless zeta potential should satisfy ψ ≤ 1. In the following discussion about the upper and lower surface mode potentials, the ranges of the amplitudes (ξ

_{1}and ξ

_{2}), constants (α and β) and mode frequencies (m and n) are 0~0.2, 0~6 and 0~6 respectively. When α = β = 0, the zeta potential are constants. The values of R and u

_{r}are assumed to be R~0.3–1 and u

_{r}~0.1–1 when in the dimensionless form.

#### 3.1. Flow Field

_{r}= 0.6, R = 0.6, m = 5.5, n = 5.8, ξ

_{1}= 0.15, ξ

_{2}= 0.18. Because of the asymmetry of the wall potentials, it can be seen from the Figure 2 that the streamlines near the upper and lower walls are also asymmetric. As the values of parameters α and β increase, the streamlines that can be observed in Figure 2b are denser than the ones seen in Figure 2a. Additionally, the characteristic of the vortexes that are near the walls in Figure 2b are significant. This means that constants α and β are the main elements that control the strength of the vortexes.

_{r}= 0.6, R = 0.6, α = 3, β = 3, ξ

_{1}= 0.15, ξ

_{2}= 0.18 when the mode frequencies m (m = 1.5, 5.5) and n (n = 1.8, 5.8) change. It can be observed in the Figure 3a,b that with the increase in mode frequencies, the density of streamlines become less obvious. However, along the direction of the x-axis, it can be seen that the periodicity becomes more pronounced as m and n increase. The reason for this phenomenon is that the cos(mx) and cos(nx) in Equation (1a,b) of the zeta potentials play important roles. The existence of cosine terms produces vertical velocity in the y-axis direction, leading to the appearance of vortexes. When m and n are larger, the 2π/m and 2π/n periods are smaller. This means that m and n are the main elements that control the periodicity of the eddy currents.

#### 3.2. Analysis of the Streaming Potential

_{r}(u

_{r}= 0.1, 0.3, 0.5, 1) on the streaming potential when x = π/4, m = 0.5, n = 0.8, ξ

_{1}= 0.02, ξ

_{2}= 0.02, α = 5, β = 5, R = 1, Ha = 1. In Figure 4a, when κ is small, the slip length does not affect the streaming potential. When κ > 4, the streaming potential decreases slightly with the increase of slip length. In Figure 4b, the influence of u

_{r}is obvious, and the streaming potential decreases with the increase of u

_{r}. The reason for this phenomenon can be explained from the perspective of physical significance. According to the equation u

_{r}= u

_{e}/u

_{p}, when u

_{r}increases, u

_{p}will decrease. A diminution in u

_{p}means that the influence of the pressure gradient is weakened, resulting in a corresponding abatement in the streaming potential. This conclusion can also be drawn from Equation (40). On the other hand, with the increase of κ, the influence of u

_{r}decreases gradually.

_{s}for different ξ

_{i}(i = 1, 2) and κ when m = 0.5, n = 0.8, α = 5, β = 5, R = 1, δ = 0.1, Ha = 1, u

_{r}= 1. In Figure 5a, the relationship between the zeta potential and the streaming potential can be analyzed by changing the amplitude of the mode potential. As the amplitude of the mode potential become more enhanced, the wall zeta potential increases, resulting in an increase in the potential in the electrolyte solution. It can be seen from Figure 5a that the streaming potential increases as the amplitude heightens. In terms of the generation mechanism of streaming potential, the increase in potential leads to an increase in the proportion of positive and negative ions that is present in the solution, while a difference in the number of positive and negative ions in the electrolyte solution is positively correlated with the streaming potential. As such, the streaming potential is positively correlated with the potential on the walls. The plot oscillation along the x-axis is caused by the emergence of the vertical velocity due to the modulated surface potential. In Figure 5b, the streaming potential decreases with the increase of the κ (κ = 4, 5, 6). As κ increases, the thickness of the EDL decreases, leading to a decrease in the number of ions in the EDL, and thus the induced streaming potential generated by pressure gradient decreases gradually.

#### 3.3. Analysis of Dimensionless Velocity

_{1}= 0.02, ξ

_{2}= 0.02, δ = 0.02, u

_{r}= 1, R = 1. It can be seen from Figure 7a that the velocity u decreases with the increase of Ha. Additionally, the velocity changes rapidly near the walls. Under the influence of the modulated potentials, the vortexes and oscillations generate in the velocity profile. The negative values maybe emerge in the velocity profile (when Ha = 0.3 in Figure 7a), which mean the backflow of the fluid. According to the definition of the Hartman number, which is the ratio between the electromagnetic force and the viscous force in physics, and only one term in the modified N-S equation contains magnetic field, which corresponds to −Ha

^{2}u in the dimensionless Equation (25), it can be seen that the Hartman number plays an obstructive role in the fluid movement process. An opposite trend can be observed in Figure 7b. In Figure 7b, the velocity v increases with the increase of Ha. The reason for this phenomenon is that the flow rate in the parallel plate is a certain amount, when the Hartmann number increases, the velocity component u of the x-direction decreases, resulting in the velocity component v of the y-direction increasing.

_{1}, ξ

_{2}, Ha). Symmetrical (in Figure 8a,c) and asymmetric modulated (in Figure 8b,d) potentials are set at the upper and lower plates, respectively. In Figure 8a,b, when α = β = 0, the potential on surfaces are uniform. By increasing the amplitude (ξ

_{1}and ξ

_{2}) and constant (α and β) of the modulated potentials, the corresponding flow rate also tends to increase. In Figure 8c,d, when Hartmann number is small (Ha = 0.3), the plot of the flow rate varies rapidly with the parameter κ. It can be found that the flow rate under the modulated potentials is larger than that under the uniform potentials.

## 4. Conclusions

_{1}, ξ

_{2}, α and β). Additionally, the streaming potential and flow rate increase as the modulated potentials increase. In the analysis of the relationship between the modulated potentials and the flow rate, three types of uniform potentials, symmetric modulated potentials and asymmetric modulated potentials are considered. When comparing these three cases, it can be found that the flow rate in the charge-modulated mode is larger than that in the uniform mode. This proves that modulated charged surfaces are beneficial for fluid transport and mixing. The influence of some non-dimensional parameters (Hartmann number Ha, slip length δ, dimensionless parameter u

_{r}and κ) are also discussed under the charge-modulated potentials. The main function of slip length δ is to add an initial velocity to the fluid at the walls, so the velocity of fluid increases with the increasing of slip length. Although the velocity is oscillating, the Hartman number Ha always hinders the flow of fluid.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Schematic of the physical model. (

**a**) 3D view of microchannel; (

**b**) The cross section of the microchannel.

**Figure 2.**Distributions of streamlines for different α and β (Ha = 1, κ = 8, δ = 0, u

_{r}= 0.6, R = 0.6, m = 5.5, n = 5.8, ξ

_{1}= 0.15, ξ

_{2}= 0.18). (

**a**) α = 1.5, β = 1.8; (

**b**) α = 5.5, β = 5.8.

**Figure 3.**Changes in streamlines for different m and n (Ha = 1, κ = 8, δ = 0, u

_{r}= 0.6, R = 0.6, α = 3, β = 3, ξ

_{1}= 0.15, ξ

_{2}= 0.18). (

**a**) m = 1.5, n = 1.8; (

**b**) m = 5.5, n = 5.8.

**Figure 4.**Variation of the dimensionless streaming potential E

_{s}with the dimensionless parameter κ for different values of δ and u

_{r}(x = π/4, m = 0.5, n = 0.8, ξ

_{1}= 0.02, ξ

_{2}= 0.02, α = 5, β = 5, R = 1, Ha = 1). (

**a**) E

_{s}at different δ (u

_{r}= 1); (

**b**) E

_{s}at different u

_{r}(δ = 0.1).

**Figure 5.**The variations of streaming potential E

_{s}for different ξ

_{i}(i = 1, 2) and κ (m = 0.5, n = 0.8, α = 5, β = 5, R = 1, δ = 0.1, Ha = 1, u

_{r}= 1). (

**a**) E

_{s}at different ξ

_{1}and ξ

_{2}(κ = 3); (

**b**) E

_{s}at different κ (ξ

_{1}= 0.02, ξ

_{2}= 0.02).

**Figure 6.**The variations of velocity with the nondimensional slip length δ (x = π/2, m = 0.5, n = 0.8, Ha = 1, ξ

_{1}= 0.02, ξ

_{2}= 0.02, α = 5, β = 5, u

_{r}= 1, R = 1). (

**a**) The variations in velocity at y = −1; (

**b**) Velocity variations in microchannel (κ = 5).

**Figure 7.**The dimensionless velocity at different positions varies with the Hartmann number Ha (x = π/4, m = 0.5, n = 0.8, κ = 7, α = 5, β = 5, ξ

_{1}= 0.02, ξ

_{2}= 0.02, δ = 0.02, u

_{r}= 1, R = 1). (

**a**) velocity u; (

**b**) velocity v.

**Figure 8.**The variations of the flow rate in microchannel of the parallel plate (δ = 0.02, u

_{r}= 1, R = 1). (

**a**) Symmetric modulated potentials with different α, β, ξ

_{1}and ξ

_{2}(m = n = 0.5, Ha = 1); (

**b**) Asymmetric modulated potentials with different α, β, ξ

_{1}and ξ

_{2}(m = 0.5, n = 0.8, Ha = 1). (

**c**) Symmetric modulated potentials with different Ha (m = n = 0.5, ξ

_{1}= ξ

_{2}= 0.02, α = β = 5); (

**d**) Asymmetric modulated potentials with different Ha (m = 0.5, n = 0.8, ξ

_{1}= 0.02, ξ

_{2}= 0.04, α = 5, β = 6).

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**MDPI and ACS Style**

Bian, X.; Li, F.; Jian, Y.
The Streaming Potential of Fluid through a Microchannel with Modulated Charged Surfaces. *Micromachines* **2022**, *13*, 66.
https://doi.org/10.3390/mi13010066

**AMA Style**

Bian X, Li F, Jian Y.
The Streaming Potential of Fluid through a Microchannel with Modulated Charged Surfaces. *Micromachines*. 2022; 13(1):66.
https://doi.org/10.3390/mi13010066

**Chicago/Turabian Style**

Bian, Xinyue, Fengqin Li, and Yongjun Jian.
2022. "The Streaming Potential of Fluid through a Microchannel with Modulated Charged Surfaces" *Micromachines* 13, no. 1: 66.
https://doi.org/10.3390/mi13010066