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A semi-analytical solution is presented using method of Laplace transform for the transient pulse electroosmotic flow (EOF) of Maxwell fluid in a circular micro-channel. The driving mode of pulse EOF here is considered as an ideal rectangle pulse. The solution involves solving the linearized Poisson-Boltzmann (P-B) equation, together with the Cauchy momentum equation and the general Maxwell constitutive equation. The results show that the profiles of pulse EOF velocity vary rapidly and gradually stabilize as the increase of time
*a*. However, as the pulse width
*a *increases, this change will be less obvious. At the same time, the different change frequency of velocity profiles will slow down, which means a long cycle time. Additionally, the time needed to attain the steady status becomes longer with the increase of relaxation time
*a*.

In recent years, microfluidic devices have been vigorously developed and applied in micro-electronic mechanical systems and microbiological sensors, and the electroosmotic flow (EOF) formed in these devices has become more and more attractive [

Among previous research, various theoretical and experimental studies on steady EOF of Newtonian fluids in micro-channels under different geometric regions and physical conditions have been carried out [

All of the above-mentioned studies are related to Newtonian fluids. However, most of solutions of industry and biopharmaceutical are fluid that has the structural characteristic of non-Newtonian fluids, for example biological fluid and other solutions of long-chain molecules, whose structural characteristics include strain force, normal shear stress, hysteresis effect, variable viscosity, memory effect and so on [

Although we have learned some basic characteristics of EOF of non-Newtonian fluids in the above research, its rich characteristics still need to be studied. 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 [

Besides, Liu et al. [

The transient pulse EOF of an incompressible Maxwell fluid through a circular micro-channel with radius R, the length of the channel is L, assumed to be much larger than the diameteri.e., L ≫ 2 R is sketched in _{0} with pulse amplitude of 1, pulse repetition period of 2 a and pulse width of a in

f ( t ) = { 1 , t ∈ [ 0 , a ) , − 1 , t ∈ ( a , 2 a ] . (1)

Due to the symmetry of the geometry, we only study the semi-section of the micro-channel. Considering the pure pulse EOF and ignoring the pressure gradient, the one-dimensional Cauchy momentum equation can be given as

ρ ∂ u ( r , t ) ∂ t = − 1 r ∂ ∂ r ( r τ r z ) + ρ e ( r ) E 0 f ( t ) (2)

where u ( r , t ) is the velocity along z axial direction, ρ is the fluid density, t is the time, τ r z is the stress tensor and ρ e ( r ) is the volume charge density, E 0 f ( t ) is the ideal rectangle pulse electric field of strength E_{0}.

The boundary condition of Equation (2) is supposed no slip and can be written as

u ( r , t ) | r = R = 0 , ∂ u ( r , t ) ∂ r | r = 0 = 0 (3)

For the Maxwell fluid, the constitutive equation satisfies [

τ r z + λ 1 ∂ ∂ t τ r z = − η 0 ∂ u ( r , t ) ∂ r (4)

where λ 1 and η 0 are the relaxation time and the zero shear rate viscosity, respectively.

The chemical interaction of electrolyte liquid and solid wall generates an electric double layer (EDL), a very thin charged liquid layer at the solid-liquid interface. A cylindrical coordinate system ( r , θ , z ) is introduced. 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 [

1 r d d r ( r d ψ ( r ) d r ) = − ρ e ( r ) ε (5)

ρ e ( r ) = − 2 n 0 z ν e 0 sinh [ z ν e 0 ψ ( r ) k b T ] (6)

where ε is the dielectric constant of the electrolyte liquid, ψ ( r ) 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 a sine function.

Combining Equations (5) and (6) gives

1 r d d r ( r d ψ ( r ) d r ) = 2 n 0 z ν e 0 ε sinh [ z ν e 0 ψ ( r ) k b T ] (7)

which is subject to the following boundary conditions

ψ ( r ) | r = R = ψ 0 , d ψ ( r ) d r | r = 0 = 0 (8)

where ψ 0 is wall zeta potential, r is radial coordinate and R is radius of the circular micro-channel.

Provided that the electrical potential is small enough, the Debye-Hückel linearization approximation can be applied, which means physically that the electrical potential is small compared to the thermal energy of the charged species [

1 r d d r ( r d ψ ( r ) d r ) = κ 2 ψ ( r ) , where κ = ( 2 n 0 z ν 2 e 0 2 ε k b T ) 1 / 2 (9)

where κ is the Debye-Hückel parameter and 1 / κ usually denotes the thickness of the EDL in physical.

The net charge density can be obtained by solving Equation (9) with boundary condition of Equation (8)

ρ e ( r ) = − ε κ 2 ψ 0 I 0 ( κ r ) I 0 ( κ R ) (10)

where I_{0} is first kind modified Bessel function of order zero.

In order to obtain the solution of velocity field, some dimensionless parameters are given as

r ¯ = r R , K = κ R , ( t ¯ , λ 1 ¯ ) = ( t , λ 1 ) ρ R 2 / η 0 u ¯ ( r ¯ , t ¯ ) = u ( r , t ) U e o , τ r z ¯ ¯ = τ r z η 0 U e o / R , U e o = − ε ψ 0 E 0 η 0 (11)

where U_{eo} denotes dimensionless steady Helmholtz-Smoluchowshi EOF velocity of Newtonian fluids, K is the ratio of the characteristic width of the micro-channel to Debye length.

Using Equation (11), the Equations of (2) and (4) and the corresponding boundary condition (3) can be written as

∂ u ¯ ( r ¯ , t ¯ ) ∂ t ¯ = − 1 r ¯ ∂ ∂ r ¯ ( r ¯ τ r z ¯ ¯ ) + K 2 f ( t ¯ ) I 0 ( K r ¯ ) I 0 ( K ) (12)

τ r z ¯ ¯ + λ 1 ¯ ∂ ∂ t ¯ τ r z ¯ ¯ = − ∂ u ¯ ( r ¯ , t ¯ ) ∂ r ¯ (13)

u ¯ ( r ¯ , t ¯ ) | r ¯ = 1 = 0 , ∂ u ¯ ( r ¯ , t ¯ ) ∂ r ¯ | r ¯ = 0 = 0 (14)

Eliminating τ r z ¯ ¯ from Equation (12) and Equation (13) yields

∂ u ¯ ( r ¯ , t ¯ ) ∂ t ¯ + λ 1 ¯ ∂ 2 u ¯ ( r ¯ , t ¯ ) ∂ t ¯ 2 = 1 r ¯ ∂ ∂ r ¯ ( r ¯ ∂ u ¯ ( r ¯ , t ¯ ) ∂ r ¯ ) + ( 1 + λ 1 ¯ ∂ ∂ t ¯ ) K 2 f ( t ¯ ) I 0 ( K r ¯ ) I 0 ( K ) (15)

Let us employ the method of Laplace transform defined by

U ( r ¯ , s ) = L [ u ¯ ( r ¯ , t ¯ ) ] = ∫ 0 ∞ u ¯ ( r ¯ , t ¯ ) e − s t ¯ d t ¯ (16)

Obviously ∂ ∂ t ¯ f ( t ¯ ) = 0 , and the Laplace transform of f ( t ¯ ) is given by the

Appendix. If initial condition satisfies u ¯ ( r ¯ , 0 ) = 0 , the Laplace transforms of Equation (15) and boundary condition (14) can be given as

λ 1 ¯ s 2 U ( r ¯ , s ) + s U ( r ¯ , s ) = ∂ 2 U ( r ¯ , s ) ∂ r ¯ 2 + 1 r ¯ ∂ U ( r ¯ , s ) ∂ r ¯ + K 2 tanh ( a s 2 ) s I 0 ( K r ¯ ) I 0 ( K ) (17)

U ( r ¯ , s ) | r ¯ = 1 = 0 , ∂ U ( r ¯ , s ) ∂ r ¯ | r ¯ = 0 = 0 (18)

here tanhis a hyperbolic tangent function.

Equation (17) can be simplified as

∂ 2 U ( r ¯ , s ) ∂ r ¯ 2 + 1 r ¯ ∂ U ( r ¯ , s ) ∂ r ¯ − β 2 U ( r ¯ , s ) = − K 2 tanh ( a s 2 ) s I 0 ( K r ¯ ) I 0 ( K ) (19)

where β = λ ¯ 1 s 2 + s .

Equation (19) is a linear and inhomogeneous ordinary differential equation, and its solution can be written as the sum of a general solution U h ( r ¯ , s ) corresponding to homogeneous equation and a special solution U s ( r ¯ , s ) .

U ( r ¯ , s ) = U h ( r ¯ , s ) + U s ( r ¯ , s ) (20)

Due to the finite of U ( r ¯ , s ) at r ¯ = 0 , the homogeneous solution of Equation (19) is written as

U h ( r ¯ , s ¯ ) = A I 0 ( β r ¯ ) (21)

here A is constant, which can be determined from boundary conditions of Equation (18).

Observing the formation of the right hand side of Equation (19), the special solution can be expressed as

U s ( r ¯ , s ) = C I 0 ( K r ¯ ) (22)

Substituting Equation (22) into Equation (19) yields

C [ d 2 I 0 ( K r ¯ ) d r ¯ 2 + 1 r ¯ d I 0 ( K r ¯ ) d r ¯ − β 2 I 0 ( K r ¯ ) ] = − K 2 tanh ( a s 2 ) s I 0 ( K r ¯ ) I 0 ( K ) (23)

From Equation (9) and Equation (11), we have

d 2 I 0 ( K r ¯ ) d r ¯ 2 + 1 r ¯ d I 0 ( K r ¯ ) d r ¯ = K 2 I 0 ( K r ¯ ) (24)

Inserting Equation (24) into Equation (23) and equalizing the coefficients in front of the modified Bessel functions I 0 ( K r ¯ ) at the two sides of the equation yields

C = − K 2 tanh ( a s 2 ) s ( K 2 − β 2 ) I 0 ( K ) (25)

Therefore, the solution of velocity U ( r ¯ , s ¯ ) can be given as

U ( r ¯ , s ) = A I 0 ( β r ¯ ) − K 2 tanh ( a s 2 ) s ( K 2 − β 2 ) I 0 ( K ) I 0 ( K r ¯ ) (26)

The coefficient A with boundary condition of Equation (18) can be determined as

A = K 2 tanh ( a s 2 ) s ( K 2 − β 2 ) I 0 ( β ) (27)

Substituting Equation (27) into Equation (26), we can get

U ( r ¯ , s ) = K 2 tanh ( a s 2 ) s ( K 2 − β 2 ) ( I 0 ( β r ¯ ) I 0 ( β ) − I 0 ( K r ¯ ) I 0 ( K ) ) (28)

The inverse Laplace transform is defined by

u ¯ ( r ¯ , t ¯ ) = L − 1 [ U ( r ¯ , s ) ] = 1 2 π i ∫ Γ U ( r ¯ , s ) e s t ¯ d s (29)

where Γ is a vertical line to the right of all singularities of U ( r ¯ , s ) in the complex s plane. Because of the complexity of the express of U ( r ¯ , s ) , the numerical computation must be performed by numerical inverse Laplace transform [

In the second section, we have obtained the semi-analytical solution of the transient pulse EOF velocity of Maxwell fluid through a circular micro-channel, which mainly relies on relevant dimensionless parameters, such as the relaxation time λ ¯ 1 , the pulse width a and the electrokinetic width K. In this section, we will discuss their influence on the normalized pulse EOF velocity in detail.

the fluid gradually tends to become a Hookean elastic solid. Thus, the flow takes longer time to attain the steady status [

The variations of normalized pulse EOF velocity with radius for different relaxation time λ ¯ 1 (0.01, 0.1, 0.3 and 0.7) are presented in

relaxation time λ ¯ 1 , it is clearly seen from

A semi-analytical solution of the transient pulse EOF of Maxwell fluid through a circular micro-channel under the Debye-Hückel approximation is presented in this work. The computational results show that the velocity profiles depend mainly on the relaxation time λ ¯ 1 , the pulse width a and the electrokinetic width K. With the aid of inverse Laplace transform, the following conclusions can be drawn:

• The profiles of normalized pulse EOF velocity vary rapidly and gradually stabilize as the increase of time t ¯ within a half period (that is, pulse width a ).

• Increasing relaxation time λ ¯ 1 will lead to larger velocity amplitude. In addition, the time needed to attain the steady status becomes longer with the increase of relaxation time λ ¯ 1 .

• The velocity profiles at the center of the micro-channel increase significantly with relaxation time λ ¯ 1 , especially for the smaller pulse width a . However, as the increase of pulse width a , this change will be less obvious. At the same time, the different change frequency of velocity profiles becomes slower, which means a long cycle time, and the time required for the fluid to reach a steady state becomes longer.

• With the increase of electrokinetic width K, the velocity variations are confined to a very narrow area close to the EDL for small relaxation time λ ¯ 1 . However, as the relaxation time λ ¯ 1 increases, the elasticity of the fluid becomes conspicuous and the velocity variations can be extended to the entire region of flow.

This work was supported by the Scientific Research Project of Inner Mongolia University of Technology (Grant No. ZZ201813).

This manuscript has not been published and is not under consideration for publication elsewhere. We have no conflicts of interest to disclose.

Li, D.S. and Li, K. (2021) Analysis of Transient Pulse Electroosmotic Flow of Maxwell Fluid through a Circular Micro-Channel Using Laplace Transform Method. Open Journal of Fluid Dynamics, 11, 67-80. https://doi.org/10.4236/ojfd.2021.112005

The Laplace transform of f ( t ¯ ) is expressed as follows

∫ 0 ∞ f ( t ¯ ) e − s t ¯ d t ¯ = ∫ 0 T f ( t ¯ ) e − s t ¯ d t ¯ + ∫ T ∞ f ( t ¯ ) e − s t ¯ d t ¯ = ∫ 0 T f ( t ¯ ) e − s t ¯ d t ¯ + e − s T ∫ 0 ∞ f ( τ ) e − s τ d τ ( Let t ¯ = T + τ ) (A.1)

here T = 2 a denotes the period of the ideal rectangle pulse wave.

By shifting the term of Equation (A.1) and using the Equation (1), we have

F ( s ) = ∫ 0 ∞ f ( t ¯ ) e − s t ¯ d t ¯ = 1 1 − e − s T ∫ 0 T f ( t ¯ ) e − s t ¯ d t ¯ = 1 1 − e − 2 a s F 1 ( s ) = 1 s tanh ( a s 2 ) (A.2)

where

F 1 ( s ) = ∫ 0 a f ( t ¯ ) e − s t ¯ d t ¯ + ∫ a 2 a f ( t ¯ ) e − s t ¯ d t ¯ = 1 − e − a s s − e − 2 a s ( e a s − 1 ) s = e − 2 a s ( e a s − 1 ) 2 s (A.3)