Effect of Slip Velocity on the Rotating Electro-Osmotic Flow of the Power-Law Fluid in a Slowly Varying Microchannel

1 Introduction

The electro-osmotic flow (EOF) is a process of fluid motion under the control of an externally applied electric field. When the solid surface of a microchannel is brought into contact with an electrolyte solution, the chemical equilibrium between them results in the interface acquiring a net fixed electrical charge, after which the free ions in the electrolyte solution gather in the region near the solution surface and form an electrical double layer (EDL) (also called the ‘Debye layer’). When an external electrical potential is applied across the microchannel, due to the viscous effect of the fluid, the resulting Coulomb force drags the bulk solution along with the induced progressing diffuse layer to one side, resulting in the flow of the solution through the microchannel.

Microfluidic devices, such as microbiological sensor and micro-electromechanical microbial sensor system (MEMS), play an important role in the fields of biomedical diagnostics and drug development. For microfluidic devices, the rotating EOF can enhance the effectiveness of blending different liquids, thereby reducing the micro-fluid mixing time. The generation of vortices in the rotating EOF depends on the geometry of the microchannel. Ajdari [1] reported the generation of vortices in the EOF with sinusoidal surface potential in a microchannel. For a dielectric fluid confined within two horizontal planes with an externally applied vertical AC electric field and vertical temperature gradient, Takashima [2] studied the effect of rotation on the onset of convective instability. Chang and Wang [3] considered the rotating EOF and found that the rotation reduces the EO flow rate in the direction of the applied electric field.

Most of the present studies are based on the following assumptions: uniform cross-section, no-slip boundary condition, constant zeta potential and Newtonian fluid. However, a general case requires that one or more of the above assumptions be not met. For some hydrophobic materials, the water droplet can roll across the surface. The Newtonian slip conditions were first described by Navier [4]. These were discussed in more detail by Maxwell when he studied the stresses in rarified gas flows [5]. The slip boundary condition also plays an important role in the EOF in the microchannel. Berli and Olivares [6] and Goswami and Chakraborty [7] studied the influence of the slip boundary conditions on the EOF of Newtonian fluid in a microchannel. Shit et al. [8] studied the same problem in an inhomogeneous microchannel, and analyzed numerically the distribution of velocity in a microchannel, whose walls are slowly varying periodically.

The microfluidic devices utilizing the EOF have already been proven to be effective and functional within medical applications. In this field, most of the fluids exhibit the non-Newtonian behavior, such as biological fluids, etc. Among those presented, the power-law fluid model is the simplest and the most common one to describe the non-Newtonian fluids, which has received special attraction from the researchers in this field. Kaushik et al. for example, investigated the transient analysis of the transport features of a power-law fluid in a rotating microfluidic channel as modulated by the electrical double-layer effect [9]. The two-dimensional electroosmotic flow of a power law fluid is studied for circular and elliptic microchannels by Srinivas [10], who found that the shear thickening fluids have larger frictional parameter values compared with the shear thinning fluids. Zhao et al. studied the dynamic characteristics of the electroosmosis of a power law liquid in a rectangular microchannel by using numerical simulations [11].

However, although many researches have reported the effect of slip boundary conditions on the EOF in a microfluidic device, very few studies have investigated the same scientific problem of non-Newtonian fluids with a rotational frame. Xie and Jian [12] discussed the rotating electro-osmosis flow of power-law fluid in a in microchannel at high zeta potential, and utilized the nonlinear Poisson-Boltzmann model to obtain the analytical solution of the potential distribution of electric double layer. They also reported the influence of the Coriolis force on the velocity of flow in a microchannel. Qi and Ng [13] comprehensively discussed the rotational EOF of the power-law fluid through a slit microchannel bounded by asymmetric walls.

Motivated by the above studies, the aim of present study is to investigate the rotational EOF of a power-law fluid with slip boundary condition in a non-uniform microchannel, whose wall varies periodically. By using the Navier-Stokes equations and the nonlinear Poisson-Boltzmann model, the potential distribution and the double layer are described, after which the velocity profile of the EOF is numerically obtained with the help of the finite difference method. Finally, the influence of the material parameters on the EOF of the power-law fluid is discussed. In order to verify the accuracy of the present study, we compare the obtained results with the known ones. When the behavior index of the power-law fluid is n = 1 and there is no slip boundary condition, the obtained results in the present study agree well with the analytical solutions given by Chang and Wang [3].

2 Model Description

Consider an unsteady rotating EOF of incompressible power-law fluid in a non-uniform symmetrical microchannel with the finite and large length L, the height of the channel is bounded by z′=±h′(x′), and the walls of the microchannel is assumed to vary cosinusoidally, which is given by

where (x′,y′,z′) are the cartesian coordinates, H is the fixed height of the channel and a is the amplitude of the wary wall.

We assume that the microchannel is full of electrolyte solution with a dielectric constant ε and that the angular velocity of the rotating system is Ω=(0,0,Ω), i.e. the system rotates about the z direction, as shown in Figure 1. The flow of fluid in the microchannel is assumed to be symmetric about the x′−axis and flows in the x′ direction. The external electric field E=(E,0,0) is applied along the x direction, and the slit wall is uniformly charged with a zeta potential Ψ_w. From the theory of electrostatics, the electrostatic potential satisfies the Poisson-Boltzmann equation, in which the net electric charge density ρ_f and the potential distribution Ψ are coupled, namely,

Figure 1:

The schematic representation of the physical problem [8].

Here, n₀ is the electrolyte ion density, z₀ is the ion valence, e is the unit charge, k_b is the Boltzmann constant and T is the absolute temperature. The corresponding boundary conditions are given by

Then the solution of the above equation, i.e. the electrical potential can be obtained analytically [14]

where α=z0e/kbT, κ2=2n0z02e2/εkbT and κ is the inverse of the thickness of the double layer (see Appendix for details).

Using the following co-ordinate transformation:

Eq. (1) is transformed to the following form:

where H is the fixed height of the channel, a is the amplitude of the wary walls of the microchannel and L is the finite length of the microchannel. The height is much smaller than the length and width of the microchannel. The whole system rotating around the z axis and a potential electric field in the x axis both provide a necessary power for the EOF.

For power-law fluid [15], the dynamic viscosity μ can be expressed as

where Δ is the magnitude of the rate of strain tensor and is defined as

Here, Γ is the shear deformation rate, η is the viscosity coefficient and n is the behavior index of the power-law fluid. According to the value of n, the power-law fluid can be divided into two types: (1) expansion plastic fluid (n > 1), in which the viscosity of fluid increases with the increasing shear rate, and (2) pseudo plastic fluid (n < 1), in which the viscosity of fluid decreases with the increasing shear rate. Due to the dynamic viscosity of the viscous power-law fluid and the shear rate of deformation, the relationship between the viscous stress tensor and the rate of strain tensor can be expressed as

where ∇ V is the velocity gradient tensor and T denotes the transpose.

For an incompressible fluid, the continuity equation is given by

where V=(u,v,w) is the velocity of the EOF. In the present case, it is obvious that w = 0. Consider the rotating system with a constant angular velocity Ω, the Cauchy momentum equation can be written as

where ρ is the density of power-law fluid; t is the time; P is the pressure corrected by centrifugal force, P = p – ρ|Ω × r|²/2, r=(x,y,z); p is pressure; F is the body force vector and τ is the stress tensor. When n = 1 is expressed, the EOF of the Newtonian fluid in a rotating environment is considered.

The initial and boundary conditions are respectively given by

In Eqs. (14), (15) and (16), b is the slip length parameter.

Similar to the methods of Chang and Wang [3], we can find the solution of the form like u=u(z,t), v=v(z,t). In this case, the continuity Eq. (12) would be satisfied automatically, and the dynamic viscosity of power-law fluid can be simplified as

Then the shear forces τzx and τzy can be expressed respectively as

Therefore, the momentum equations can be simplified as

3 Numerical Algorithm

Now, we use the finite difference method to solve the EOF of the power-law fluid in a rotating system. A rectangular computational domain [0,T]×[0,h] is considered, and the domain is discretized by the mesh points given by

where Δt=T/(Nt−1) is the grid spacing in the t direction and Δz=H/(N−1) is the grid spacing in the z direction. The velocity components u and v at the grid points (i,m) are respectively denoted by

We use the central difference scheme to discretize Eq. (17), then the dynamic viscosity of the power-law fluid can be written as

Considering that Eqs. (23) and (24) are parabolic equations, by using the forward difference formula for the partial derivatives with respect to time, we obtain

The viscosity term in the momentum equation is the second-order derivative with respect to space. Here, we use the backward difference formula to approximate ∂u∂z, then the forward difference is used to approximate ∂∂z(⋅), namely,

Finally, the momentum equations can be discretized into

The corresponding discretized initial conditions and boundary conditions are, respectively given by

where β=b/h is the slip parameter.

4 Results and Discussion

As shown in Eq. (9), the viscosity of the power-law fluid depends on the velocity gradient of the fluid, so the momentum Eqs. (30) and (31) are obviously nonlinear equations. The governing equations can be solved numerically by use of the aforementioned explicit difference scheme. It is conditionally stable, and the stability condition is |ηΔt/(ρΔzn+1)|≤0.5, which is same as that for the case of the Newtonian fluid. The spatial and time step sizes are chosen as Δz=10−6 and Δt=5×10−8, respectively. To obtain the value of the velocity at point j and time level i + 1, the known date at the three adjacent points on the time line i should be used. Thus, the iterative technique must be done until the condition |(uim+1−uim)/uim+1|≤10−8 is satisfied.

In the present study, all the parameters in the governing equations are dimensional, and they are chosen as follows for the EOF in a microchannel [12]: the permittivity of the electrolyte solution ε=7.09×10−10F/m; the half-height H = 100 μm; the electric field strength E=104V/m; the absolute temperature T = 293 K; the valence of ions z₀ = 1; the wall electrical potential ψw=−0.1V; the flow consistency, η=0.9×10−3Pa⋅sn; the concentration of the electrolyte solution n0=NAc, where c = 10⁻⁵mol/L is the molar concentration; NA=6.02×1023/mol; e=1.6×10−19C; kb=1.38×10−23J/K and ρ=1.06×103kg/m3. Furthermore, the electrodynamic width K is defined as the ratio of the half-height H of the microchannel to the EDL thickness κ−1. As a matter of convenience, all the figures of the numerical results are shown non-dimensionally, and the x- and y-components of the velocity and the height of the microchannel are normalized by Ueo=−εEΨw/η and H, respectively.

In order to demonstrate the accuracy of the numerical method and the theoretical model in the present study, when n = 1 and β = 0, i.e. for the case of Newtonian fluid and K = 10, Ω = 100 rad/s, Ψ_w = −0.025 V, the corresponding obtained numerical results of the velocity components u and the analytical results given by Chang and Wang [3] are shown in Figure 2. As can be seen, the numerical results are in complete agreement with those analytical ones, indicating that the present numerical method is correct and effective.

Figure 2:

The comparison of analytical results [3] with the present numerical results when β = 0, n = 1, K = 10, Ω = 100 rad/s, Ψ_w = −0.025 V.

Figure 3 illustrates the effect of the power-law index n on the normalized velocity profiles of u and v when the slip parameter β = 0.1. As shown in Figure 3a, in the region near the wall of microchannel, the velocity profile is steep. This is in agreement with the finding obtained by Xie and Jian [12], except for the slip distance on the walls. The figure also shows that the increasing power-law index n decreases the velocity magnitude. This can be attributed to the fact that the increasing power-law index indicates a larger viscosity of the fluids, which makes it is more difficult to flow in the microchannel. The same phenomenon can also be found in Figure 3b.

Figure 3:

The effect of n on the velocity profiles when β = 0.1, K = 10, Ω = 100 rad/s and Ψ_w = −0.025 V.

The variations of both the velocity components for the different values of the slip constant β are shown in Figures 4 and 5, respectively. As shown in the figures, the axial velocity modestly increases with the increasing value of β. The effect of β on the axial velocity profile is more obvious in the region near the wall than that in the region far away the wall, because the larger slip parameter suggests the weaker adhesion effect of the boundary. For the same physical reason, an interesting phenomenon is noted: the peak velocity is closer to the wall of the microchannel when the value of β is larger.

Figure 4:

The velocity profiles for the different values of β when n = 0.8, K = 10, Ω = 100 rad/s and Ψ_w = −0.025 V.

Figure 5:

The velocity profiles for the different values of β when n = 1.2, K = 10, Ω = 100 rad/s and Ψ_w = −0.025 V.

5 Conclusion

This study presents a mathematical model to describe the rotating EOF of the power-law fluid in a non-uniform microchannel with slip boundary condition. Using the nonlinear Poisson-Boltzmann theory and the finite difference method, the numerical solution for the velocity components are obtained. By conducting a comparison with the known analytical results for the Newtonian fluid, the present numerical scheme is proved to be effective and correct. The effects of the slip parameter β and the flow behavior index n on the velocity distribution are also discussed. Results indicate that the increasing slip parameter enhances the flow in the microchannel and makes the peak velocity close to the wall of the microchannel. The increasing flow behavior index also enhances the effect of adhesion and decreases the magnitude of the velocity profile.