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In this paper, a coordinate transformation method (CTM) is employed to numerically solve the Poisson–Nernst–Planck (PNP) equation and Navier–Stokes (NS) equations for studying the traveling-wave electroosmotic flow (TWEF) in a two-dimensional microchannel. Numerical solutions indicate that the numerical solutions of TWEF with and without the coordinate transformation are in good agreement, while CTM effectively improves stability and convergence rate of the numerical solution, and saves computational cost. It is found that the averaged flow velocity of TWEF in a micro-channel strongly depends on frequency of the electric field. Flow rate achieves a maximum around the charge frequency of the electric double layer. The approximate solutions of TWEF with slip boundary conditions are also presented for comparison. It is shown that the NS solution with slip boundary conditions agree well with those of complete PNP-NS equations in the cases of small ratios of Electric double layer(EDL) thickness to channel depth( λD/ H). The NS solution with slip boundary conditions over-estimates the electroosmotic flow velocity as this ratio( λD/ H) is large.

Microfluidics has emerged as a new area of multiphysical research associated with fluid mechanics, biology, chemistry and electricity [_{e}E is a nonlinear interaction between the applied electric field and induced charge density. It is called induced charge electroosmosis (ICEO) [^{ }showed that the averaged electroosmotic flow velocity of TWEF in a microchannel achieves a maximum around the charge frequency of EDL. Exact analytic solutions of TWEF in a microchannel have not been available. Numerical solution is also difficult because of the locally high gradient near the solid wall and multiphysical interactions of fluid flow, electricity and ion migration. Furthermore, the gradient is much greater in the direction perpendicular to the wall (transverse direction) than that in the direction tangent to the wall (longitudinal direction). Therefore, a refined grid is needed in the region near the wall. However, the grid size in the transverse direction may be much smaller than that in the longitudinal direction. Such kind of grid often leads to poor numerical solutions. It is difficult to manage numerical convergence and grid refinement. The coordinate transformation method (CTM) [

A two-dimensional microchannel with embedded electrodes is shown in _{0}, ω, k are the amplitude, angle frequency and wave number of the traveling wave, respectively. is the physical frequency, and, where L is the wave length. It is expected that the traveling wave electroosmotic flow behaves periodically in the longitudinal direction of the channel (x), thus a part of the channel (L × H) is needed for numerical analysis, as shown in

The continuity equation and Navier-Stokes (NS) equation for incompressible fluid flow read as:

where, V is velocity, ρ is fluid density, μ is dynamic viscosity of the fluid, p is pressure, and ρ_{e} is the volume charge density of the solution. The last term in Equation (2) (–ρ_{e}ΔΨ) is the electric force acting on the fluid, where Ψ is electrical potential. No-ship boundary conditions on channel walls and periodical flow conditions at the inlet and outlet of the channel are imposed. The electric potential of TWEF is governed by the Poisson equation [

where c_{+}, c_{–} are mole concentrations of positive and negative ions, respectively, ε is solution permittivity, and F is Faraday’s constant. The traveling wave electric potential on the bottom wall of the channel is

and

on the insulated top wall. Periodic conditions of the potential Ψ at inlet and outlet of the channel are imposed. The ion concentrations are governed by the PoissonNernst-Planck (PNP) equations reflecting charge conservation law.

where D is ion diffusivity, R is the gas constant, T is the absolute temperature, z_{i} is the ion valence, and J_{i} is the ion flux. Furthermore, n·J_{i} = 0 on the channel wall, and periodic conditions of ion concentration are imposed at inlet and outlet of the channel. In general, the dimensionless flow variables are defined as follows:

, , , ,

, , , ,

,. (8)

, , , ,

, ,. (9)

where, is the characteristic thickness of the electric double layer, ε and σ are permittivity and electric conductivity of the solution, respectively, and c_{0} is the ion concentration of the bulk solution.

Dimensionless Poisson equation is written as:

Dimensionless Nernst-Planck equation reduces to

Dimensionless Navier-Stokes equation is written as:

where, ,. Equations (10)-(14) are the dimensionless coupled governing equations of TWEF in a microchannel.

Coordinate Transformation Method for TWEF in a MicrochannelCoordinate transformation is introduced by defining

, ,. (15)

where, , in Equations (10)-(14) are expressed as:

,. (16)

where, , ,

Poisson Equation (10) in the transformed coordinate system is written as:

where,.

Nernst-Planck Equations (11)-(12) in the system are written as:

Flow Equations (13)-(14) in the system are written as:

Typical microchannel geometric data and physica1 parameters are given as follows. The computational channel length and depth are, as shown in ^{3}, c_{0} = 10^{–2} mol/m^{3}, μ = 6.919 × 10^{–4} Pa/s, T = 310 K, ε = 7.2036 × 10^{–10} F/m, F = 96,490 C/mol, R= 8.314 J/(k·mol), λ_{D} = 100 nm, D = 2 × 10^{–9} m^{2}/s, c_{0} = 10^{–2} mol/m^{3}, U_{0} = 2 × 10^{–4} m^{2}/s, σ = 1.45 × 10^{–4} (mS)/m, a = 0.2 in coordinate transformation Equation (15). Averaged TWEF velocity is defined as

, where P is the period of TWEF. Dimensionless averaged velocity is defined as. COMSOL Multiphysics was used to solve TWEF Equations (18)-(23) in the transformed system () and Equations (10)-(14) in the physical system (). To validate our numerical solutions, an approximate solution of the NS equations with electroosmotic slip velocity boundary conditions is also presented for comparison with that of the complete TWEF equations. When the voltage amplitude of the traveling wave electric field is low, and the ratio of EDL thickness to channel depth () is small, the effects of EDL on the electrokinetic flow can be replaced by a local slip velocity [9-11]. It is called Helmholtz-Smoluchowski velocity, where ξ and E are the local wall zeta potential and wall electric field intensity, respectively. This velocity serves as the slip boundary condition when solving the NS equations without solving Poisson-NernstPlanck (PNP) equations [

channel. The averaged velocity behavior is consistent with experimental results [

In the case of large voltage applied on the electrodes () in

The effects of EDL thickness (dependent on fluid properties) on the electroosmotic flow velocity is illustrated in

The effects of wave length of TWEF on the electroosmotic flow velocity is shown in

The numerical investigation of traveling wave electroosmotic flows (TWEF) in a microchannel is carried out in this study using a coordinate transformation method, based on fully coupled nonlinear Poisson-Nernst-Planck equations and Navier-Stokes equations. Conclutions are summarized as follows.

1) Grid refinement is required for accurate numerical solution due to high gradients of TWEF near the solid wall and multiphysical interactions between fluid flow, electricity and ion transport. The coordinate transformation method adopted in this study effectively decreases gradients of flow variables and improves stability and convergence. The numerical solutions in the transformed system with a coarse grid can be as accurate as those in the physical system with a refined grid.

2) Averaged velocity of TWEF in the microchannel behaves like a Gaussian distribution with respect to the electric field frequency, and achieves a maximum when the frequency of TWEF approaches the charge frequency of the electric double layer, i.e.,. The averaged velocity is small as the electric frequency is far away from the charge frequency. The averaged velocity decreases when EDL thickness in the microcannel or the wave length of TWEF increases.

3) Slip boundary model simplifies numerical computation of TWEF in the microchannel. The slip boundary solution is accurate in the case of small ratio of EDL thickness to channel depth, and over-estimates averaged TWEF velocity when the ratio () is large.