Simulating Research Of Electro-osmotic Flow With Finite Element Method

With rapid development of micro-fluidic chip technology, it is widely used in many subjects, such as, biology, chemistry, medicine and so on. Doing research on electrical phenomena, optimizing the parameters, we can improve the capability of the micro-fluidic devices, and provide guidance to electro-osmotic pumps and micro- fluidic chips.By the aid of computer simulation, the electro-osmotic influencing factors and dynamic properties are investigated. Progress is made as follows:In the first place, electro-osmotic research state at home and abroad is summed up, and the outline of this paper is to take up with finite element method (FEM) and multi-scale finite element method (MsFEM), according to numerical simulation solution, to analyze the dynamic aspects of electro-osmotic flow in micro-channels.Secondly, theoretical basis of electro-osmotic flow is studied. Based on the analysis of electro-osmotic coupling field, the mathematical models of parallel plates and rectangular micro-channels are set up, and also governing equations are given. They are Poisson-Boltzmann equation, Laplace equation and modified Navier-Stokes equation. To solve the equations conveniently, the dimensionless process is carried out. The electro-osmotic flows with homogeneous and heterogeneous surface potential are computed particularly, and also stable and periodic electro-osmosis is analyzed concretely.In the next place, dynamic aspects of electro-osmotic flows are analyzed by the aid of computer simulation. It is can be seen that on application of the electric field, electro-osmotic flows begin to develop in electrical double layer (EDL). Due to viscous shear diffusion, the flow outside the EDL region develops soon after. However, in a closed-end micro-channel, different from the electro-osmotic flow in the open-end micro-channel, there exists a reverse flow in the bulk liquid region,the flow pattern in micro-channels is composed of electro-osmotic flow and Poiseuille flow. When the flow pattern reaches stable state, the pressure gradient gets to minimum value. It is found that the pressure takes on linear distribution along the micro-channel and the pressure gradient decreases along with time. When the system is under ac electric field, the simulation is done on application of external sinusoidal electric field, it can be seen that the time period decreases gradually. The velocity within the EDL region changes quickly along with the increment of electric field frequency. However, the corresponding frequency at the central region shows relatively large delay. The dynamic characteristics of electro-osmosis can be used in electro-acoustic and dielectrophoretic theory, and it can be applied to measure the particle electrophoresis velocity, furthermore, it has important usages in DNA or protein separation.Fourthly, boundary condition of Navier-Stokes equation is studied. In microchannels, steady and fully developed flow pattern is assumed, and atmospheric pressure or zero pressure gradient is assumed at the inlet and outlet. In this paper, the pressure loss is considered at the inlet and outlet region, after numerical simulation, uniform result is got compared with analytical solution.In the next place, FEM is used to solve the governing equations. Poisson-Boltzmann equation is a nonlinear two-dimensional equation that must be linearized by Taylor series expansion, and the high-order term is neglect. With this treatment, the numerical solution reaches to astringency of 10-9 .The fully form of Navier-Stokes equation is solved. In the solving process, creeping flow solution without convection term is used as the beginning solution for iteration. By the aid of Newton-Raphson algorithm, the steady electroosmotic solution is got. Compared with the creeping solution, the steady solution reaches convergence with‖v_steady－v_creep‖_∞<0.0075, which is to say, in low Reynolds number and laminar flows conditions the inertia term can be neglected.Numerical simulation is carried out in homogeneous and heterogeneous surface potential conditions. It has been seen that the ionic concentration, theelectric potential and the externally applied electric field have varying effect on electroosmotic velocity. The simulation result shows some conclusions as follows:EDL thickness increase gradually along with decrement of ionic concentration; when EDL thickness gets at enough length, the EDL overlaps, meanwhile, the flow pattern changes to parabolic shape; the electroosmotic velocity increase in company withζpotential and external electric strength; in heterogeneous surface condition, positive and negative potential (Ψ+ andΨ-) distribution is assumed;Ψ-=-30mv,Ψ+ is set to vary from 0 to 30mv in a 10mv increment; by the aid of velocity contour line, it is found that vortex appears near the interface ofΨ+ andΨ-; in positive area ,the vortex strength increase in company with potential increment; whenΨ+=|Ψ-|, the vortex strength reaches peak.Ultimately, MsFEM is applied to electroosmosis. MsFEM is developed to solve the elliptic or parabolic equations. If the mesh size h>>εthe small scale, MsFEM is better than FEM in computation; if h<<ε, the two methods have similar astringency. In this paper, the solution structure of MsFEM is analyzed, and the astringency is proved, furthermore, base function with linear and oscillatory boundary conditions is constructed. The methods for over sampling element and area coordinate conversion are firstly provided.The MsFEM is applied in the electroosmotic simulation for the first time. Because of resonance between the grid size and the small scale, the over sampling method is taken to improve the rate of convergence. Based on the analysis in theory, Poisson-Boltzmann equation with oscillating coefficient which represents the dielectric constant of the solution is solved numerically. It is found that the results of FEM and MsFEM are uniform. With the valid potential solution, electroosmotic flow in the rectangular microchannel is simulated and studied with MsFEM.