A Fourier-Chebyshev non-interpolating method for the direct numerical simulation of two-dimensional wall-induced electrokinetic flow*

Transport of samples in microfluidic systems relies on electrokinetic pumps or actuators. The working principle of these actuators is induced charge electroosmosis. The traditional approach to predict the behavior of such systems utilizes the Smoluchowski slip condition. Thus, the Poisson-Nernst-Planck system can be solved separately from the flow using the boundary conditions for coupling.

Yet, this approach works in the small-Debye-layer limit δ → 0, breaking down for δ ∼ O(1). The latter can be important for miniaturization or development of new systems. Full-electrokinetic effects also play a role in the behavior of vesicles under the influence of an electric field.

We present a two-dimensional direct numerical solver for electrokinetic phenomena. As an example, we use an electrokinetic pump working with traveling wave electroosmosis. The high accuracy of the code is based on the non-interpolating spectral method using a combination of Fourier and Chebyshev modes. The basic spectral operations where implemented in a spectral module that was specifically designed for this code. We will discuss a number of challenges that we encountered, such as the handling of nonlinear boundary conditions, the coupling mechanism between the Poisson-Nernst-Planck system and the Stokes flow, and the solution of the fourth-order streamfunction equation.