Two-dimensional analysis of electrokinetically driven out-of-plane vortices in a microchannel liquid flow using patterned surface charge

The technology developed for photolithographically patterning the electric surface charge to be negative, positive, or neutral enables the realization of complex liquid flows even in straight and uniform microchannels with extremely small Reynolds number. A theoretical model to analyze a steady incompressible electrokinetically driven two-dimensional liquid flow in a microchannel with an inhomogeneous surface charge under externally applied electric field is derived. The flow field is obtained analytically by solving the biharmonic equation with the Helmholtz-Smoluchowski slip boundary condition using the Fourier series expansion method. The model has been applied to study three basic out-of-plane vortical flow fields: single vortex and a train of corotating and a series of counterrotating vortex pairs. For model verification, the solution for the single vortex has been tested against numerical computations based on the full Navier-Stokes equations revealing the dominant control parameters. Two interesting phenomena have been observed in out-of-plane multivortex dynamics: merging of corotating vortices and splitting of counterrotating vortices. The criteria for the onset of both phenomena are discussed.