A Discussion\non the Critical Electric Rayleigh Number\nfor AC Electrokinetic Flow of Binary Fluids in a Divergent Microchannel

Electrokinetic (EK) flow is a type of flow driven or\nmanipulated\nby electric body forces, influenced by various factors such as electric\nfield intensity, electric field form, frequency, electric permittivity/conductivity,\nfluid viscosity, etc. The diversity of dimensionless parameters, such\nas the electric Rayleigh number, complicates the comparison of the\nEK flow stability. Consequently, comparing the performance and cost\nof micromixers or reactors based on EK flow is challenging, posing\nan obstacle to their industrial and engineering applications. In this\ninvestigation, we theoretically derived a new electric Rayleigh number\n(<i>Ra</i><sub>e</sub>) that quantifies the relationship\namong electric body forces, fluid viscosity, and ion diffusivity,\nbased on a tanh model of electric conductivity distribution. The calculation\nresults indicate that the new <i>Ra</i><sub>e</sub> exhibits\nricher variation with the control parameters and better consistency\nwith previous experimental reports. We further conducted experimental\nstudies on the critical electric Rayleigh number (<i>Ra</i><sub>ec</sub>) of the AC EK flow of binary fluids in a divergent\nmicrochannel. The experimental variations align well with the theoretical\npredictions, particularly the existence of an optimal AC frequency\nand electric conductivity ratio, demonstrating that the tanh model\ncan better elucidate the underlying physics of EK flow. With the new\nelectric Rayleigh number, we found that EK flow in the designed divergent\nmicrochannel has a much smaller <i>Ra</i><sub>ec</sub> than\npreviously reported, indicating that EK flow is more unstable and\nthus more suitable for applications in micromixers or reactors in\nindustry and engineering.