Mechanism of sinuous and varicose modes in electrokinetic instability

The flow of miscible electrolyte streams with mismatched electrical conductivities in the presence of a parallel applied electric field is known to exhibit electrokinetic instability (EKI). This paper deals with EKI in a configuration where the base state is established by electro-osmotic flow (EOF) of three coflowing streams, with the center stream having different conductivity than the sheath streams. All reported experiments of this EKI have shown that the instability exhibits either sinuous or varicose modes depending upon whether the center stream has higher or lower conductivity than the sheath streams, respectively. In this paper we elucidate the physical mechanism underlying the selection of these unstable modes in EKI using linear stability analysis. The stability analysis shows that the EOF simply convects the unstable modes besides establishing the base state. The instability occurs due to stationary convection cells, in the reference frame moving with the EOF, resulting from the coupling of the applied electric field with free charge in the regions with conductivity gradients. Importantly, we show that the unstable and stable disturbances for the configuration with a higher conductivity center stream have opposite stability characteristics when the center stream has lower conductivity than the sheath streams. Our analysis correctly explains the numerous experimental observations showing the consistent appearance of sinuous modes for higher conductivity and varicose modes for lower conductivity center streams.