Scaling Effects of the Weissenberg Number in Electrokinetic Oldroyd-B Fluid Flow Within a Microchannel

This study attempts to extend previous research on electrokinetic turbulence (EKT) in Oldroyd-B fluid by investigating the relationship between the Weissenberg number ( $Wi$ ) and the second-order velocity structure function ( $S_v^2$ ) under applied electric fields. Inspired by Sasmal's demonstration in Sasmal (2022) of how heterogeneous zeta potentials induce turbulence above a critical $Wi$ , we develop a mathematical framework linking $Wi$ to turbulent phenomena. Our analysis incorporates recent findings on AC (Zhao & Wang, 2017) and DC (Zhao & Wang 2019) EKT, which have defined scaling laws for velocity and scalar structure functions in the forced cascade region. Our finding shows that $S_v^2\left(l\right)\sim {\lambda }_1^{4/5}{l}^{2/5}$ and $S_{\sigma }^2\left(l\right)\sim {\lambda }_1^{-2/5}{l}^{4/5}$ , for a length scale $l$ , and $Wi=\frac{{\lambda }_1{u}_l}{l}$ , where $u_l=\sqrt{S_u^2\left(l\right)}$ is a velocity fluctuations quantity and ${\lambda }_1$ denotes the time relaxation parameter. This work establishes a positive correlation between ${\lambda }_1$ and turbulent flow phenomena through a rigorous analysis of velocity structure functions, thereby offering a mathematical foundation for building the design and optimization of EKT-based microfluidic devices.