Scaling Laws and Design Principles for Electroosmotic Flow of Power-Law Fluids in Tree-like Networks with Rectangular Cross-Section

Electroosmotic flow (EOF) plays a critical role in microfluidic transport, especially for ionic solutions in confined geometries. Driven by electric fields and resisted by viscous forces, EOF is especially relevant for microfluidic applications. In this paper, we present a theoretical framework for EOF of power-law fluids in fractal-like rectangular branching microchannel networks, considering both volume and surface-area constraints. Extending beyond single-channel analyses and previous numerical studies, we derive scaling laws and optimal branching strategies that integrate fluid rheology with hierarchical network geometry. The flow is modeled under the Debye–Hückel approximation for fully developed, steady, incompressible, and pressure-gradient free conditions. Our results reveal that shear-thinning fluids enhance electroosmotic flow rate compared to Newtonian or shear-thickening fluids, consistent with prior numerical studies. Under volume constraints, the optimal height ratio scales as \( \beta^* \sim N^{-1} \alpha^{-1} \), where $N$ and $\alpha$ are the branch splitting at each junction and the width ratio, respectively. We also find uniform velocity across generations at optimal volume-constraint condition. Further, the maximum normalized conductance \( E_{\text{vol}} \) reaches unity, independent of geometric or rheological parameters at optimal conditions. Furthermore, under surface-area constraints, \( \beta^* \sim N^{-1} \alpha^{-n/(n+1)} \), where \(n\) is the power-law index. Also, optimal \( E_{\text{surf}} \) decreases with increasing \( n \), \( m \), or \( N \), and with increasing $\alpha$. Notably, the voltage drop remains uniform across generations under both constraints at optimality, and the volume or surface area distribution are also uniform per generation is conserved at respective constraints. These novel scaling laws, reported for the first time for electroosmotic flow of power-law fluids in rectangular cross-section branching networks, underscore the fundamental differences between electroosmotic and pressure-driven flows. The framework offers predictive insights for designing efficient electrokinetic systems, including lab-on-a-chip platforms, biomimetic transport devices, and electroosmotic micropumps. This work bridges fluid rheology with network geometry for EOF, offering a rigorous theoretical foundation for efficient EOF transport.