Scaling Laws for Electroosmotic Flow of Power-Law Fluids in Fractal Branching Networks

Electroosmotic flow (EOF) plays a vital role in fluid transport within micro- and nano-scale systems handling ionic fluids. Driven by electric fields and resisted by viscous forces, EOF is especially relevant for microfluidic applications. This study presents the theoretical framework for EOF of power-law fluids in fractal-like branching networks, addressing both volume and surface-area constraints—a domain unexplored in existing literature on flow optimization. Prior EOF analyses have focused on Newtonian fluids in fractal network or numerical analysis of power-law fluid flows in complex geometries; here, we extend the scope to non-Newtonian fluids and complex hierarchies using theory and derived scaling laws. Assuming fully developed, steady, axisymmetric, and incompressible EOF in cylindrical microchannels, the model incorporates the Debye–Hückel approximation to linearize electrokinetic behavior and neglects pressure-driven components. The resulting electroosmotic flow rates $Q$ for power-law fluid enhances for shear-thinning fluids (lower $n$) compared to Newtonian or shear-thickening fluids. Under volume constraints, we show that the optimal branching radius ratio \(\beta^*\) scales as \(N^{-1/2}\), yielding uniform mean velocity across all generations. This configuration yields a maximum normalized conductance \(E_{\text{vol}} = 1\), independent of the number of bifurcations \(N\), length ratio \(\gamma\), or generation count \(m\). Under surface-area constraints, \(\beta^*\) scales as \(N^{-(n+1)/(2n+1)}\), where \(n\) is the power-law index. Here, optimal transport depends on \(n\) and \(N\), with conductance \(E_{\text{surf}}\) decreasing as \(\gamma\), \(n\), \(m\), or \(N\) increases. These novel scaling laws, reported for the first time for electroosmotic flow of power-law fluids in branching networks, underscore the fundamental differences between electroosmotic and pressure-driven flows. The results offer novel valuable insights for designing bioinspired microfluidic designs, electrokinetic pumps, and lab-on-a-chip devices. This work bridges fluid rheology with network geometry, offering a rigorous theoretical foundation for efficient EOF transport.