Optimization analysis of fractal tree-like microchannel network for electroviscous flow to realize minimum hydraulic resistance

As a significant principle to guide the optimization design of fractal tree-like network, Murray's law is obtained under the assumption of neglecting surface charge at the solid-liquid interface. However, surface charge should be considered due to the surface charge-induced electroviscous effect on the fluid flow over the micro/nanoscale. The present work carries out the optimization analysis of fractal tree-like microchannel network for electroviscous flow to realize minimum hydraulic resistance under the constraint of constant channel volume by considering the surface charge. It is found that the surface charge significantly modifies the conventional Murray's law. Both zeta potential to manifest surface charge and microchannel radius make non-monotonic effects on the optimal radius ratio between the daughter channel and the parent channel for the electroviscous flow in the fractal tree-like microchannel network to achieve a minimum hydraulic resistance. These non-monotonic effects are related to the apparent electro-viscosity in the microchannel with different channel radius and different zeta potential. Additionally, the optimal radius ratio is found to be linearly dependent on both the ratio of apparent electro-viscosities of electroviscous flow in the microchannels at the two successive branching levels and the length ratio of the microchannels at two successive branching levels.