The present work presents a simplified mathematical model to calculate the flowrate of the combined electroosmotic flow (EOF) and pressure driven laminar flow (PDLF) in the symmetric tree-like microchannel network under the assumptions of small zeta potential and thin electrical double layer. A numerical analysis of the combined EOF and PDLF in symmetric Y-shaped microchannel is also carried out to validate the mathematical model. The analytical results and numerical results are found to be in good agreement with each other. Using the mathematical model, the present work further investigates the effect of diameter ratio of the tree-like network on the flowrate of the combined EOF and PDLF to recognize a possible conclusion being similar to the Murray’s law. Based on the present work, it is found that the symmetric tree-like network has an optimal diameter ratio to achieve the maximum flowrate for the combined EOF and PDLF when the total microchannel volume is constant; however, this optimal diameter ratio for the combined flow disobeys the generalized Murray’s law in a simple form of power function of the branching number N, and it is not only related on the branching number, but also depends on the branching level and channel length ratio of the tree-like network. Furthermore, the optimal diameter ratio shows a monotonous transition from N−1/3 for the pure PDLF to N−1/2 for the pure EOF with the increasing ratio of the driven voltage and driven pressure. The present work discusses the effects of these parameters on the optimal diameter ratio for the combined EOF and PDLF.

中文翻译：

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树状微通道网络中的电渗流和压力驱动流组合

本工作提出了一个简化的数学模型来计算在小 zeta 电位和薄双电层假设下对称树状微通道网络中组合电渗流 (EOF) 和压力驱动层流 (PDLF) 的流量。还对对称 Y 形微通道中的 EOF 和 PDLF 组合进行了数值分析，以验证数学模型。分析结果和数值结果相互吻合。利用数学模型，本工作进一步研究了树状网络的直径比对EOF和PDLF组合流量的影响，以识别出类似于默里定律的可能结论。根据目前的工作，发现当总微通道体积恒定时，对称树状网络具有最佳直径比，以实现 EOF 和 PDLF 组合的最大流量；然而，这种组合流的最佳直径比以分支数的幂函数的简单形式违反了广义默里定律ñ，它不仅与分支数有关，还取决于树状网络的分支水平和通道长度比。此外，最佳直径比显示出从ñ-1/3对于纯 PDLFñ-1/2对于纯EOF，驱动电压和驱动压力的比率增加。目前的工作讨论了这些参数对 EOF 和 PDLF 组合的最佳直径比的影响。