Derivation of governing equation describing time-dependent penetration length in channel flows driven by non-mechanical forces.
Derivation of governing equation describing time-dependent penetration length in channel flows driven by non-mechanical forces.
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DOI：
10.1016/j.aca.2010.02.019
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发表时间：
2010-05
影响因子：
6.2
通讯作者：
S. Bhattacharya;D. Gurung
中科院分区：
文献类型：
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作者：
S. Bhattacharya;D. Gurung
过去曾多次尝试将微通道中的穿透长度描述为电渗或毛细管流的时间函数。在所有这些研究中，通过考虑惯性贡献以及其他力，推导出复杂的控制方程。该系统是不稳定的，这种推导需要考虑通道内适当的非稳定流动剖面。然而，过去的理论通过使用仅对稳定系统有效的抛物线速度场引入了不一致性。这种差异会产生与惯性贡献相同数量级的误差，从而导致两个后果。首先，固有的不准确性使得过去的理论在雷诺数大于1时不适用。其次，它使得包含惯性项对于低雷诺数系统毫无意义——惯性带来的额外复杂性没有任何作用，因为公式中已经存在类似阶次的误差。在本文中，我们通过纠正数学推导来解决这个问题，其中使用非定常流的正确解析表达式代替抛物线速度剖面。因此，对于任意雷诺数的层流系统，可以导出正确的穿透长度控制方程。我们还通过考虑适当的限制条件来简化低雷诺数情况的方程。低雷诺数流动方程虽然与早期研究中获得的方程一样准确，但比以前的版本简单得多。
Several past attempts were made to describe the penetration length in microchannels as a function of time for electroosmotic or capillary flow. In all of these studies, a complex governing equation is derived by taking into account the inertial contributions along with other forces. The system being unsteady, this derivation requires the consideration of proper unsteady flow-profile inside the channel. However, the past theories introduced an inconsistency by using parabolic velocity-field valid only for steady systems. This discrepancy creates an error of the same order as the inertial contribution leading to two consequences. Firstly, the inherent inaccuracy makes the past theories inapplicable when Reynolds number is greater than unity. Secondly, it renders the inclusion of inertial terms pointless for low Reynolds number systems—the additional complexity due to inertia does not serve any purpose because error of similar order is already present in the formulation. In this paper, we address this problem by rectifying the mathematical derivation where instead of the parabolic velocity-profile, proper analytical expression for unsteady flow is used. As a result, the correct governing equation is derived for the penetration length for any laminar system with arbitrary Reynolds number. We also simplify the equation for low Reynolds number cases by considering appropriate limiting conditions. The equation for low Reynolds number flow, while being as accurate as the ones obtained in the earlier studies, is much simpler than the previous versions.