Derivation of governing equation describing time-dependent penetration length in channel flows driven by non-mechanical forces.

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.