A simplified model for unsteady pressure driven flows in circular microchannels of variable cross-section

In this paper, we present a fast and accurate model for unsteady pressure-driven flows in circular microchannels of variable cross-section. The model is developed for channels of small diameter to length ratio, but allows for large variations in the channel's diameter along the axis. A key feature of the model is that it puts no restriction on the time dependence of the forcing, in terms of shape and frequency. The only condition on the forcing is such that the advective component of the inertia term is small. This is a major departure from many previous expositions which assume harmonic forcing. The model is based on an extended and unsteady lubrication approximation in the aspect ratio of the channel. The resulting equations for each order are solved analytically using a finite Hankel transform, except for the implicit pressure profile, which is solved numerically with a recursive time scheme. Compared to classical CFD simulations, the reduced order semi-analytic method is two orders of magnitude faster, owing in part to the fact that the number of modes required for the convergence of these expressions is not too large. The numerical simulations reveal that the model is accurate for a large class of channels and a fairly wide range of Reynolds numbers. This, combined with the fact that it imposes no conditions on the shape and frequency of the unsteady forcing, renders the model a valuable tool for rapidly simulating large fluidic circuits (as in lab-on-a-chip, μTAS and the human body), thereby allowing significant reduction in the design parameters space.