System-Level Modeling of Microflows in Circular and Rectangular Channels

System-level models for unsteady, incompressible, low Reynolds number flow in channels of slightly varying cross sections of different geometries are presented. The models are based on approximate solution of the unsteady Navier-Stokes equation subject to no-slip and first-order slip boundary conditions. The proposed model, relating the volume rate to the average pressure drop across the channel, is cast into an electric circuit model that consists essentially of an infinite number of parallel branches in series with a resistor and a nonlinear component. The resistor and the nonlinear element in series account respectively for viscous dissipation in the direction of the flow and for the convective part of the inertia term. The set of parallel branches captures the unsteady behavior as well as viscous dissipation normal to flow direction. Previous channel models available in the literature proved to be a special case of the models proposed in this paper at steady state or in the limit of vanishing Reynolds number. The proposed models offer superior accuracy when transient behavior and associated dynamic characteristics are of interest. The models also become more accurate for flows in slightly divergent channels, flows at larger Reynolds number, and in flows that experience sudden changes or that are subjected to forced oscillations with large values of the Strouhal number. The proposed models are flexible in the sense that accuracy and cost can be easily traded by increasing or decreasing the number of branches included in the model. The derived models are compared with previous models and with numerical solutions of the full Navier-Stokes equations.