The previous chapters introduce the governing equations for fluid flow and provide solutions for simple unidirectional flows. Although we can only rarely use these simple solutions to exactly describe real flows, these unidirectional flows provide a framework for generating engineering estimates for many flows, and these estimation techniques reduce complex systems with large numbers of components into a relatively simple, approximate, linear description. This is relevant for flow in microdevices both (1) because microdevices can straightforwardly be made with large numbers of microchannels and (2) because those microchannels often have long, thin geometries that lead to largely unidirectional flows that are well approximated by the solutions presented in Chapter 2. Our approach, called hydraulic circuit analysis, involves assuming that the Poiseuille flow derived in Chapter 2 provides a sound engineering estimate of the pressure drops and flow rates through long, straight channels, even if the cross section is not exactly circular and the channels are neither perfectly straight nor infinite in extent. We thus write an approximate linear relation between pressure drops and flow rates through these channels – the Hagen – Poiseuille law. This law, combined with conservation of mass, approximately prescribes fluid flow through complex rigid networks by solution of sets of algebraic equations. By adding a hydraulic capacitance or compliance, this linearized analysis also allows prediction of unsteady flow through channels with a finite flexibility.