Chapter 2
Unidirectional flow

While the Navier-Stokes solutions cannot be solved analytically in the general case, we can still obtain solutions that guide engineering analysis of fluid systems. If we make certain geometric simplifications, specifically that the flow is unidirectional through a channel of infinite extent, the Navier-Stokes equations can be simplified and solved by direct integration. The key simplification enabled by this assumption is that the convective term of the Navier-Stokes equations can be neglected, because the fluid velocity and the velocity gradients are orthogonal. The solutions in this limit include laminar flow between two flat plates (Couette flow) and laminar flow in a pipe (Poiseuille flow). These flows are simultaneously the simplest solutions of the Navier-Stokes equations and the most common types of flows observed in long, narrow channels. Many microchannel flows are described by these solutions, their superposition, or a small perturbation of these flows. This chapter presents these solutions and interprets these solutions in terms of flow kinematics, viscous stresses, and Reynolds number.

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