The Navier–Stokes equations have not been solved analytically in the general case, and the only available analytical solutions arise from simple geometries (for example, the 1D flow geometries discussed in Chapter 2). Because of this, our analytical approach for solving fluid flow problems is often to solve a simpler equation that applies in a specific limit. Some examples of these simplified equations include the Stokes equations (applicable when the Reynolds number is low, as is usually the case in microfluidic devices) and the Laplace equation (applicable when the flow has no vorticity, as is the case for purely electrokinetic flows in certain limits). These simplified equations guide engineering analysis of fluid systems.

In this chapter, we discuss Stokes flow (equivalently termed creeping flow), in which case the Reynolds number is so low that viscous forces dominate over inertial forces. The approximation that leads from the Navier–Stokes equations to the Stokes equations is shown, and analytical results are discussed. The Stokes flow equations provide useful solutions to describe the fluid forces on small particles in micro- and nanofluidic systems, because these particles are often well approximated by simple geometries (for example, spheres) for which the Stokes flow equations can be solved analytically. The Stokes flow equations also lead to simple solutions (Hele-Shaw flows) for wide, shallow microchannels of uniform depths.