For bounded flows (i.e., flows within finite domains), the Reynolds number that determines the applicability of the
Stokes approximation uses a mean flow rate through a channel as the characteristic velocity and a channel
diameter or depth as the characteristic length. In this section, we focus primarily on the Hele-Shaw
solution, an analytical solution that applies to wide, shallow channels typical of many microfabrication
processes.

8.2.1 Hele-Shaw flows

VIDEO: Hele-Shaw flows.

One flow configuration that is analytically tractable and relevant to microfluidic devices is the Hele-Shawcell. A
Hele-Shaw cell consists of a domain with a small and uniform depth d in the z-dimension, and much larger
dimensions in the x- and y-directions. Historically, these devices have been fabricated by bringing two flat plates
close together, with spacers that served as obstacles for the flow. In micro- or nanofluidic devices, this geometry
occurs naturally, since most microfabrication techniques involve etching a relatively shallow channel into a flat
substrate and affixing a flat lid, resulting in a device with x- and y-dimensions ranging from 5 μm to 1 cm and a
uniform z-dimension somewhere between 10 nm and 100 μm.

If the depth of the device is uniform and much smaller than the other dimensions, then it is reasonable to
assume that p = p(x,y), in which case we can separate variables in , writing it as the product of a
z-function and an xy-function. For a channel of depth d with walls at z = 0 and z = d, this leads to the
solution

At any given z, the velocity solution in the xy-plane corresponds to a 2D potential flow where the velocity potential
is proportional to p. The same can be said for the z-averaged velocity. Hele-Shaw flows have no z-vorticity, and the
streamlines in a Hele-Shaw flow are identical to the streamlines in a potential flow with the same geometry.
However, the pressure distribution is different in the two cases. Along the z-axis, the flow has the parabolic
distribution typical of pressure-driven flow between flat plates.

Computation of a Hele-Shaw flow requires that a 2D Laplace equation be solved for the pressure, at which point
Equation 8.11 can be used to calculate the velocity field. An example of the streamlines for Hele-Shaw flow over an
obstacle are shown in Figure 8.1.

Figure 8.1: Hele-Shaw flow over a circular spacer. The velocity distribution, when viewed from the top,
appears as a potential flow for all regions in which the distance from the sidewalls is large as compared to
the depth.

8.2.2 Numerical solution of general bounded Stokes flow problems

For bounded flows, i.e., flows of fluid within solid walls, the solution approach used by engineers to predict the flow
inside a microfluidic device is largely unchanged. Unless the solution is geometrically quite simple (as was the case
in Chapter 2) such that direct integration provides a solution, Stokes flows in microchannels are typically solved
using numerical techniques.