Because the thin electrical double layer assumption implies scale separation between the electrical double layer and
the bulk flowfield, we often find it useful to analyze electroosmotic flow in terms ofmatched asymptotics. In this
case, we find two asymptotic solutions: (1) theinner solution, corresponding to the electrical double layer, in which
we keep track of the details of the Coulomb forces and the resulting velocity gradients and vorticity, but
we assume that the extrinsic electric field is uniform, and (2) theouter solution, where we assume
that the fluid is electroneutral and irrotational, but allow the extrinsic electric field to vary spatially
(Figure 6.2). The inner solution is valid in the electrical double layer, but gives incorrect results far from
the wall if the extrinsic field varies with distance from the wall. The outer solution is valid outside
the electrical double layer, but gives incorrect results near the wall and violates the no-slip condition.

Figure 6.2: Left: Inner and outer solutions for electroosmotic flow. The inner solution is valid in the
electrical double layer, but is generally invalid in the bulk. The outer solution is valid in the bulk, but invalid
in the electrical double layer and at the wall. Right: The composite solution smoothly transitions from the
inner solution to the outer solution.