* the acceleration of the wall,
* the tangential pressure gradients along the boundary,
* the tangential electric field along the boundary, or
* the tangential velocity gradient of the boundary.
The first and the last sources are zero when the boundary is stationery in most situations,
such as the EOF in the wavy channel discussed in this work. Hence, vorticity is generated at the
surface by the induced pressure gradient and the wall electric field, and they generate equal
amount but opposite tone of vorticity, resulting the net production of vorticity zero at the channel
surface. Hence the vorticity inside the channel remains zero.
When the boundary is moving, the vorticity generation at the channel wall surface has to
include the tangential acceleration and the tangential velocity gradient. For example in the EOF
with the effective slipping boundary condition, the Helmholtz-Smoluchowski velocity is
assumed at the edge of the electric double layer, and the entire fluid is assumed electrically
neutral (pe = 0). The wall pressure gradient and the velocity gradient of boundary become the
sources of vorticity in EOF. Again, their vorticity production cancels each other and zero
vorticity enter into the EOF from the channel surface. As a result, vorticity is zero everywhere in
the EOF.
4.4.5 Electroosmotic Flow with Pressure Gradients
The discussion in Sections 4.4.1-4.4.4 pertains to an electroosmosis-driven flow with no
external pressure applied. When an external pressure drop is present, the resulting flow can be
viewed as superposition of a Poiseuille flow and the EOF, since the governing equation in a
creeping flow is a linear ordinary differential equation. Figure 4-9a shows the velocity profile of
an EOF in a wavy channel of scaled width h = 2.84 and scaled wave amplitude a = 0.45 while
the velocity profile of the pressure driven flow in the same channel is illustrated in Figure 4-9b.