The hypothesized solution (V = E) are examined by plugging it back to the governing
equations and the boundary conditions in Equations 4-11 to 4-18. Employing the assumptions
and the properties of E in Equations 4-47 to 4-50, it is found that the governing equations are
satisfied and the boundary conditions are met. Hence the solution V = E is a valid solution, and
as a resultV x = 0. The electroosmotic flow velocity field is thus a conservative vector field,
meaning the EOF is irrotational. This explains why a flow separation is absent in EOF in a wavy
channel whereas a creeping Poiseuille or Couette flow in the same wavy channel may experience
as mentioned in Section 4.4.2. However, the similarity fails to hold when heterogeneity is
present in the zeta potential of channel walls or when an external pressure is applied, where flow
recirculation may occur.34'35,77, 105
4.4.4 Vorticity in Electroosmotic Flow
As pointed in previous section that V x V = 0 is valid everywhere in EOF, implying
vorticity is zero throughout the electroosmotic flow (except inside EDL). This is true not only to
EOF in the wavy channel, but to EOF in any shape of channels. To explain this, let's consider a
2D, incompressible, barotropic flow of fluid with constant viscosity and permittivity, and subject
to a slipping wall (Vwa, = Vwat ). The momentum equation for EOF is
DV 1 PV
p+vV2V V
Dt p p (4-51)
At the channel surface, the momentum equation in tangential direction is rewritten as
following, where ) = is the vorticity at the surface.
an
-V -a + -- 1- + +I v + wal
[all at i,, P a wall P ai w alt Wail
acceleration of surface wall pressure gradient wall electric field velocity gradient of surface (4-52)
The formula implies that vorticity that diffuses toward the EOF at the channels comes from