V2 = 0 (4-9)
Since the double layer thickness is much smaller than the channel width, the effect of the
flow in the double layer (also called as the inner flow region) on the outer flow can be
approximated by the slip velocity.74'76, 78, 80, 100 The effective slip boundary condition is
described by the Helmholtz-Smoluchowski equation,14
walls = 'Vals, (4-10)
84pot
where 7 is the electroosmotic mobility, defined in Equation 1-21 as r =
pv
To non-dimensionalize x, y, u, v, p, and # in Equations 4-1, and 4-4 to 4-10, the following
x y *- u v
is introduced: [x = u =, v ,
L/271 L/271 2;T o / L 2,7r0/L
p = P = 2 ]. It should be noted that the scale of the convection term in the
pvrDo (2r/L)2 0
uL
momentum equation is on the order of Reynolds number, Re = As EOF is typically in the
v
regime of a creeping flow where Re << 1, the inertial term is then dropped from the momentum
equation (Equation 4-8),
Vp* V2*. (4-11)
For clarity, the superscript stars in the dimensionless variables are dropped hereinafter.101
To characterize the effects of the channel geometry, two dimensionless parameters related to the
channel geometry are introduced: scaled wave amplitude a = 2.iA/L (the ratio of the wave
amplitude to the wavelength, the characteristic slope of the sinusoidal wall) and scaled channel
width h = 27riH/L (the ratio of the channel width to the wavelength).96 By using a stream
function yf(x,y) and taking the curl of the momentum equation (Equation 4-11), a biharmonic
equation for Vf is obtained,