# Interdisciplinary Applied Mathematics

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du    d 2u

p/— =    + peEx sm(m),    (7.32)

where Ex is the magnitude, and Q is the frequency of the unsteady external electric field E. We note here that due to the straight channel geometry, the cross-stream velocity and electric field components are zero, and equation (7.32) is used to determine the streamwise velocity component due to the time-periodic electric field. The electric charge density pe can be expressed as

Pe = —2n0ez sinh (

kB T

(7.33)

2noez sinh(^*),

Assuming that the EDL thickness is much smaller than the channel-half gap, the electroosmotic potential variation can be written as a function of the distance from the wall (x = v/Ad), using equation (7.10). Utilizing the Deby-Hiickel parameter (w), we can rewrite the electroosmotic body force on the fluid in the following form:

-2noezEx sin Qt

2

и puns a

sin Qt,

(7.34)

where uHS = —^ZEx/p, is the Helmholtz-Smoluchowski velocity. Therefore, equation (7.32) can be written as

du

Pfm

32u

##### ‘‘э? +

We are interested in the solution of the above equation under no-slip and symmetry boundary conditions on the wall and the channel center, respectively. Nondimensionalization of equation (7.35) using characteristic time (1/Q) and length (AD) results in the following equation: