Electrokinetic Flow Between Two Parallel Soft Plates

5 ELECTROKINETIC FLOW BETWEEN TWO PARALLEL SOFT PLATES 

FIGURE 21.4 Schematic representation of a channel of two parallel similar plates 1 and 2 at separation h covered hy ion-penetrable surface charge layers of thickness d under the influence of an electric field E and a pressure gradient P and potential distribution. 

We can write the electric potential (x,y) as the sum of the equilibrium electric potential j/ x) and the potential of the applied field —Ey, namely. 

We assume that charged groups of valence Z are distributed in the surface charge layers at a uniform density N. Since end effects are neglected, the fluid velocity is in the y-direction, depending only on x, and the present system can be considered to be at thermodynamic equilibrium with respect to the x-direction. We define m(x) as the fluid velocity in the y-drrection and Pei(- ) as the volume charge density of electrolyte ions, both independent of y. 

Let us consider the situation where P = 0 and h is inhnitely large, and obtain the liquid velocity at a large distance from plate 1. This velocity is called the electro-osmotic velocity, which we denote by Ugo- By setting P = Q and — oo in Eqs. (21.92) and (21.93) and noting that i/ (x) 0 far from the plate, we find that 

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