# Interdisciplinary Applied Mathematics

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In the rest of this section we present a brief review of historical developments and recent research on electrokinetic flows. Liquid flows in capillary porous systems under the influence of external electric fields have attracted the attention of many scientists since the discovery of electrokinetic transport (Reuss, 1809). In 1870, Helmholtz developed the electric double layer theory, which relates the electrical and flow parameters for electrokinetic transport. Electroosmosis has been used for chemistry applications

 Parameter Parameter range Typical channel thickness, h (pm) О о 1—1 г со о о Electrolyte concentration, na (mM) 1—1 о г о о о 1—1 Debye length, Xjj (nm) 1 — 100 Zeta potential, ( (mV) ±1 — ±100 Electric field, E (V/mm) 1 — 100 Electroosmotic Velocity, U (mm/s) < 2 Reynolds number, Re 10-4 — 1

TABLE 7.1. Typical physical and electrochemical parameters utilized in elec-trokinetically driven phenomena.

since the late 1930s. Modern theoretical developments include solution of mixed electroosmotic/pressure-driven flows in very thin two-dimensional slits (Burgreen and Nakache, 1964; Ohshima and Kondo, 1990), as well as in thin cylindrical capillaries (Rice and Whitehead, 1965; Lo and Chan, 1994; Keh and Liu, 1995). In 1952, Overbeek proposed irrotationality of internal electroosmotic flows for arbitrarily shaped geometries (Overbeek, 1952). This    was    followed    by    the    ideal    electroosmosis    concept,    i.e.,    elec

troosmotic flow in the absence of externally imposed pressure gradients, which results in the similarity between the electric and velocity fields under specific outer field boundary conditions (Cummings et al., 2000). Also, Santiago has shown that ideal electroosmosis is observed for low Reynolds number steady flows. However, unsteady or high Reynolds number flows violate this condition (Santiago, 2001). The analytical solution of unsteady electroosmotic flows obtained in Section 7.4.2 confirms these predictions.

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