doi:10.1016/j.colsurfb.2006.08.012 
Copyright © 2006 Elsevier B.V. All rights reserved.
Electrokinetic flow in an elliptic microchannel covered by ion-penetrable membrane
Jyh-Ping Hsua, 
, 
, Ying-Lun Wenga, 1, Duu-Jong Leea, 1, Shiojenn Tsengb, Ay Suc and Chur-Jen Chend
aDepartment of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan
bDepartment of Mathematics, Tamkang University, Tamsui, Taipei 25137, Taiwan
cDepartment of Mechanical Engineering & Fuel Cells Research Center, Yuan Ze University, 135 Yuan-Tung Road, Chung Li, Tao Yuan 320, Taiwan
dDepartment of Mathematics, Tunghai University, Taichung 403, Taiwan
Received 6 June 2006;
revised 22 July 2006;
accepted 4 August 2006.
Available online 22 August 2006.
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Abstract
The electrokinetic flow of an electrolyte solution in an elliptical microchannel covered by an ion-penetrable, charged membrane layer is examined theoretically. The present analysis extends previous results in that a two-dimensional problem is considered, and the system under consideration simulates the flow of a fluid, for example, in a microchannel of biological nature such as vein. The electroosmostic volumetric flow rate, the total electric current, the streaming potential, and the electroviscous effect of the system under consideration are evaluated. We show that, for a constant hydraulic diameter, the variations of these quantities as a function of the aspect ratio of a microchannel may have a local minimum or a local maximum at a medium level of ionic strength, which depends on the thickness of the membrane layer. For a constant cross-sectional area, the electroosmostic volumetric flow rate, the total electric current, and the streaming potential increase monotonically with the increase in the aspect ratio, but the reverse is true for the electroviscous effect.
Keywords: Electrokinetic flow; Elliptic microchannel; Ion-penetrable membrane; Streaming current; Electroosmotic velocity
Fig. 1. Schematic representation of the system under consideration (a) and computational domain (b).
Fig. 2. Simulated spatial variation of scaled potential at I = 10−4 M, P = 107 N/m3, Dh = 10−6 m, and Ez = 0 V/m. (a) H/W = 1 and d = Dh/10, (b) H/W = 1 and d = Dh/40, and (c) H/W = 1/4 and d = Dh/10.
Fig. 3. Simulated spatial variation of scaled velocity U for the case of Fig. 2.
Fig. 4. Variation of semi-minor axis H (a) and cross-sectional area (b) as a function of aspect ratio for the case when Dh = 10−6 m.
Fig. 5. Variation of total current It as a function of aspect ratio (H/W) for various ionic strength I and membrane thickness d at Dh = 10−6 m, Ez = 1000 V/m, and P = 0 N/m3. (a) I = 10−3 M, (b) I = 10−4 M and (c) I = 10−5 M. Curve 1, d = Dh/40; 2, d = Dh/20; 3, d = Dh/10; 4, d = Dh/4.
Fig. 6. Variation of total volumetric flow rate current Vt as a function of aspect ratio (H/W) at various I and d for the case of Fig. 5.
Fig. 7. Variation of streaming potential Est as a function of aspect ratio (H/W) for various ionic strength I and membrane thickness d at Dh = 10−6 m and P = 107 N/m3. (a) I = 10−3 M, (b) I = 10−4 M and (c) I = 10−5 M. Curve 1, d = Dh/40; 2, d = Dh/20; 3, d = Dh/10.
Fig. 8. Variation of streaming potential Rvis as a function of aspect ratio (H/W) at various I and d for the case of Fig. 7.
Fig. 9. Variation of electroviscous parameter Rvis as a function of electrokinetic diameter De for various aspect ratio (H/W) at Dh = 10−6 m, d = Dh/10, and P = 107 N/m3. Curve 1, H/W = 1; 2, H/W = 1/4; 3, H/W = 1/20.
Fig. 10. Variation of (Vt/It)P=0 as a function of (|Est|/P)It=0 at Dh = 10−6 m, d = Dh/10, and I = 10−3 M. Discrete symbols, from right to left, represent the values for H/W = 1, 1/4, 1/8, 1/12, 1/16, 1/20.
Fig. 11. Variation of semi-minor axis H as a function of aspect ratio (H/W) for the case when the cross-sectional area is fixed at 7.85 × 10−13 m2.
Fig. 12. Variation of total current It as a function of aspect ratio (H/W) at various membrane thickness dc and ionic strength I for the case when cross-sectional area is fixed, Ez = 1000 V/m, P = 0 N/m3, and Dhc = 10−6 m. (a) I = 10−3 M, (b) I = 10−4 M and (c) I = 10−5 M. Curve 1, dc = Dhc/50; 2, dc = Dhc/40; 3, dc = Dhc/20.
Fig. 13. Variation of total volumetric flow rate Vt as a function of aspect ratio (H/W) at various dc and I for the case of Fig. 12.
Fig. 14. Variation of streaming potential Est as a function of aspect ratio (H/W) at various membrane thickness dc and ionic strength I for the case when cross-sectional area is fixed and P = 107 N/m3, Dhc = 10−6 m. (a) I = 10−3 M, (b) I = 10−4 M and (c) I = 10−5 M. Curve 1, dc = Dhc/50; 2, dc = Dhc/40; 3, dc = Dhc/20.
Fig. 15. Variation of electroviscous parameter Rvis as a function of aspect ratio (H/W) at various membrane thickness dc and ionic strength I for the case of Fig. 14.
Fig. 16. Variation of total volumetric flow rate Vt (a) and total current It (b) as a function of aspect ratio (H/W) at various membrane thickness dc for the case when cross-sectional area is fixed, Ez = 0 V/m, P = 107 N/m3, Dhc = 10−6 m, and I = 10−3 M. Curve 1, dc = Dhc/50; 2, dc = Dhc/40; 3, dc = Dhc/20.
Fig. 17. Variation of total volumetric flow rate Vt as a function of aspect ratio (H/W) at various membrane thickness dc for the case when cross-sectional area is fixed, Ez = Est, P = 107 N/m3, Dhc = 10−6 m, and I = 10−3 M. Curve 1, dc = Dhc/50; 2, dc = Dhc/40; 3, dc = Dhc/20.
Table 1.
Variation of |
w| at various combinations of ionic strength I and De (=κDh) for the case when Dh = 10−6 m, Ez = 1000 V/m, P = 0 N/m3, d = Dh/10, and H/W = 1
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