Electroosmotic flow in a rectangular channel with variable wall zeta ...

614 S. Datta et al. Electrophoresis 2006, 27, 611–6195 Results of computationFigure 2 shows the electroosmotic mobility m eof normalizedby the reference mobility m refeoffor Case A. Thesequantities are defined in Section 2 and may be calculatedusing the formulas presented in Section 3. The monotonicincrease of m eof /m refeof with z 1/z 0 is expected, since a higherzeta-potential corresponds to higher slip velocities. Thesensitivity of the flow to changes in z 1 /z 0 depends on theaspect ratio c/b, decreasing with increasing values of c/b.This is because when c/b is large, the face with the differentz potential, z = z 1 is further removed from the bulk ofthe solution and consequently has a smaller effect on theoverall fluid flux. The numerical results agree very wellwith the theory. This is to be expected, since when thereare no axial variations at all, the asymptotic solutionbased on “slow” axial variations becomes exact.The dimensionless form of the negative pressure gradientp , (defined by Eq. 3) is shown in Fig. 3 for Case B. The p calculated from the asymptotic theory using Eq. (8) is alsoin good overall agreement. Figure 4 shows the correspondingmobility data m eof /m refeofboth for Case B as wellas Case C as a function of the dimensionless parameteraL. The asymptotic theory is expected to be reliable untilabout a , (2p)/(2b), that is, for aL , 2p(L/2b) ^ 120, andindeed this is consistent with Fig. 4.Figure 3. Dimensionless negative pressure gradient (p )as a function of x/2b for Case B. The curves labelled “a”,“b” and “c” are obtained for aL = 115.13, 23.03 and 12.79,respectively. The symbol “o” denotes the numericallycalculated result and the solid line denotes the correspondingresult from asymptotic theory.Figures 5 and 6 show the normalized pressure gradient,p , for the long (l = L) and short (l = L/10) wavelengthcases of Case D with Dz/z 0 = 0.5. The short wavelengthFigure 4. Electroosmotic mobility for Case B, indicatedas curve (a) and for Case C indicated as curve (b) as afunction of aL. The symbol “o” denotes the numericallycalculated result and the solid line denotes the correspondingresult from asymptotic theory.Figure 2. Electroosmotic mobility for Case A as a functionof z 1 /z 0 for various aspect ratios c/b. The curveslabelled “a”, ”b”, “c”, “d” and “e” are obtained for aspectratio c/b = 0.25,0.5,1, 2 and 4, respectively. The symbol“o” denotes the numerically calculated result and thesolid line denotes the corresponding result from asymptotictheory.case corresponds to l = L/10 = 4b, so that the lubricationapproximation is only marginally satisfied. In both caseshowever the induced pressure gradient is very accuratelypredicted. Figures 7 and 8 show the normalized velocityu/u 0 (Panel A) at four different axial locations indicated bythe symbols in Figs. 5 and 6. In the long wavelength case,the agreement with the theory is excellent but in the shortwavelength case, slight discrepancies – of the order of1% – is seen.© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com

Electrophoresis 2006, 27, 611–619 CE and CEC 615If Eq. (6) is used to evaluate c and the computed numericalresults are used for u and p in Eq. (13), then thisquantity, which can be given the physical meaning of“pressure driven part of the flow per unit pressure gradient”is a useful quantity to compare with theory. Indeed,in the lubrication limit, u p is expected to be independent ofeverything but the channel geometry and should be givenby Eq. (5). Panel B in Figs. 7 and 8 compares u p extractedfrom the data [Case D with Dz/z 0 = 0.5] in this way with Eq.(5). In both cases there is collapse of the data onto thecurve, though the agreement is much better in the longwavelength limit (Fig. 7) compared to the short wavelengthcase (Fig. 8). Since the smallness of l/(2b) is ameasure of the accuracy of the lubrication approximation,this is consistent with expectations.Figure 5. Dimensionless negative pressure gradient (p )as a function of x/2b for Case D with l = L and Dz/z 0 = 0.5.The dots indicate numerically calculated results and thesolid line corresponds to the calculation based onasymptotic theory. The points marked *, n, e and udenote the streamwise locations x/2b = 3.3,8.9,11.1,16.7for which velocity profiles are indicated by the correspondingsymbols in Fig. 7.Figure 9 is the same as Panel B in Figs. 7 and 8 exceptthat we vary the amplitude of the sine wave in Case Dwhile keeping the wavelength fixed at l = L/5. The amplitudeDz/z 0 is varied from 0.25 to 4. The quantity u p definedin Eq. (13) is plotted as a function of z/(2b) for y = 0 at twoaxial locations: x/2b = 9.8,10.3. The theoretical curve,Eq. (5) is shown by the solid line. The collapse of the dataonto the theoretical curve is very good, even though theamplitude Dz/z 0 is not small. This is to be expected, sincethe lubrication theory is based on long axial length scales,not small fluctuations in z. The deviation from the lowestorder asymptotic theory however does depend on theamplitude, and generally the error increases with theamplitude of the fluctuations. This is indeed what is seenin Fig. 9.It is easily verified from Eq. (7) that for Case D, the fluctuatingpart of the zeta-potential does not contribute to thevolume flux of fluid. Thus, for Case D, we expect m eof /m refeof= 1. This was indeed observed in all cases run for Case D.Deviations from unity only appeared in the third or fourthdecimal places and consequently could not be distinguishedfrom numerical truncation errors.Figure 6. Same as Fig. 5 except that l = L/10 and correspondingvelocity profiles are in Fig. 8.Equation (4) in Section 3 could be easily inverted (notingthat if p does not depend on y and z then p = p) and writtenin the form:u p ¼ m eE 0c dp 1u(13)4pm dxIn Figs. 10–12 the symbols indicate the computed velocityprofiles for Case E at several axial locations for l/(2b) =1, 1/3 and 1/12. In Panel A the computed solution iscompared with the lubrication theory result, Eq. (5). It isseen that while for l/(2b) = 1 there is reasonably goodagreement, the accuracy of the lubrication solutionrapidly deteriorates with decreasing values of l/(2b). Inparticular, when l/(2b) = 1/12, the lubrication solution isnot even qualitatively correct. For such short wavelengthoscillations in the z potential, the velocity perturbation isconfined to a narrow region next to the wall. Since in theshort wavelength case one would expect that thepresence of the side walls at y = 6b would be irrelevant,one might expect that b ? ? might be a more usefulapproximation than the lubrication limit in describing thissituation. Fortunately, an exact analytical solution isknown for Stokes flow between parallel plates in the infinitelythin EDL limit [10]. In Panel B of Figs. 10–12 thecomputed solution is compared with the solution repre-© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com

616 S. Datta et al. Electrophoresis 2006, 27, 611–619Figure 7. Panel (A) shows thestreamwise velocity profiles u(y= 0, z)/u 0 for Case D with l = Land Dz/z 0 = 0.5 as a function ofz/(2b). Panel (B) shows the correspondingu p (y = 0, z)/(2b) 2 . Thesymbols: *, n, e and h, denotenumerically calculated profilesat the streamwise locationsindicated in Fig. 5. The solid linerepresents the correspondingprofiles obtained from asymptotictheory.Figure 8. Same as Fig. 7 exceptthat l = L/10.sented by Eqs. (9), (27) and (28) of [10]. These equationsgive the stream function for the velocity perturbation for asinusoidal charge distribution in the cases where the sinewaves on the upper and lower plates are (1) in phase and(2) out of phase. The present case of a single chargedplate can be generated by a linear superposition of thesetwo solutions. It is seen that for l/(2b) = 1, the computedsolution agrees better with Ajdari’s parallel plate solutionthan the lubrication theory of Section 3. Further, as l/(2b)is further reduced, the agreement with lubrication theory© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com

Electrophoresis 2006, 27, 611–619 CE and CEC 617Figure 9. u p (0, z)/(2b) 2 on the planes x/2b = 9.8 and x/2b =10.3 for two different values of Dz/z 0 with l = L/5 in CaseD. The hollow and solid symbol of a given shape correspondsto Dz/z 0 = 0.25 and Dz/z 0 = 4, respectively. Theround symbol corresponds to u p (0,z) on the plane x/2b =9.8 and the triangular symbol corresponds to u p (0,z) onthe plane x/2b = 10.3.gets progressively worse while remaining in excellentaccord with Ajdari’s solution. This is exactly the behaviorone would expect, since the two theories are complementaryin the sense that their respective zones ofasymptotic convergence are l (2b) (lubrication theory)and l (2b) (Ajdari’s solution).6 ConclusionNumerical simulations of the incompressible Navier-Stokes equations were performed in a rectangular ductwith spatially varying wall zeta-potential, constant electricfield and zero pressure drop between channel inlet andoutlet. The limit of infinitely thin EDL was assumed so thatthe effect of the wall zeta-potential could be handled in asimple manner through the artifice of the Helmholtz-Smoluchowski “slip boundary conditions”. Since Debyelengths are typically on the order of 10 nm compared to achannel width of 20 — 100 mm this is usually an excellentapproximation in microfluidics. The effect of a spatialvariation of the wall zeta potential on the electroosmoticflow was studied. Exponential and sinusoidal axial variationsin the wall zeta potential was examined in the symmetric(all walls have the same zeta potential at a givenaxial location) as well as nonsymmetric cases.Numerically calculated distributions of induced pressuregradients, flow rates and velocity profiles were comparedwith the corresponding analytical predictions based onlubrication theory with the objective of determining thelimits of validity of the lubrication approximation. It wasfound that even though lubrication theory has formalvalidity only when the characteristic length scale for axialvariations is much larger than the channel width, the predictedresults were accurate to within a few percent evenwhen the characteristic length scale was of the sameorder as the channel width. When fluctuations on aFigure 10. Panel (A) shows thestreamwise velocity profiles u(y= 0, z) for Case E with l/(2b) = 1,c/b = 0.2 and Dz/z 0 = 0.1 as afunction of z/(2b) compared withthe theoretical values calculatedusing Eq. (4). Panel (B) showsu(y = 0,z) compared with thetheoretical results presented byAjdari [10]. The symbols: *, n, eand h denote numericallycalculated profiles at thestreamwise locations x/2b =0.1,0.2,0.6,0.7. The solid linerepresents the correspondingprofiles obtained from the relevanttheory.© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com

618 S. Datta et al. Electrophoresis 2006, 27, 611–619Figure 11. Same as Fig. 10except that l/(2b) = 1/3.Figure 12. Same as Fig. 10except that l/(2b) = 1/12.scale smaller than the channel width is considered, lubricationtheory fails. This is also consistent with earlierstudies of cylindrical capillaries [2]. The observed accuracyof lubrication theory, even when the formal conditionsfor its validity are only marginally satisfied, is veryencouraging because it suggests that lubrication theorycould be used for obtaining analytical solutions for a widevariety of problems in microfluidics.© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com

Electrophoresis 2006, 27, 611–619 CE and CEC 619This “better than expected” performance however is neithersurprising nor rare among asymptotic theories. In anasymptotic approximation, one only knows how the errorscales with the expansion parameter but not the magnitudeof the error itself [12]. Examples of asymptotic approximationsexist on both sides of the spectrum: thosethat are accurate beyond all expectations as well as somethat are so inaccurate as to be essentially useless forpractical purposes.SG was supported by the NSF under grant CTS-0330604.NAP was supported by the NSF through the CAREERaward (CTS-0134546)7 References[1] Batchelor, G. B., An Introduction to Fluid Dynamics, CambridgeUniversity Press, New York 1973, pp. 219–222.[2] Ghosal, S., J. Fluid Mech. 2002, 459, 103–128.[3] Ghosal, S., Anal. Chem. 2002, 74, 71–775.[4] Ghosal, S., Electrophoresis 2004, 25, 214–228.[5] Long, D., Stone, H. A., Ajdari, A., J. Coll. Int. Sci., 1999, 212,228–349.[6] Morrison, F. A. Jr, J. Coll. Int. Sei. 1970, 34, 45–54.[7] Patankar, S.V., Numerical Heat Transfer and Fluid Flow,Hemisphere Publishing Corporation, Washington 1980,p. 126.[8] Patankar, N. A., Hu, H. H., Anal. Chem. 1998, 70, 1870–1881.[9] Stone, H. A., Stroock, D. A., Ajdari, A., Ann. Rev. Fluid Mech.2004, 36, 381–411.[10] Ajdari, A., Phys. Rev. E, 1996, 53, 4996–5005.[11] Versteeg, H. K., Malalasekera, W., An Introduction to ComputationalFluid Dynamics: The Finite Volume Method,Prentice Hall, Harlow 1995, pp. 203–205.[12] Van Dyke, M., Perturbation Methods in Fluid Mechanics, TheParabolic Press, Stanford 1975, pp. 30–32.© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com