Oscillatory Flow Through a Channel With Stick-Slip Walls: Complex Navier s Slip Length

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1 Oscillatory Flow Troug a Cannel Wit Stick-Slip Walls: Complex Navier s Slip Lengt Ciu-On Ng cong@ku.k Department of ecanical Engineering, Te University of Hong Kong, Pokfulam Road, Hong Kong C. Y. Wang Department of atematics, icigan State University, East Lansing, I Effective slip lengts for pressure-driven oscillatory flow troug a parallel-plate cannel wit boundary slip are deduced using a semi-analytic metod of eigenfunction expansions and point matcing. Te cannel walls are eac a superydropobic surface micropatterned wit no-sear alternating wit no-slip stripes, wic are aligned eiter parallel or normal to te flow. Te slip lengts are complex quantities tat are functions of te oscillation frequency, te cannel eigt, and te no-sear area fraction of te wall. Te dependence of te complex nature of te slip lengt on te oscillation frequency is investigated in particular. DOI:.5/.4329 Keywords: oscillatory microcannel flow, superydropobic surface, effective slip lengt Introduction In a classical paper, Pilip presented several basic solutions of te steady Navier Stokes equation subject to mixed no-slip and no-sear boundary conditions. Of particular interest are te solutions for sear flow over a plate wit a regular array of longitudinal or transverse no-sear slots. In te case of longitudinal slots, Pilip presented an expression for te flow velocity at large distances from te plate. In recent years, tis expression by Pilip as been extensively quoted by te microfluidics community, as it is also an expression for te so-called effective or macroscopic slip lengt for sear flow over a superydropobic surface made up of no-slip alternating wit no-sear stripes e.g., Cottin- Bizonne et al. 2 and Ng and Wang 3. By Navier s 4 slip condition, te slip velocity on a boundary is proportional to te near-boundary sear stress. For a Newtonian fluid, te constant of proportionality in te slip condition relating slip velocity and boundary velocity gradient is known as te slip lengt. Slip lengt can be interpreted as te dept into te boundary at wic te velocity profile would extrapolate to zero. Slip lengts tat ave been acieved in practice are of te order of m, and boundary slip is important to microfluidic applications. Superydropobicity refers to a state at wic a liquid does not wet a surface. A common type of superydropobic surface is a surface micropatterned wit grooves and ribs. Tese grooves are so small in widt tat, owing to te ydropobicity of te solid pase and for a pressure below te capillary pressure, te liquid Corresponding autor. Contributed by te Fluids Engineering Division of ASE for publication in te JOURNAL OF FLUIDS ENGINEERING. anuscript received August 28, 2; final manuscript received December 8, 2; publised online January 3, 2. Assoc. Editor: Neeles A. Patankar. flowing over te surface cannot penetrate te cavity between adjacent ribs, wic is ten filled wit gas or vapor. Te liquid flow encounters eterogeneous conditions on te surface: zero slip on te liquid-solid interface and little sear as gas is muc less viscous tan liquid on te liquid-gas interface. If te liquid-gas interface is idealized to be flat by ignoring te meniscus curvature, and to be perfect slip by ignoring te gas viscosity, te surface amounts to a flat plate wit alternate stick-slip slots, te one studied by Pilip. Hence, under tese idealized conditions, Pilip s formula can be applied to determine te effective slip lengts for longitudinal and transverse stripes P =2 P = 2L ln sec a 2 were L is alf te periodic lengt of te pattern, and a is te no-sear area fraction of te surface. Pilip s formula is, in principle, valid only for steady flow. It remains an unanswered question weter or not it still works wen te flow becomes time-varying or oscillatory. Tere exist a number of studies tat ave examined microcannel flows driven by time-oscillating forcings under a slip condition e.g., Yang and Kwok 5, Kaled and Vafai 6, Wuetal.7, and attews and Hill 8. Despite te flow being oscillatory, Navier s slip lengt is assumed by many autors to be a real quantity. Tis assumption is subject to scrutiny, owever. It is possible tat, for oscillatory flow, te slip velocity is different in pase from te velocity gradient near te boundary, as can be inferred from Stokes second problem. Hence, te slip lengt needs to be a complex quantity in order to account for suc a pase difference. Te possibility of a complex Navier s slip lengt in oscillating flow, wic as been pointed out by Willmott and Tallon 9, is yet to be understood in detail. Te objective of tis note is to look into te complex nature of te effective slippage arising from pressure-driven oscillatory flow troug a parallel-plate cannel wit stick-slip walls. Te walls are eac made up of a periodic array of alternate no-slip and no-sear stripes, wic are aligned eiter parallel or normal to te flow. Te problems are solved by a semi-analytic metod using eigenfunction expansions and point matcing. Expressions for te slip lengts, wic are complex in general, are deduced as functions of te cannel eigt, te no-sear area fraction of te wall, and an oscillation parameter. We sall examine, from te calculated results, under wat condition te slip lengt will materially deviate from Pilip s formula for steady flow. We sall also look into te fact tat te boundary slip will lose its effect on te flow as te frequency increases. Te condition under wic te flow enancement due to boundary slip may become insignificant is presented. 2 Formulation Consider fully-developed oscillatory viscous flow driven by a time-periodic axial pressure gradient K cost troug a parallelplate cannel, were K is te magnitude of te pressure gradient, is te angular frequency, and t is time. Eac of te cannel wall is patterned wit periodic no-sear alternating wit no-slip stripes, were te pattern periodic lengt is 2L and te widt of a no-sear stripe is 2aL. Te no-sear area fraction of te wall is terefore given by a. Te cannel walls, wic are at a distance 2L apart, ave teir patterns aligned in pase so tat te flow is symmetrical about te centerline of te cannel. In Fig., te coordinates x,y,z and te corresponding velocity components u,v,w are defined. In tis work, low-reynolds-number flow is considered and te flow inertia is ignored. 2. Longitudinal Flow. We first consider flow being in a direction parallel to te stripes i.e., purely in te z-direction. Te momentum equation reads as follows: Journal of Fluids Engineering Copyrigt 2 by ASE JANUARY 2, Vol. 33 / 452- Downloaded 3 Jan 2 to Redistribution subject to ASE license or copyrigt; see ttp://

2 no-sear no-slip no-sear no-slip A sin + A n n tan n cos n x = n= x a transverse flow longitudinal flow y z + 2 w 2 w t = p x w 2 were and are te density and te dynamic viscosity of te fluid, respectively. Let us introduce p z = K Reeit w =ReWe it 3 were Re denotes te real part, i is te complex unit, and W is a complex function of x,y. We furter introduce normalized variables: W=KL 2 /Ŵ and x,y=lxˆ,ŷ. On dropping te carets, te nondimensional momentum equation can be written as 2 W x W 2 2 W = 4 were 2 =il 2 / or = +i 5 in wic =/ is te kinematic viscosity of te fluid, and = L/ 2/ 6 One sould note tat 2/ is te tickness of te Stokes layer, wic is te boundary layer near te wall due to te oscillatory flow. Terefore, is te ratio of alf te pattern lengt to te Stokes layer tickness; it is small/large wen te oscillation frequency is low/ig. By geometry, W is even in bot x and y, and te solution to Eq. 4 can be written as Wx,y = 2 + A cos + A n cos n x cos ny 7 n= cos n were A and A n are unknown coefficients, n = n and n = 2 n Te solution is to satisfy te following mixed boundary conditions on te wall y=: free slip W/= for xa and zero slip W= for ax. On truncating A n to terms, tese conditions are a x= z x y= v y=- w u Fig. Oscillatory flow troug a parallel-plate cannel wit walls tat are patterned wit a periodic array of longitudinal or transverse no-sear stripes. Te patterns of te two walls are arranged in-pase wit eac oter so tat te flow is symmetrical about te centerline of te cannel. Te x- and z-axes are normal and parallel to te stripes, respectively, wile te y-axis is perpendicular to te cannel walls. Te lengt dimensions are normalized wit respect to alf te period of te wall pattern. 2 + A cos + A n cos n x = a x n= Te coefficients A and A,, are found by te metod of point collocation. Te conditions above are to be imposed at + equidistant discrete points on x, tereby forming a system of + equations for te unknowns. In tis work, we used te FOR- TRAN subroutine ISL-DLSARG ig-precision solver to solve te system of equations, were =4 was considered. Te effective slip lengt is found by relating te mean velocity to tat of a macroscopically equivalent one-dimensional problem. Te section-mean velocity is w =ReW expit, were by Eq. 7, W = Wdxdy = 2 + A sin In an equivalent problem, we consider one-dimensional oscillatory flow w e =ReW e expit 2 troug a cannel wit a constant wall slip lengt. Te boundary conditions are W e = W e at y=. Te section-mean velocity, for tis equivalent problem, can be readily found to be W e = 2 sin 3 cos + sin On relating Eq. to Eq. 3, we can deduce te effective slip lengt for te longitudinal flow as = sin wic is complex in nature. 2 A + cos Transverse Flow. We next consider flow tat is in a direction normal to te stripes. Te momentum and continuity equations for te two-dimensional flow are 2 u t = p x + 2 u x u 5 v t = p + 2 v 2 x v 6 u x + v = 7 We first introduce p =ReKx Pe it u,v =ReU,Ve it 8 were K is a real constant equal to te applied pressure gradient, and U, V, P are complex functions of x,y. We furter introduce te normalized variables: U,V=KL 2 /Û,Vˆ, x,y =Lxˆ,ŷ, and P=KLPˆ. By substituting tese variables into Eqs. 5 7, tese equations become 2 U x U 2 2 U = P x 2 V x V 2 2 V = P / Vol. 33, JANUARY 2 Transactions of te ASE Downloaded 3 Jan 2 to Redistribution subject to ASE license or copyrigt; see ttp://

3 .4 (a (b.2 Re(δ.8.6 a =.9 Im(δ a = (c (d W _ a =.9 W 2 φ a = Fig. 2 For longitudinal flow, a te real part of te slip lengt, b te imaginary part of te slip lengt, c te magnitude of te mean velocity W relative to tat witout wall slip W, and d te pase of te mean velocity are plotted as functions of te oscillation parameter, and te no-sear area fraction of te wall a, were te cannel eigt =2. In a, te dotted lines are te limiting steady-state values of te slip lengt P given by Eq.. In d, te pase is in degrees, and te dased line is for te case witout wall slip or a=. U x + V = 2 were te carets ave been dropped, and is given in Eq. 5. By geometry, U is even in bot x and y, wile V is odd in bot x and y. Te general solution to Eqs. 9 2 tat satisfies V = at y= can be written as Ux,y = 2 + B cos cos + n x B n n y n= cos n cos n sin n n sin n ny cos 22 Vx,y = n= B n sin n x cos n sin n y sin n sin n sin ny 23 Px,y = 2 Bn sin n x n= n cos n cos ny 24 were B and B n are unknown coefficients and n and n are given in Eq. 8. Te solution is to furter satisfy te mixed boundary conditions on te wall y=: free slip U/= for xa and zero slip U= for ax. On truncating B n to terms, tese conditions are B sin 2 Bn tan n cos n x = n= n 2 + B cos + n= a x x a B n n tan n n tan n cos n x = As in te longitudinal flow problem, te coefficients B and B,, are found by te metod of point collocation. Te conditions are to be imposed at + equidistant discrete points on x, tereby forming a system of + equations for te unknowns. Again, te ISL-DLSARG ig-precision solver was used to solve te system of equations, were =4 was considered. Te section-mean streamwise velocity is ū=reū expit, were by Eq. 22, Journal of Fluids Engineering JANUARY 2, Vol. 33 / Downloaded 3 Jan 2 to Redistribution subject to ASE license or copyrigt; see ttp://

4 .75 (a a =.9 -. (b Re(δ Im(δ a =.9.4 (c -6 (d -65 W _ a =.9 W.2.7 φ a =.9 Fig. 3 For longitudinal flow, a te real part of te slip lengt, b te imaginary part of te slip lengt, c te magnitude of te mean velocity W relative to tat witout wall slip W, and d te pase of te mean velocity are plotted as functions of te cannel eigt, and te no-sear area fraction of te wall a, were te oscillation parameter =5. In d, te pase is in degrees, and te dased line is for te case witout wall slip or a=. Ū = Udy = 2 + B sin 27 Hence, te effective slip lengt for transverse flow can be obtained, on relating te mean velocity to tat of te macroscopically equivalent flow given in Eq. 3, as follows: = sin wic is also complex in nature. 2 B + cos 28 3 Results Te two slip lengts, given in Eqs. 4 and 28, are functions of te cannel eigt, te no-sear area fraction of te wall a, and te oscillation parameter. As noted earlier, te parameter is te ratio of alf te wall pattern lengt to te Stokes layer tickness. Te frequency of oscillatory flow in microcannels can be in te order of Hz or even iger 2. For suc an order of frequency, te Stokes boundary layer will ave a tickness of te order of micrometer, wic is comparable to te lengt scale of te micropatterns. In tis work, is considered. Figure 2 sows results as functions of for te case of longitudinal flow, were =2 and a=.9,.7,,. Te cannel eigt, =2, is one wic is large enoug for te cannel confinement to be negligible, and te effective slip lengt is essentially equal to tat for flow over a single flat plate. Te real and imaginary parts of are sown in Figs. 2a and 2b, respectively. We ave cecked tat in te limit of steady flow i.e., zero frequency, Re will tend to Pilip s slip lengt P given in Eq., wile Im will tend to zero. Te figure reveals tat, for a sufficiently low frequency suc tat, Re does not deviate muc from P, and Im remains muc smaller in magnitude tan Re. Hence, as long as, it is safe to assume tat te effective slip lengt is real and approximately equal to te one for steady flow. Wen, Re decreases materially, wile Im increases in magnitude. Te real and imaginary parts become comparable wit eac oter in magnitude as te frequency increases. Hence, for sufficiently fast oscillation, te slip lengt can no longer be treated as a real quantity. Te boundary slip reduces te wall drag and terefore enances te flow rate or te tidal flux in te case of oscillatory flow. Figures 2c and 2d sow te magnitude ratio and te pase of te mean velocity, wic can be expressed as W =W expi, as functions of. Te magnitude ratio W /W, were W is te mean velocity wen te boundary slip is zero, can be regarded as an enancement factor. One can see tat, for a given a, te enancement factor is te maximum wen te flow is steady, =. Te enancement drops significantly as te frequency increases. For example, for a=.9, te enancement will drop below % wen 2.5. Tis penomenon can be readily understood as follows. Wen te oscillation is so fast tat te Stokes boundary layer becomes muc tinner tan te cannel eigt, te flow in / Vol. 33, JANUARY 2 Transactions of te ASE Downloaded 3 Jan 2 to Redistribution subject to ASE license or copyrigt; see ttp://

5 .6 (a a =.9 (b Re(δ.4 Im(δ a = (c (d - -2 U _ a =.9 U φ a = Fig. 4 For transverse flow, a te real part of te slip lengt, b te imaginary part of te slip lengt, c te magnitude of te mean velocity Ū relative to tat witout wall slip Ū, and d te pase of te mean velocity are plotted as functions of te oscillation parameter, and te no-sear area fraction of te wall a, were te cannel eigt =2. In a, te dotted lines are te limiting steady-state values of te slip lengt P given by Eq.. In d, te pase is in degrees, and te dased line is for te case witout wall slip or a=. te core of te cannel will be a plug flow tat is independent of te boundary condition, weter stick or slip. Hence, in te limit of very fast oscillation, te boundary condition will ave no effect on te mean velocity. Te present results suggest tat te flow enancement due to boundary slip will become practically insignificant wen 5, were is te ratio of alf te cannel eigt to te Stokes layer tickness. Te tendency to approac plug flow can be reflected from te pase lag between te mean velocity and te forcing. Tis is because te pase as, but 9 deg as. Figure 2d sows tat for a larger a, te flow can reac te limiting plug flow at smaller. Tis is expected since flow in te no-sear region of te cannel is by nature a plug flow. Hence, a larger no-sear area fraction of te wall will make te mean flow come closer to a plug flow at te same frequency. Results of longitudinal flow as functions of te cannel eigt are sown in Fig. 3, were =5. As long as is not too small, te slip lengt is nearly a constant, independent of te cannel eigt. Te cannel confinement will lose its effect on te slip lengt wen. On te oter and, te enancement in te mean flow due to boundary slip is appreciable only for sufficiently tin cannels, in tis case. Hence, consistent wit te observation above, te flow enancement becomes insignificant wen te frequency is so ig tat 5. As can be inferred from te pase, te mean flow is already nearly a plug flow for a cannel eigt as small as wen a=.9. In contrast, in te absence of wall slip te dases in Fig. 3d, te flow does not come close to a plug flow until te cannel eigt is large. Figure 4 sows results as functions of for te case of transverse flow, were =2 and a=.9,.7,,. Te curves ere display qualitatively te same beaviors as tose sown in Fig. 2. Terefore, te observations tat we ave made above for longitudinal flow can be applied to transverse flow as well. Namely, te effective slip lengt is approximately equal to te limiting value for steady flow as long as. Also, te enancement in te mean flow due to boundary slip will diminis substantially wen 5. 4 Concluding Remarks In tis note, we ave seen ow te oscillation frequency may turn te effective slip lengt into a complex quantity for te problem of oscillatory flow troug a cannel bounded by plates patterned wit no-sear and no-slip stripes. Te oscillation frequency ere is represented by a dimensionless parameter, wic is te ratio of alf te wall pattern lengt to te Stokes layer tickness. We ave examined cases for a.9, were a is te no-sear area fraction of te wall. Te cannel eigt is found to ave little effect on te effective slip lengts wen. Two conditions ave been inferred from te results: i wen, te effective slip lengt is practically te same as te one for steady flow, tereby a real quantity; ii wen 5, te flow in te cannel core is essentially a plug flow unaffected by te boundary slip. Journal of Fluids Engineering JANUARY 2, Vol. 33 / Downloaded 3 Jan 2 to Redistribution subject to ASE license or copyrigt; see ttp://

6 Conversely speaking, under te conditions tat and 5, te effective wall slip needs to be taken as a complex quantity, wile its effect on te flow is appreciable. By a stocastic analysis, Cakraborty 3,4 presented a generalized model for assessing te competing aspects of te stickslip influences of random surface rougness on te frictional caracteristics of steady pressure-driven flow in microcannels. It is wort extending teir stocastic model to time-varying flow in order to examine te effects of flow oscillation on te effective slip in a more general formalism. Oter penomena tat ave been recently reported in te context of steady microflow, e.g., te generation of nanoscale vortices in cannels wit patterned substrates 5, or flow troug a periodically grooved tube 6, are also wort re-examining for time-oscillating flow. Acknowledgment Te work was supported by te Researc Grants Council of te Hong Kong Special Administrative Region, Cina troug Project No. HKU 7569E and No. HKU 755E and also by te University of Hong Kong troug te Seed Funding Programme for Basic Researc under Project Code References Pilip, J. R., 972, Flows Satisfying ixed No-Slip and No-Sear Conditions, Z. Angew. at. Pys., 23, pp Cottin-Bizonne, C., Barentin, C., Carlaix, E., Bocquet, L., and Barrat, J.-L., 24, Dynamics of Simple Liquids at Heterogeneous Surface: olecular- Dynamics Simulations and Hydrodynamic Description, Eur. Pys. J. E, 5, pp Ng, C. O., and Wang, C. Y., 29, Stokes Sear Flow Over a Grating: Implications for Superydropobic Slip, Pys. Fluids, 2, pp Navier, C. L.. H., 823, émoire sur les Lois du ouvement des Fluides, emoires de l Academie Royale des Sciences de l Institut de France, VI, pp Yang, J., and Kwok, D. Y., 23, Effect of Liquid Slip in Electrokinetic Parallel-Plate icrocannel Flow, J. Colloid Interface Sci., 26, pp Kaled, A.-R. A., and Vafai, K., 24, Te Effect of te Slip Condition on Stokes and Couette Flows due to an Oscillating Wall: Exact Solutions, Int. J. Non-Linear ec., 39, pp Wu, Y. H., Wiwatanapatapee, B., and Hu,., 28, Pressure-Driven Transient Flows of Newtonian Fluids Troug icrotubes Wit Slip Boundary, Pysica A, 387, pp attews,. T., and Hill, J.., 29, On Tree Simple Experiments to Determine Slip Lengts, icrofluid. Nanofluid., 6, pp Willmott, G. R., and Tallon, J. L., 27, easurement of Newtonian Fluid Slip Using a Torsional Ultrasonic Oscillator, Pys. Rev. E, 76, pp Green, N. G., Ramos, A., Gonzalez, A., organ, H., and Castellanos, A., 2, Fluid Flow Induced by Nonuniform AC Electric Fields in Electrolytes on icroelectrodes. I. Experimental easurements, Pys. Rev. E, 64, pp arcos, C. Y., Wong, T. N., and Ooi, K. T., 24, Dynamic Aspects of Electroosmotic Flow in Rectangular icrocannels, Int. J. Eng. Sci., 42, pp Huang, H. F., and Lai, C. L., 26, Enancement of ass Transport and Separation of Species by Oscillatory Electroosmotic Flows, Proc. R. Soc. London, Ser. A, 462, pp Cakraborty, S., 27, Towards a Generalized Representation of Surface Effects on Pressure-Driven Liquid Flow in icrocannels, Appl. Pys. Lett., 9, pp Cakraborty, S., Das, T., and Cattoraj, S., 27, A Generalized odel for Probing Frictional Caracteristics of Pressure-Driven Liquid icroflows, J. Appl. Pys., 2, pp Karakare, S., Kar, A., Kumar, A., and Cakraborty, S., 2, Patterning Nanoscale Flow Vortices in Nanocannels Wit Patterned Substrates, Pys. Rev. E, 8, pp Ng, C. O., and Wang, C. Y., 2, Stokes Flow Troug a Periodically Grooved Tube, ASE J. Fluids Eng., 32, pp / Vol. 33, JANUARY 2 Transactions of te ASE Downloaded 3 Jan 2 to Redistribution subject to ASE license or copyrigt; see ttp://