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Effective slip boundary conditions for arbitrary periodic surfaces: the surface mobility tensor

Published online by Cambridge University Press:  16 July 2010

KEN KAMRIN*
Affiliation:
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 01238, USA
MARTIN Z. BAZANT
Affiliation:
Departments of Chemical Engineering and Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
HOWARD A. STONE
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
Email address for correspondence: kkamrin@seas.harvard.edu

Abstract

In a variety of applications, most notably microfluidics design, slip-based boundary conditions have been sought to characterize fluid flow over patterned surfaces. We focus on laminar shear flows over surfaces with periodic height fluctuations and/or fluctuating Navier scalar slip properties. We derive a general formula for the ‘effective slip’, which describes equivalent fluid motion at the mean surface as depicted by the linear velocity profile that arises far from it. We show that the slip and the applied stress are related linearly through a tensorial mobility matrix, and the method of domain perturbation is then used to derive an approximate formula for the mobility law directly in terms of surface properties. The specific accuracy of the approximation is detailed, and the mobility relation is then utilized to address several questions, such as the determination of optimal surface shapes and the effect of random surface fluctuations on fluid slip.

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Papers
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Ajdari, A. 2002 Transverse electrokinetic and microfluidic effects in micropatterned channels: lubrication analysis for slab geometries. Phys. Rev. E 65 (1), 016301.CrossRefGoogle ScholarPubMed
Batchelor, G. K. 1970 Slender-body theory for particles of arbitary cross-section in Stokes flow. J. Fluid Mech. 44, 419440.CrossRefGoogle Scholar
Bazant, M. Z. & Vinogradova, O. I. 2008 Tensorial hydrodynamic slip. J. Fluid Mech. 613, 125134.CrossRefGoogle Scholar
Bechert, D. W. & Bartenwerfer, M. 1989 The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105129.CrossRefGoogle Scholar
Bocquet, L. & Barrat, J. L. 2007 Flow boundary conditions from nano- to micro-scales. Soft Matter 3, 685693.CrossRefGoogle Scholar
Davis, A. M. J. & Lauga, E. 2009 Geometric transition in friction for flow over a bubble mattress. Phys. Fluids 21, 011701.CrossRefGoogle Scholar
Feuillebois, F., Bazant, M. Z. & Vinogradova, O. I. 2009 Effective slip over superhydrophobic surfaces in thin channels. Phys. Rev. Lett. 102, 026001.CrossRefGoogle ScholarPubMed
Higdon, J. J. L. 1985 Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities. J. Fluid Mech. 159, 195226.CrossRefGoogle Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Hocking, L. M. 1976 A moving fluid interface on a rough surface. J. Fluid Mech 76, 801817.CrossRefGoogle Scholar
Jung, Y. & Torquato, S. 2005 Fluid permeabilities of triply periodic minimal surfaces. Phys. Rev. E 72, 056319.CrossRefGoogle ScholarPubMed
Kunert, C. & Harting, J. 2008 Simulation of fluid flow in hydrophobic rough microchannels. Intl J. Comput. Fluid D 27 (7), 475480.CrossRefGoogle Scholar
Lauga, E., Brenner, M. P. & Stone, H. A. 2007 Handbook of Experimental Fluid Dynamics, Ch. 19, pp. 12191240. Springer.Google Scholar
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 5577.CrossRefGoogle Scholar
Luchini, P., Manzo, F. & Pozzi, A. 1991 Resistance of a grooved surface to parallel flow and cross-flow. J. Fluid Mech. 228, 87109.Google Scholar
Miksis, M. J. & Davis, S. H. 1994 Slip over rough and coated surfaces. J. Fluid Mech. 273, 125139.CrossRefGoogle Scholar
Panzer, P., Liu, M. & Einzel, D. 1992 The effects of boundary curvature on hydrodynamic fluid flow: calculation of slip lengths. Intl J. Mod. Phys. B 6 (20), 3251.CrossRefGoogle Scholar
Priezjev, N. V. & Troian, S. M. 2006 Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular-scale simulations versus continuum predictions. J. Fluid Mech. 554, 2546.CrossRefGoogle Scholar
Sarkar, K. & Prosperetti, A. 1996 Effective boundary conditions for Stokes flow over a rough surface. J. Fluid Mech. 316, 223240.CrossRefGoogle Scholar
Sbragaglia, M. & Prosperetti, A. 2007 Effective velocity boundary condition at a mixed slip surface. J. Fluid Mech. 578, 435451.CrossRefGoogle Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices. Annu. Rev. Fluid Mech. 36, 381411.CrossRefGoogle Scholar
Stroock, A. D., Dertinger, S. K. W., Ajdari, A., Mezić, I., Stone, H. A. & Whitesides, G. M. 2002 a Chaotic mixer for microchannels. Science 295, 647651.CrossRefGoogle ScholarPubMed
Stroock, A. D., Dertinger, S. K., Whitesides, G. M. & Ajdari, A. 2002 b Patterning flows using grooved surfaces. Anal. Chem. 74, 53065312.CrossRefGoogle ScholarPubMed
Torquato, S. 2002 Random Heterogeneous Materials. Springer.CrossRefGoogle Scholar
Tuck, E. O. & Kouzoubov, A. 1995 A laminar roughness boundary condition. J. Fluid Mech. 300, 5970.CrossRefGoogle Scholar
Vinogradova, O. I. 1999 Slippage of water over hydrophobic surfaces. Intl J. Miner. Proc. 56, 3160.CrossRefGoogle Scholar
Wang, C.-Y. 1978 Drag due to a striated boundary in slow Couette flow. Phys. Fluids 21 (4), 697698.CrossRefGoogle Scholar
Wang, C.-Y. 1994 The Stokes drag clue to the sliding of a smooth plate over a finned plate. Phys. Fluids 6 (7), 22482252.CrossRefGoogle Scholar
Wang, C.-Y 2003 Flow over a surface with parallel grooves. Phys. Fluids 15, 11141121.CrossRefGoogle Scholar
Wang, C.-Y 2004 Stokes flow through a channel with three-dimensional bumpy walls. Phys. Fluids 16 (6), 21362139.CrossRefGoogle Scholar
Wilkening, J. 2009 Practical error estimates for Reynolds' lubrication approximation and its higher order corrections. SIAM J. Math. Anal. 41, 588630.CrossRefGoogle Scholar
Zhou, H., Martinuzzi, R. J. & Straatman, A. G. 1995 On the validity of the perturbation approach for the flow inside weakly modulated channels. Intl J. Numer. Methods Fluids 39, 11391159.CrossRefGoogle Scholar
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