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Effective permeability tensor of confined flows with wall grooves of arbitrary shape

  • Mainendra Kumar Dewangan (a1) and Subhra Datta (a1)


Pressure and shear-driven flows of a confined film of fluid overlying a periodic one-dimensional topography of arbitrary shape are considered for prediction of the effective hydraulic permeability in the Stokes flow regime. The other surface confining the fluid may be a planar no-slip wall, an identically patterned wall, a free surface or a surface with prescribed shear. Analytical predictions are obtained using spectral analysis and the domain perturbation method under the assumption of small pattern size to pitch ratio. Using a novel decomposition of the channel height effects into exponentially and algebraically decaying components, a simple surface-metrology-dependent relationship which connects the eigenvalues of the effective permeability tensor is obtained. Two representative topographies are assessed numerically: the infinitely differentiable topography of a phase-modulated sinusoid which has multiple local extrema and zero crossings and the non-differentiable triangular-wave topography. Non-differentiability in the form of corners of triangular patterns and the cusps of scalloped patterns are not found to be an impediment to meaningful and numerically accurate asymptotic predictions of effective permeability and effective slip, contradicting an earlier suggestion from the literature. Several distinct applications of the theory to the friction-reduction and shear-stability performance of the recently developed lubricant impregnated patterned surfaces as well as to scalloped and trapezoidal drag-reduction riblets are discussed, with comparison to experimental data from the literature for the last application. Analytical approximations which have an extended domain of numerical accuracy are also proposed.


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Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Courier Corporation.
Ajdari, A. 2001 Transverse electrokinetic and microfluidic effects in micropatterned channels: lubrication analysis for slab geometries. Phys. Rev. E 65 (1), 016301.
Alamé, K. & Mahesh, K. 2019 Wall-bounded flow over a realistically rough superhydrophobic surface. J. Fluid Mech. 873, 9771019.
Annepu, H., Sarkar, J. & Basu, S. 2014 Pattern formation in soft elastic films cast on periodically corrugated surfaces – a linear stability and finite element analysis. Model. Simul. Mater. Sci. Engng 22 (5), 055003.
Asako, Y. & Faghri, M. 1987 Finite-volume solutions for laminar flow and heat transfer in a corrugated duct. Trans. ASME J. Heat Transfer 109 (3), 627634.
Asmolov, E. S., Belyaev, A. V. & Vinogradova, O. I. 2011 Drag force on a sphere moving toward an anisotropic superhydrophobic plane. Phys. Rev. E 84 (2), 026330.
Asmolov, E. S., Dubov, A. L., Nizkaya, T. V., Kuehne, A. J. C. & Vinogradova, O. I. 2015 Principles of transverse flow fractionation of microparticles in superhydrophobic channels. Lab on a Chip 15 (13), 28352841.
Asmolov, E. S., Nizkaya, T. V. & Vinogradova, O. I. 2018 Enhanced slip properties of lubricant-infused grooves. Phys. Rev. E 98, 033103.
Asmolov, E. S., Schmieschek, S., Harting, J. & Vinogradova, O. I. 2013a Flow past superhydrophobic surfaces with cosine variation in local slip length. Phys. Rev. E 87, 023005.
Asmolov, E. S. & Vinogradova, O. I. 2012 Effective slip boundary conditions for arbitrary one-dimensional surfaces. J. Fluid Mech. 706, 108117.
Asmolov, E. S., Zhou, J., Schmid, F. & Vinogradova, O. I. 2013b Effective slip-length tensor for a flow over weakly slipping stripes. Phys. Rev. E 88, 023004.
Bazant, M. Z. & Vinogradova, O. I. 2008 Tensorial hydrodynamic slip. J. Fluid Mech. 613, 125134.
Bechert, D. W. & Bartenwerfer, M. 1989 The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105129.
Canuto, C., Hussaini, M., Quarteroni, A. & Zang, T. A. J. 1987 Spectral Methods in Fluid Dynamics (Scientific Computation). Springer.
Chakraborty, S. 2007 Towards a generalized representation of surface effects on pressure-driven liquid flow in microchannels. Appl. Phys. Lett. 90 (3), 034108.
Chang, J., Jung, T., Choi, H. & Kim, J. 2019 Predictions of the effective slip length and drag reduction with a lubricated micro-groove surface in a turbulent channel flow. J. Fluid Mech. 874, 797820.
Choi, H., Moin, P. & Kim, J. 1993 Direct numerical simulation of turbulent flow over riblets. J. Fluid Mech. 255, 503539.
Choi, C.-H., Ulmanella, U., Kim, J., Ho, C.-M. & Kim, C.-J. 2006 Effective slip and friction reduction in nanograted superhydrophobic microchannels. Phys. Fluids 18 (8), 087105.
Choudhary, R., Bhakat, T., Singh, R. K., Ghubade, A., Mandal, S., Ghosh, A., Rammohan, A., Sharma, A. & Bhattacharya, S. 2011 Bilayer staggered herringbone micro-mixers with symmetric and asymmetric geometries. Microfluid Nanofluid 10 (2), 271286.
Chu, K.-H. W. 1996 Stokes slip flow between corrugated walls. Z. Angew. Math. Phys. 47 (4), 591599.
Datta, S. & Choudhary, J. N. 2013 Effect of hydrodynamic slippage on electro-osmotic flow in zeta potential patterned nanochannels. Fluid Dyn. Res. 45 (5), 055502.
Devasenathipathy, S., Santiago, J. G. & Takehara, K. 2002 Particle tracking techniques for electrokinetic microchannel flows. Analyt. Chem. 74 (15), 37043713.
Dewangan, M. K. & Datta, S. 2018 Flow through microchannels with topographically patterned wall: a spectral theory for arbitrary groove depths. Eur. J. Mech. (B/Fluids) 70, 7384.
Dewangan, M. K. & Datta, S. 2019 Improved asymptotic predictions for the effective slip over a corrugated topography. Appl. Math. Model. 72, 247258.
Dey, P., Saha, S. K. & Chakraborty, S. 2018 Microgroove geometry dictates slippery hydrodynamics on superhydrophobic substrates. Phys. Fluids 30 (12), 122007.
Dubov, A. L., Nizkaya, T. V., Asmolov, E. S. & Vinogradova, O. I. 2018 Boundary conditions at the gas sectors of superhydrophobic grooves. Phys. Rev. Fluids 3 (1), 014002.
Dunlop, J. 2017 Telecommunications Engineering. Routledge.
Dyke, M. V. 1984 Computer-extended series. Annu. Rev. Fluid Mech. 16 (1), 287309.
Einzel, D., Panzer, P. & Liu, M. 1990 Boundary condition for fluid flow: curved or rough surfaces. Phys. Rev. Lett. 64, 22692272.
Feuillebois, F., Bazant, M. Z. & Vinogradova, O. I. 2009 Effective slip over superhydrophobic surfaces in thin channels. Phys. Rev. Lett. 102, 026001.
Feuillebois, F., Bazant, M. Z. & Vinogradova, O. I. 2010 Transverse flow in thin superhydrophobic channels. Phys. Rev. E 82, 055301.
Gamrat, G., Favre-Marinet, M., Le Person, S., Baviere, R. & Ayela, F. 2008 An experimental study and modelling of roughness effects on laminar flow in microchannels. J. Fluid Mech. 594, 399423.
Ghosal, S. 2002 Lubrication theory for electroosmotic flow in a channel of slowly varying cross-section and wall charge. J. Fluid Mech. 459, 103128.
Ghosh, U. & Chakraborty, S. 2015 Electroosmosis of viscoelastic fluids over charge modulated surfaces in narrow confinements. Phys. Fluids 27 (6), 062004.
Guo, L., Chen, S. & Robbins, M. O. 2016 Effective slip boundary conditions for sinusoidally corrugated surfaces. Phys. Rev. Fluids 1 (7), 074102.
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.
Hocking, L. M. 1976 A moving fluid interface on a rough surface. J. Fluid Mech. 76, 801817.
Imani, B. M. & Elbestawi, M. A. 2001 Geometric simulation of ball-end milling operations. J. Manufacturing Sci. Engng 123 (2), 177184.
Kamrin, K., Bazant, M. Z. & Stone, H. A. 2010 Effective slip boundary conditions for arbitrary periodic surfaces: the surface mobility tensor. J. Fluid Mech. 658, 409437.
Kumar, A., Datta, S. & Kalyanasundaram, D. 2016 Permeability and effective slip in confined flows transverse to wall slippage patterns. Phys. Fluids 28 (8), 082002.
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 5577.
Lecoq, N., Anthore, R., Cichocki, B., Szymczak, P. & Feuillebois, F. 2004 Drag force on a sphere moving towards a corrugated wall. J. Fluid Mech. 513, 247264.
Li, C. & Chen, T. 2005 Simulation and optimization of chaotic micromixer using lattice Boltzmann method. Sensors Actuators B 106 (2), 871877.
Luchini, P. 2013 Linearized no-slip boundary conditions at a rough surface. J. Fluid Mech. 737, 349367.
Luchini, P., Manzo, F. & Pozzi, A. 1991 Resistance of a grooved surface to parallel flow and cross-flow. J. Fluid Mech. 228, 87109.
McCarthy, K. P., Ring, L. & Rana, B. S. 2010 Anatomy of the mitral valve: understanding the mitral valve complex in mitral regurgitation. Eur. J. Echocardiography 11 (10), i3i9.
Miksis, M. J. & Davis, S. H. 1994 Slip over rough and coated surfaces. J. Fluid Mech. 273, 125139.
Mongruel, A., Chastel, T., Asmolov, E. S. & Vinogradova, O. I. 2013 Effective hydrodynamic boundary conditions for microtextured surfaces. Phys. Rev. E 87 (1), 011002.
Nizkaya, T. V., Asmolov, E. S., Zhou, J., Schmid, F. & Vinogradova, O. I. 2015 Flows and mixing in channels with misaligned superhydrophobic walls. Phys. Rev. E 91 (3), 033020.
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), 32513278.
Parmar, V., Kumar, A., Sankar, M. M., Datta, S., Prakash, G. V., Mohanty, S. & Kalyanasundaram, D. 2018 Oxidation facilitated antimicrobial ability of laser micro-textured titanium alloy against gram-positive staphylococcus aureus for biomedical applications. J. Laser Appl. 30, 032001.
Qi, C. & Ng, C.-O. 2015 Electroosmotic flow of a power-law fluid in a slit microchannel with gradually varying channel height and wall potential. Eur. J. Mech. (B/Fluids) 52, 160168.
Rangsten, P., Hedlund, C., Katardjiev, I. V. & Bäcklund, Y. 1998 Etch rates of crystallographic planes in Z-cut quartz-experiments and simulation. J. Micromech. Microengng 8 (1), 1.
Richardson, S. 1973 On the no-slip boundary condition. J. Fluid Mech. 59 (4), 707719.
Sarkar, K. & Prosperetti, A. 1996 Effective boundary conditions for Stokes flow over a rough surface. J. Fluid Mech. 316, 223240.
Schmieschek, S., Belyaev, A. V., Harting, J. & Vinogradova, O. I. 2012 Tensorial slip of superhydrophobic channels. Phys. Rev. E 85, 016324.
Scholle, M., Wierschem, A. & Aksel, N. 2004 Creeping films with vortices over strongly undulated bottoms. Acta Mechanica 168 (3–4), 167193.
Schönfeld, F. & Hardt, S. 2004 Simulation of helical flows in microchannels. AIChE J. 50 (4), 771778.
Sharma, H., Gaddam, A., Agrawal, A. & Joshi, S. S. 2019 Slip flow through microchannels with lubricant-infused bi-dimensional textured surfaces. Microfluid Nanofluid 23 (2), 28.
Sisavath, S., Al-Yaarubi, A., Pain, C. C. & Zimmerman, R. W. 2003 A simple model for deviations from the cubic law for a fracture undergoing dilation or closure. In Thermo-Hydro-Mechanical Coupling in Fractured Rock, pp. 10091022. Springer.
Sloane, N. J. A.2019 The on-line encyclopedia of integer sequences. Available at:
Solomon, B. R., Chen, X., Rapoport, L., Helal, A., McKinley, G. H., Chiang, Y.-M. & Varanasi, K. K. 2018 Enhancing the performance of viscous electrode-based flow batteries using lubricant-impregnated surfaces. ACS Appl. Energy Mater. 1 (8), 36143621.
Song, D., Daniello, R. J. & Rothstein, J. P. 2014 Drag reduction using superhydrophobic sanded Teflon surfaces. Exp. Fluids 55 (8), 1783.
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices. Annu. Rev. Fluid Mech. 36, 381411.
Stroock, A. D., Dertinger, S. K., Whitesides, G. M. & Ajdari, A. 2002 Patterning flows using grooved surfaces. Analyt. Chem. 74 (20), 53065312.
Stroock, A. D. & McGraw, G. J. 2004 Investigation of the staggered herringbone mixer with a simple analytical model. Phil. Trans. R. Soc. Lond. A 362 (1818), 971986.
Sun, R. & Ng, C.-O. 2017 Effective slip for flow through a channel bounded by lubricant-impregnated grooved surfaces. Theor. Comput. Fluid Dyn. 31 (2), 189209.
Tavakol, B., Froehlicher, G., Holmes, D. P. & Stone, H. A. 2017 Extended lubrication theory: improved estimates of flow in channels with variable geometry. Proc. R. Soc. Lond. A 473 (2206), 20170234.
Taylor, G. I. 1971 A model for the boundary condition of a porous material. Part 1. J. Fluid Mech. 49 (2), 319326.
Tuck, E. O. & Kouzoubov, A. 1995 A laminar roughness boundary condition. J. Fluid Mech. 300, 5970.
Van Dyke, M. 1987 Slow variations in continuum mechanics. Adv. Appl. Mech. 25, 145.
Vasudeviah, M. & Balamurugan, K. 1999 Stokes slip flow in a corrugated pipe. Intl J. Engng Sci. 37 (12), 16291641.
Vinogradova, O. I. & Belyaev, A. V. 2011 Wetting, roughness and flow boundary conditions. J. Phys.: Condens. Matter 23 (18), 184104.
Wang, C. Y. 2003 Flow over a surface with parallel grooves. Phys. Fluids 15 (5), 11141121.
Wang, C. Y. 2010 Shear flow over a wavy surface with partial slip. Trans. ASME J. Fluids Engng 132 (8), 084503.
Wang, C. Y. 2011 On Stokes slip flow through a transversely wavy channel. Mech. Res. Commun. 38 (3), 249254.
Ware, C. S., Smith-Palmer, T., Peppou-Chapman, S., Scarratt, L. R. J., Humphries, E. M., Balzer, D. & Neto, C. 2018 Marine antifouling behavior of lubricant-infused nanowrinkled polymeric surfaces. ACS Appl. Mater. Interfaces 10 (4), 41734182.
Watson, G. N. 1995 A Treatise on the Theory of Bessel Functions. Cambridge University Press.
Wexler, J. S., Jacobi, I. & Stone, H. A. 2015 Shear-driven failure of liquid-infused surfaces. Phys. Rev. Lett. 114, 168301.
Whitehouse, D. J. 1994 Handbook of Surface Metrology. CRC Press.
Wierschem, A., Scholle, M. & Aksel, N. 2003 Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers. Phys. Fluids 15 (2), 426435.
Wong, T.-S., Kang, S. H., Tang, S. K. Y., Smythe, E. J., Hatton, B. D., Grinthal, A. & Aizenberg, J. 2011 Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity. Nature 477 (7365), 443447.
Ybert, C., Barentin, C., Cottin-Bizonne, C., Joseph, P. & Bocquet, L. 2007 Achieving large slip with superhydrophobic surfaces: scaling laws for generic geometries. Phys. Fluids 19 (12), 123601.
Yutaka, A., Hiroshi, N. & Faghri, M. 1988 Heat transfer and pressure drop characteristics in a corrugated duct with rounded corners. Intl J. Heat Mass Transfer 31 (6), 12371245.
Zhou, J., Asmolov, E. S., Schmid, F. & Vinogradova, O. I. 2013 Effective slippage on superhydrophobic trapezoidal grooves. J. Chem. Phys. 139 (17), 174708.
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Effective permeability tensor of confined flows with wall grooves of arbitrary shape

  • Mainendra Kumar Dewangan (a1) and Subhra Datta (a1)


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