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Linear instability of lid- and pressure-driven flows in channels textured with longitudinal superhydrophobic grooves

Published online by Cambridge University Press:  02 December 2021

Samuel D. Tomlinson Open the ORCID record for Samuel D. Tomlinson [Opens in a new window]  and
Demetrios T. Papageorgiou Open the ORCID record for Demetrios T. Papageorgiou [Opens in a new window]
Samuel D. Tomlinson*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Demetrios T. Papageorgiou
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

It is known that an increased flow rate can be achieved in channel flows when smooth walls are replaced by superhydrophobic surfaces. This reduces friction and increases the flux for a given driving force. Applications include thermal management in microelectronics, where a competition between convective and conductive resistance must be accounted for in order to evaluate any advantages of these surfaces. Of particular interest is the hydrodynamic stability of the underlying basic flows, something that has been largely overlooked in the literature, but is of key relevance to applications that typically base design on steady states or apparent-slip models that approximate them. We consider the global stability problem in the case where the longitudinal grooves are periodic in the spanwise direction. The flow is driven along the grooves by either the motion of a smooth upper lid or a constant pressure gradient. In the case of smooth walls, the former problem (plane Couette flow) is linearly stable at all Reynolds numbers whereas the latter (plane Poiseuille flow) becomes unstable above a relatively large Reynolds number. When grooves are present our work shows that additional instabilities arise in both cases, with critical Reynolds numbers small enough to be achievable in applications. Generally, for lid-driven flows one unstable mode is found that becomes neutral as the Reynolds number increases, indicating that the flows are inviscidly stable. For pressure-driven flows, two modes can coexist and exchange stability depending on the channel height and slip fraction. The first mode remains unstable as the Reynolds number increases and corresponds to an unstable mode of the two-dimensional Rayleigh equation, while the second mode becomes neutrally stable at infinite Reynolds numbers. Comparisons of critical Reynolds numbers with the experimental observations for pressure-driven flows of Daniello et al. (Phys. Fluids, vol. 21, issue 8, 2009, p. 085103) are encouraging.

JFM classification

Mathematical Foundations: Computational methods Micro-/Nano-fluid dynamics: Microfluidics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 932 , 10 February 2022 , A12
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Asmolov, E.S. & Vinogradova, O.I. 2012 Effective slip boundary conditions for arbitrary one-dimensional surfaces. J. Fluid Mech. 706, 108–117.CrossRefGoogle Scholar
Bagheri, S., Schlatter, P., Schmid, P.J. & Henningson, D.S. 2009 Global stability of a jet in crossflow. J. Fluid Mech. 624, 33–44.CrossRefGoogle Scholar
Bucci, M.A., Puckert, D.K., Andriano, C., Loiseau, J.-Ch., Cherubini, S., Robinet, J.-Ch. & Rist, U. 2018 Roughness-induced transition by quasi-resonance of a varicose global mode. J. Fluid Mech. 836, 167–191.CrossRefGoogle Scholar
Cassie, A.B.D. & Baxter, S. 1944 Wettability of porous surfaces. Trans. Faraday Soc. 40, 546–551.CrossRefGoogle Scholar
Choi, H. & Kim, J. 2006 Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface. Phys. Rev. Lett. 96, 066001.CrossRefGoogle Scholar
Cottin-Bizonne, C., Barentin, C., Charlaix, É., Bocquet, L. & Barrat, J.-L. 2004 Dynamics of simple liquids at heterogeneous surfaces: molecular-dynamics simulations and hydrodynamic description. Eur. Phys. J. E 15 (4), 427–438.CrossRefGoogle ScholarPubMed
Crowdy, D.G. 2015 A transform method for Laplace's equation in multiply connected circular domains. IMA J. Appl. Maths 80, 1902–1931.CrossRefGoogle Scholar
Crowdy, D.G. 2016 Analytical formulae for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci. J. Fluid Mech. 791, R7.CrossRefGoogle Scholar
Crowdy, D.G. 2017 a Perturbation analysis of subphase gas and meniscus curvature effects for longitudinal flows over superhydrophobic surfaces. J. Fluid Mech. 822, 307–326.CrossRefGoogle Scholar
Crowdy, D.G. 2017 b Slip length for transverse shear flow over a periodic array of weakly curved menisci. Phys. Fluids 29, 091702.CrossRefGoogle Scholar
Daniello, R.J., Waterhouse, N.E. & Rothstein, J.P. 2009 Drag reduction in turbulent flows over superhydrophobic surfaces. Phys. Fluids 21 (8), 085103.CrossRefGoogle Scholar
Davies, J., Maynes, D., Webb, B.W. & Woolford, B. 2006 Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs. Phys. Fluids 18 (8), 087110.CrossRefGoogle Scholar
Davis, A.M.J. & Lauga, E. 2009 Geometric transition in friction for flow over a bubble mattress. Phys. Fluids 21 (1), 011701.CrossRefGoogle Scholar
Duck, P.W. 2011 On the downstream development and breakup of systems of trailing-line vortices. Theor. Comput. Fluid Dyn. 25 (1), 43–52.CrossRefGoogle Scholar
Ehrenstein, U. 1996 On the linear stability of channel flow over riblets. Phys. Fluids 8 (11), 3194–3196.CrossRefGoogle Scholar
Floryan, J.M. 1997 Stability of wall-bounded shear layers in the presence of simulated distributed surface roughness. J. Fluid Mech. 335, 29–55.CrossRefGoogle Scholar
Game, S., Hodes, M., Kirk, T. & Papageorgiou, D.T. 2018 Nusselt numbers for poiseuille flow over isoflux parallel ridges for arbitrary meniscus curvature. Trans. ASME J. Heat Transfer 140 (8), 081701.CrossRefGoogle Scholar
Game, S.E., Hodes, M., Keaveny, E.E. & Papageorgiou, D.T. 2017 Physical mechanisms relevant to flow resistance in textured microchannels. Phys. Rev. Fluids 2 (9), 094102.CrossRefGoogle Scholar
Game, S.E., Hodes, M. & Papageorgiou, D.T. 2019 Effects of slowly varying meniscus curvature on internal flows in the Cassie state. J. Fluid Mech. 872, 272–307.CrossRefGoogle Scholar
Hall, P. & Horseman, N.J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375.CrossRefGoogle Scholar
Hall, P. & Sherwyn, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.CrossRefGoogle Scholar
Hodes, M., Kirk, T.L., Karamanis, G. & MacLachlan, S. 2017 Effect of thermocapillary stress on slip length for a channel textured with parallel ridges. J. Fluid Mech. 814, 301–324.CrossRefGoogle Scholar
Kirk, T.L., Hodes, M. & Papageorgiou, D.T. 2017 Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature. J. Fluid Mech. 811, 315–349.CrossRefGoogle Scholar
Landel, J.R., Peaudecerf, F.J., Temprano-Coleto, F., Gibou, F., Goldstein, R.E. & Luzzatto-Fegiz, P. 2020 A theory for the slip and drag of superhydrophobic surfaces with surfactant. J. Fluid Mech. 883, A18.CrossRefGoogle ScholarPubMed
Lauga, E. & Cossu, C. 2005 A note on the stability of slip channel flows. Phys. Fluids 17 (8), 088106.CrossRefGoogle Scholar
Lauga, E. & Stone, H.A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77.CrossRefGoogle Scholar
Loiseau, J.-Ch., Robinet, J.-Ch., Cherubini, S. & Leriche, E. 2014 Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations. J. Fluid Mech. 760, 175–211.CrossRefGoogle Scholar
Luca, E., Marshall, J. & Karamanis, G. 2018 Longitudinal shear flow over a bubble mattress with curved menisci: arbitrary protrusion angle and solid fraction. IMA J. Appl. Maths 83, 917–941.Google Scholar
Marmur, A. 2003 Wetting on hydrophobic rough surfaces: to be heterogeneous or not to be? Langmuir 19 (20), 8343–8348.CrossRefGoogle Scholar
Marshall, J. 2017 Exact formulae for the effective slip length of a symmetric superhydrophobic channel with flat or weakly curved menisci. SIAM J. Appl. Maths 77, 1606–1630.CrossRefGoogle Scholar
Martell, M.B., Perot, J.B. & Rothstein, J.P. 2009 Direct numerical simulations of turbulent flows over superhydrophobic surfaces. J. Fluid Mech. 620, 31–41.CrossRefGoogle Scholar
Maynes, D., Webb, B.W. & Davies, J. 2008 Thermal transport in a microchannel exhibiting ultrahydrophobic microribs maintained at constant temperature. Trans. ASME J. Heat Transfer 130 (2), 022402.CrossRefGoogle Scholar
Min, T. & Kim, J. 2004 Effects of hydrophobic surface on skin-friction drag. Phys. Fluids 16 (7), L55–L58.CrossRefGoogle Scholar
Mohammadi, A., Moradi, H.V. & Floryan, J.M. 2015 New instability mode in a grooved channel. J. Fluid Mech. 778, 691–720.CrossRefGoogle Scholar
Moradi, H.V. & Floryan, J.M. 2014 Stability of flow in a channel with longitudinal grooves. J. Fluid Mech. 757, 613–648.CrossRefGoogle Scholar
Motz, H. 1947 The treatment of singularities of partial differential equations by relaxation methods. Q. Appl. Maths 4, 371–377.CrossRefGoogle Scholar
Ng, C.O. & Wang, C.Y. 2009 Stokes shear flow over a grating: implications for superhydrophobic slip. Phys. Fluids 21, 013602.CrossRefGoogle Scholar
Nizkaya, T.V., Asmolov, E.S. & Vinogradova, O.I. 2014 Gas cushion model and hydrodynamic boundary conditions for superhydrophobic textures. Phys. Rev. E 90, 043017.CrossRefGoogle ScholarPubMed
Ou, J., Perot, B. & Rothstein, J.P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16 (12), 4635–4643.CrossRefGoogle Scholar
Ou, J. & Rothstein, J.P. 2005 Drag reduction and $\mu$-PIV measurements of the flow past ultrahydrophobic surfaces. Phys. Fluids 17 (12), 103606.CrossRefGoogle Scholar
Park, H., Sun, G. & Kim, J. 2013 A numerical study of the effects of superhydrophobic surface on skin-friction drag in turbulent channel flow. Phys. Fluids 25 (11), 110815.CrossRefGoogle Scholar
Peaudecerf, F.J., Landel, J.R., Goldstein, R.E. & Luzzatto-Fegiz, P. 2017 Traces of surfactants can severely limit the drag reduction of superhydrophobic surfaces. Proc. Natl Acad. Sci. USA 114 (28), 7254–7259.CrossRefGoogle ScholarPubMed
Peyret, R. 2013 Spectral Methods for Incompressible Viscous Flow, vol. 148. Springer Science & Business Media.Google Scholar
Philip, J.R. 1972 Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23, 353–372.CrossRefGoogle Scholar
Picella, F., Robinet, J.-Ch. & Cherubini, S. 2019 Laminar–turbulent transition in channel flow with superhydrophobic surfaces modelled as a partial slip wall. J. Fluid Mech. 881, 462–497.CrossRefGoogle Scholar
Picella, F., Robinet, J.-Ch. & Cherubini, S. 2020 On the influence of the modelling of superhydrophobic surfaces on laminar–turbulent transition. J. Fluid Mech. 901, A15.CrossRefGoogle Scholar
Pralits, J.O., Alinovi, E. & Bottaro, A. 2017 Stability of the flow in a plane microchannel with one or two superhydrophobic walls. Phys. Rev. Fluids 2 (1), 013901.CrossRefGoogle Scholar
Rothstein, J.P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42, 89–109.CrossRefGoogle Scholar
Sbragaglia, M. & Prosperetti, A. 2007 A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys. Fluids 19 (4), 043603.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2012 Stability and Transition in Shear Flows, vol. 142. Springer Science & Business Media.Google Scholar
Schönecker, C., Baier, T. & Hardt, S. 2014 Influence of the enclosed fluid on the flow over a microstructured surface in the Cassie state. J. Fluid Mech. 740, 168–195.CrossRefGoogle Scholar
Schönecker, C. & Hardt, S. 2013 Longitudinal and transverse flow over a cavity containing a second immiscible fluid. J. Fluid Mech. 717, 376–394.CrossRefGoogle Scholar
Song, D., Song, B., Hu, H., Du, X., Du, P., Choi, C.-H. & Rothstein, J.P. 2018 Effect of a surface tension gradient on the slip flow along a superhydrophobic air-water interface. Phys. Rev. Fluids 3, 033303.CrossRefGoogle Scholar
Tatsumi, T. & Yoshimura, T. 1990 Stability of the laminar flow in a rectangular duct. J. Fluid Mech. 212, 437–449.CrossRefGoogle Scholar
Teo, C.J. & Khoo, B.C. 2009 Analysis of stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves. Microfluid Nanofluid 7 (3), 353.CrossRefGoogle Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39 (4), 249–315.CrossRefGoogle Scholar
Theofilis, V., Duck, P.W. & Owen, J. 2004 Viscous linear stability analysis of rectangular duct and cavity flows. J. Fluid Mech. 505, 249–286.CrossRefGoogle Scholar
Waleffe, F. 1997 On self-sustaining processes in shear flows. Phys. Fluids 9 (4), 883–900.CrossRefGoogle Scholar
Wang, S. & Jiang, L. 2007 Definition of superhydrophobic states. Adv. Mater. 19 (21), 3423–3424.CrossRefGoogle Scholar
Woods, L.C. 1953 The relaxation treatment of singular points in Poisson's equation. Q. J. Mech. Appl. Maths 6, 163–185.CrossRefGoogle Scholar
Woolford, B., Maynes, D. & Webb, B.W. 2009 Liquid flow through microchannels with grooved walls under wetting and superhydrophobic conditions. Microfluid Nanofluid 7, 121–135.CrossRefGoogle Scholar
Yariv, E. & Crowdy, D.G. 2020 Longitudinal thermocapillary flow over a dense bubble mattress. SIAM J. Appl. Maths 80 (1), 1–19.CrossRefGoogle Scholar
Yariv, E. & Kirk, T.L. 2021 Longitudinal thermocapillary slip about a dilute periodic mattress of protruding bubbles. IMA J. Appl. Maths 86, 490–501.CrossRefGoogle Scholar
Yariv, E. & Schnitzer, O. 2018 Pressure-driven plug flows between superhydrophobic surfaces of closely spaced circular bubbles. J. Engng Maths 111, 15–22.CrossRefGoogle Scholar
Yu, K.H., Teo, C.J. & Khoo, B.C. 2016 Linear stability of pressure-driven flow over longitudinal superhydrophobic grooves. Phys. Fluids 28 (2), 022001.CrossRefGoogle Scholar