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Effects of slowly varying meniscus curvature on internal flows in the Cassie state

Published online by Cambridge University Press:  10 June 2019

Simon E. Game ,
Marc Hodes  and
Demetrios T. Papageorgiou Open the ORCID record for Demetrios T. Papageorgiou [Opens in a new window]
Simon E. Game
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Marc Hodes
Affiliation:
Department of Mechanical Engineering, Tufts University, Medford, MA 02138, USA
Demetrios T. Papageorgiou*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: d.papageorgiou@imperial.ac.uk
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Abstract

The flow rate of a pressure-driven liquid through a microchannel may be enhanced by texturing its no-slip boundaries with grooves aligned with the flow. In such cases, the grooves may contain vapour and/or an inert gas and the liquid is trapped in the Cassie state, resulting in (apparent) slip. The flow-rate enhancement is of benefit to different applications including the increase of throughput of a liquid in a lab-on-a-chip, and the reduction of thermal resistance associated with liquid metal cooling of microelectronics. At any given cross-section, the meniscus takes the approximate shape of a circular arc whose curvature is determined by the pressure difference across it. Hence, it typically protrudes into the grooves near the inlet of a microchannel and is gradually drawn into the microchannel as it is traversed and the liquid pressure decreases. For sufficiently large Reynolds numbers, the variation of the meniscus shape and hence the flow geometry necessitates the inclusion of inertial (non-parallel) flow effects. We capture them for a slender microchannel, where our small parameter is the ratio of ridge pitch-to-microchannel height, and order-one Reynolds numbers. This is done by using a hybrid analytical–numerical method to resolve the nonlinear three-dimensional (3-D) problem as a sequence of two-dimensional (2-D) linear ones in the microchannel cross-section, allied with non-local conditions that determine the slowly varying pressure distribution at leading and first orders. When the pressure difference across the microchannel is constrained by the advancing contact angle of the liquid on the ridges and its surface tension (which is high for liquid metals), inertial effects can significantly reduce the flow rate for realistic parameter values. For example, when the solid fraction of the ridges is 0.1, the microchannel height-to-(half) ridge pitch ratio is 6, the Reynolds number of the flow is 1 and the small parameter is 0.1, they reduce the flow rate of a liquid metal (Galinstan) by approximately 50 %. Conversely, for sufficiently large microchannel heights, they enhance it. Physical explanations of both of these phenomena are given.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Mathematical Foundations: Mathematical Foundations Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 872 , 10 August 2019 , pp. 272 - 307
Copyright
© 2019 Cambridge University Press 

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References

Ahuja, A., Taylor, J. A., Lifton, V., Sidorenko, A. A., Salamon, T. R., Lobaton, E. J., Kolodner, P. & Krupenkin, T. N. 2008 Nanonails: a simple geometrical approach to electrically tunable superlyophobic surfaces. Langmuir 24 (1), 9–14.Google Scholar
Akbari, M., Sinton, D. & Bahrami, M. 2011 Viscous flow in variable cross-section microchannels of arbitrary shapes. Intl J. Heat Mass Transfer 54 (17), 3970–3978.Google Scholar
Asmolov, E. S., Nizkaya, T. V. & Vinogradova, O. I. 2018 Enhanced slip properties of lubricant-infused grooves. Phys. Rev. E 98 (3), 033103.Google Scholar
Barber, R. 2002 Elasticity. Kluwer Academic Publishers.Google Scholar
Blasius, H. 1910 Laminare stromung in kanalen wechselnder breite. Z. Angew. Math. Phys. 58, 225–233.Google Scholar
Chadwick, R. S. 1985 Asymptotic analysis of stokes flow in a tortuous vessel. Q. Appl. Maths 43 (3), 325–336.Google Scholar
Crowdy, D. G. 2016 Analytical formulae for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci. J. Fluid Mech. 791, R7.Google 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, 094102.Google 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.Google Scholar
Ghosal, S. 2002 Lubrication theory for electro-osmotic flow in a microfluidic channel of slowly varying cross-section and wall charge. J. Fluid Mech. 459, 103–128.Google 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.Google Scholar
Hodes, M., Zhang, R., Lam, L. S., Wilcoxon, R. & Lower, N. 2014 On the potential of galinstan-based minichannel and minigap cooling. IEEE Trans. Compon. Packag. Manufacturing Technol. 4 (1), 46–56.Google Scholar
Kirk, T. L. 2018 Asymptotic formulae for flow in superhydrophobic channels with longitudinal ridges and protruding menisci. J. Fluid Mech. 839, R3.Google 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.Google Scholar
Kotorynski, W. P. 1979 Slowly varying channel flows in three dimensions. J. Inst. Maths Applics. 24, 71–80.Google Scholar
Kotorynski, W. P. 1995 Viscous flow in axisymmetric pipes with slow variations. Computers Fluids 24 (6), 685–717.Google Scholar
Lam, L. S., Hodes, M. & Enright, R. 2015 Analysis of galinstan-based microgap cooling enhancement using structured surfaces. Trans. ASME J. Heat Transfer 137 (9), 091003.Google Scholar
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven stokes flow. J. Fluid Mech. 489, 55–77.Google Scholar
Lauga, E., Stroock, A. D. & Stone, H. A. 2004 Three-dimensional flows in slowly varying planar geometries. Phys. Fluids 16 (8), 3051–3062.Google Scholar
Lee, C., Choi, C.-H. & Kim, C.-J. 2016 Superhydrophobic drag reduction in laminar flows: a critical review. Exp. Fluids 57 (12), 176.Google Scholar
Liu, T., Sen, P. & Kim, C.-J. 2012 Characterization of nontoxic liquid-metal alloy galinstan for applications in microdevices. J. Microelectromech. Syst. 21 (2), 443–450.Google Scholar
Manton, M. J. 1971 Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech. 49 (3), 451–459.Google Scholar
Marshall, J. S. 2017 Exact formulae for the effective slip length of a symmetric superhydrophobic channel with flat or weakly curved menisci. SIAM J. Appl. Maths 77 (5), 1606–1630.Google Scholar
Maynes, D., Jeffs, K., Woolford, B. & Webb, B. W. 2007 Laminar flow in a microchannel with hydrophobic surface patterned microribs oriented parallel to the flow direction. Phys. Fluids 19 (9), 093603.Google Scholar
Ng, C. O., Chu, H. C. W. & Wang, C. Y. 2010 On the effects of liquid–gas interfacial shear on slip flow through a parallel-plate channel with superhydrophobic grooved walls. Phys. Fluids 22 (10), 102002.Google 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 (4), 043017.Google 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.Google Scholar
Peyret, J. R. & Taylor, T. D. 1983 Numerical Methods for Fluid Flow. Springer.Google Scholar
Philip, J. R. 1972a Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (3), 353–372.Google Scholar
Philip, J. R. 1972b Integral properties of flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (6), 960–968.Google Scholar
Sbragaglia, M. & Prosperetti, A. 2007 A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys. Fluids 19 (4), 043603.Google 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 (3), 033303.Google Scholar
Tanner, R. I. 1966 Pressure losses in viscometric capillary tubes of slowly varying diameter. British J. Appl. Phys. 17 (5), 663–670.Google 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), 353–382.Google Scholar
Teo, C. J. & Khoo, B. C. 2010 Flow past superhydrophobic surfaces containing longitudinal grooves: effects of interface curvature. Microfluid. Nanofluid. 9 (2–3), 499–511.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB, vol. 10. SIAM.Google Scholar
Tuteja, A., Choi, W., Mabry, J. M., McKinley, G. H. & Cohen, R. E. 2008 Robust omniphobic surfaces. Proc. Natl Acad. Sci. USA 105 (47), 18200–18205.Google Scholar
Van Dyke, M. 1983 Laminar flow in a meandering channel. SIAM J. Appl. Maths 43 (4), 696–702.Google Scholar
Van Dyke, M. 1987 Slow variations in continuum mechanics. Adv. Appl. Mech. 25, 1–45.Google Scholar
Wild, R., Pedley, T. J. & Riley, D. S. 1977 Viscous flow in collapsible tubes of slowly varying elliptical cross-section. J. Fluid Mech. 81 (02), 273–294.Google Scholar
Xu, Q., Oudalov, N., Guo, Q., Jaeger, H. M. & Brown, E. 2012 Effect of oxidation on the mechanical properties of liquid gallium and eutectic gallium-indium. Phys. Fluids 24 (6), 063101.Google Scholar

Game Supplementary Movie 1

Animation of the steady state leading order streamwise velocity $w_0$ as the channel is traversed from the entry $z=0$ where the meniscus curvature is largest, to the exit $z=1$ where the meniscus curvature is zero. Other parameters are: surface tension parameter $\Gamma=1$, channel height $H=0.5$ and trench width $\delta=0.8$.

Download Game Supplementary Movie 1(Video)
Video 392 KB

Game Supplementary Movie 2

Animation of the normalized steady state first order streamwise velocity $w_1/\mathrm{Re}$ as the channel is traversed from the entry $z=0$ where the meniscus curvature is largest, to the exit $z=1$ where the meniscus curvature is zero. Other parameters are: surface tension parameter $\Gamma=1$, channel height $H=0.5$ and trench width $\delta=0.8$.

Download Game Supplementary Movie 2(Video)
Video 384.7 KB

Game et al. supplementary movie 3

Animation of the steady state leading order cross-plane spanwise velocity $u_0$ as the channel is traversed from the entry $z=0$ where the meniscus curvature is largest, to the exit $z=1$ where the meniscus curvature is zero. Other parameters are: surface tension parameter $\Gamma=1$, channel height $H=0.5$ and trench width $\delta=0.8$.

Download Game et al. supplementary movie 3(Video)
Video 388.5 KB

Game Supplementary Movie 4

Animation of the steady state leading order cross-plane vertical velocity $v_0$ as the channel is traversed from the entry $z=0$ where the meniscus curvature is largest, to the exit $z=1$ where the meniscus curvature is zero. Other parameters are: surface tension parameter $\Gamma=1$, channel height $H=0.5$ and trench width $\delta=0.8$.

Download Game Supplementary Movie 4(Video)
Video 358 KB