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Thermocapillary stress and meniscus curvature effects on slip lengths in ridged microchannels

Published online by Cambridge University Press:  04 May 2020

Toby L. Kirk Open the ORCID record for Toby L. Kirk [Opens in a new window] ,
Georgios Karamanis ,
Darren G. Crowdy Open the ORCID record for Darren G. Crowdy [Opens in a new window]  and
Marc Hodes
Toby L. Kirk*
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, OxfordOX2 6GG, UK
Georgios Karamanis
Affiliation:
Department of Mechanical Engineering, Tufts University, Medford, MA02155, USA
Darren G. Crowdy
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
Marc Hodes
Affiliation:
Department of Mechanical Engineering, Tufts University, Medford, MA02155, USA
*
†Email address for correspondence: [email protected]
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Abstract

Pressure-driven flow in the presence of heat transfer through a microchannel patterned with parallel ridges is considered. The coupled effects of curvature and thermocapillary stress along the menisci are captured. Streamwise and transverse thermocapillary stresses along menisci cause the flow to be three-dimensional, but when the Reynolds number based on the transverse flow is small the streamwise and transverse flows decouple. In this limit, we solve the streamwise flow problem, i.e. that in the direction parallel to the ridges, using a suite of asymptotic limits and techniques – each previously shown to have wide ranges of validity thereby extending results by Hodes et al. (J. Fluid Mech., vol. 814, 2017, pp. 301–324) for a flat meniscus. First, we take the small-ridge-period limit, and then we account for the curvature of the menisci with two further complementary limits: (i) small meniscus curvature using boundary perturbation; (ii) arbitrary meniscus curvature but for small slip (or cavity) fractions using conformal mapping and the Poisson integral formula. Heating and cooling the liquid always degrade and enhance (apparent) slip, respectively, but their effect is greatest for large meniscus protrusions, with positive protrusion (into the liquid) being the most sensitive. For strong enough heating the solutions become complex, suggesting instability, with large positive protrusions transitioning first.

JFM classification

Drops and Bubbles: Thermocapillarity Micro-/Nano-fluid dynamics: Microfluidics Flow Control: Drag reduction
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 894 , 10 July 2020 , A15
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Ageev, A. I., Golubkina, I. V. & Osiptsov, A. N. 2018 Application of boundary element method to Stokes flows over a striped superhydrophobic surface with trapped gas bubbles. Phys. Fluids 30 (1), 012102.CrossRefGoogle Scholar
Alinovi, E. & Bottaro, A. 2018 Apparent slip and drag reduction for the flow over superhydrophobic and lubricant-impregnated surfaces. Phys. Rev. Fluids 3, 124002.CrossRefGoogle Scholar
Baier, T., Steffes, C. & Hardt, S. 2010 Thermocapillary flow on superhydrophobic surfaces. Phys. Rev. E 82 (3), 037301.Google ScholarPubMed
Belyaev, A. V. & Vinogradova, O. I. 2010 Effective slip in pressure-driven flow past super-hydrophobic stripes. J. Fluid Mech. 652, 489–499.CrossRefGoogle Scholar
Bergman, T. L., Lavine, A. S., Incropera, F. P. & DeWitt, D. P. 2011 Fundamentals of Heat and Mass Transfer, 7th edn. John Wiley & Sons.Google Scholar
Bertrand, J. 1878 Sur l’homogénéité dans les formules de physique. Comptes Rendus 86 (15), 915–920.Google Scholar
Cottin-Bizonne, C., Barentin, C., Charlaix, E., 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.Google ScholarPubMed
Crowdy, D. G. 2010 Slip length for longitudinal shear flow over a dilute periodic mattress of protruding bubbles. Phys. Fluids 22 (12), 121703.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 Perturbation analysis of subphase gas and meniscus curvature effects for longitudinal flows over superhydrophobic surfaces. J. Fluid Mech. 822, 307–326.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
Enright, R., Hodes, M., Salamon, T. & Muzychka, Y. 2014 Isoflux Nusselt number and slip length formulae for superhydrophobic microchannels. Trans. ASME J. Heat Transfer 136 (1), 012402.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., Kirk, T. L. & Papageorgiou, D. T. 2018 Nusselt numbers for Poiseuille flow over isoflux parallel ridges for arbitrary meniscus curvature. Trans. ASME J. Heat Transfer 140, 081701.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
Geratherm2009. Safety Data Sheet for Galinstan, Geschwenda, Germany. Available at: http://www.geratherm.com/wp-content/uploads/2010/02/Safety-Data-Sheet-Galinstan-2010-EN.pdf.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.CrossRefGoogle Scholar
Hodes, M., Lam, L. S., Cowley, A., Enright, R. & MacLachlan, S. 2015 Effect of evaporation and condensation at menisci on apparent thermal slip. Trans. ASME J. Heat Transfer 137 (7), 071502.CrossRefGoogle 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. Technol. 4 (1), 46–56.Google Scholar
Hyväluoma, J. & Harting, J. 2008 Slip flow over structured surfaces with entrapped microbubbles. Phys. Rev. Lett. 100, 246001.CrossRefGoogle ScholarPubMed
Kirk, T. L. 2018 Asymptotic formulae for flow in a channel with ridged walls and curved menisci. J. Fluid Mech. 839, R3.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
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.CrossRefGoogle Scholar
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77.CrossRefGoogle Scholar
Li, Y., Alame, K. & Mahesh, K. 2017 Feature-resolved computational and analytical study of laminar drag reduction by superhydrophobic surfaces. Phys. Rev. Fluids 2, 054002.CrossRefGoogle 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.CrossRefGoogle Scholar
Maynes, D. & Crockett, J. 2014 Apparent temperature jump and thermal transport in channels with streamwise rib and cavity featured superhydrophobic walls at constant heat flux. Trans. ASME J. Heat Transfer 136 (1), 011701.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.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
Ng, C.-O. & Wang, C. Y. 2011 Effective slip for Stokes flow over a surface patterned with two- or three-dimensional protrusions. Fluid Dyn. Res. 43 (6), 065504.Google Scholar
NIST 2020 NIST Chemistry WebBook, NIST Standard Reference Database Number 69. National Institute of Standards and Technology.Google Scholar
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 Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces. Phys. Fluids 17 (10), 103606.CrossRefGoogle Scholar
Patankar, S., Liu, C. & Sparrow, E. 1977 Fully developed flow and heat transfer in ducts having streamwise-periodic variations of cross-sectional area. Trans. ASME J. Heat Transfer 99 (2), 180–186.CrossRefGoogle Scholar
Peaudecerf, F., Landel, J., Goldstein, R. & 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
Philip, J. R. 1972a Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (3), 353–372.CrossRefGoogle Scholar
Philip, J. R. 1972b Integral properties of flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23, 960–968.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
Schnitzer, O. 2016 Singular effective slip length for longitudinal flow over a dense bubble mattress. Phys. Rev. Fluids 1, 052101.CrossRefGoogle Scholar
Schnitzer, O. 2017 Slip length for longitudinal shear flow over an arbitrary-protrusion-angle bubble mattress: the small-solid-fraction singularity. J. Fluid Mech. 820, 580–603.CrossRefGoogle Scholar
Sneddon, I. N. 1966 Mixed Boundary Value Problems in Potential Theory. North-Holland.Google Scholar
Song, D., Song, B., Hu, H., Du, X., Du, P., Choi, C. & 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.CrossRefGoogle Scholar
Steinberger, A., Cottin-Bizonne, C., Kleimann, P. & Charlaix, E. 2007 High friction on a bubble mattress. Nat. Mater. 6 (9), 665–668.CrossRefGoogle ScholarPubMed
Tam, D., von Arnim, V., McKinley, G. H. & Hosoi, A. E. 2009 Marangoni convection in droplets on superhydrophobic surfaces. J. Fluid Mech. 624, 101–123.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), 353–382.CrossRefGoogle 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.CrossRefGoogle Scholar
Tsai, P., Peters, A. M., Pirat, C., Wessling, M., Lammertink, R. G. H. & Lohse, D. 2009 Quantifying effective slip length over micropatterned hydrophobic surfaces. Phys. Fluids 21 (11), 112002.CrossRefGoogle Scholar
Tuckerman, D. B. & Pease, R. F. W. 1981 High-performance heat sinking for VLSI. IEEE Electron Device Lett. 2 (5), 126–129.CrossRefGoogle Scholar
Vinogradova, O. I. 1995 Drainage of a thin liquid film confined between hydrophobic surfaces. Langmuir 11 (6), 2213–2220.CrossRefGoogle Scholar
Wang, L. P., Teo, C. J. & Khoo, B. C. 2014 Effects of interface deformation on flow through microtubes containing superhydrophobic surfaces with longitudinal ribs and grooves. Microfluid Nanofluid 16 (1–2), 225–236.CrossRefGoogle Scholar
Yariv, E. 2018 Thermocapillary flow between longitudinally grooved superhydrophobic surfaces. J. Fluid Mech. 855, 574–594.CrossRefGoogle Scholar
Yariv, E. & Crowdy, D. 2020 Longitudinal thermocapillary flow over a dense bubble mattress. SIAM J. Appl. Maths 80 (1), 1–19.CrossRefGoogle Scholar
Yariv, E. & Crowdy, D. G. 2019 Thermocapillary flow between grooved superhydrophobic surfaces: transverse temperature gradients. J. Fluid Mech. 871, 775–798.CrossRefGoogle Scholar
Yariv, E. & Schnitzer, O. 2018 Pressure-driven plug flows between superhydrophobic surfaces of closely spaced circular bubbles. J. Engng Maths (1), 1–8.Google Scholar
Zhang, R., Hodes, M., Lower, N. & Wilcoxon, R. 2015 Water-based microchannel and Galinstan-based minichannel cooling beyond 1 kw cm-2 heat flux. IEEE Trans. Compon. Packag. Manuf. Technol. 5 (6), 762–770.CrossRefGoogle Scholar