Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-24T15:05:03.550Z Has data issue: false hasContentIssue false
  • English
  • Français

Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature

Published online by Cambridge University Press:  07 December 2016

Toby L. Kirk ,
Marc Hodes  and
Demetrios T. Papageorgiou
Toby L. Kirk*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Marc Hodes
Affiliation:
Department of Mechanical Engineering, Tufts University, Medford, MA 02155, USA
Demetrios T. Papageorgiou
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: tlk12@ic.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate forced convection in a parallel-plate-geometry microchannel with superhydrophobic walls consisting of a periodic array of ridges aligned parallel to the direction of a Poiseuille flow. In the dewetted (Cassie) state, the liquid contacts the channel walls only at the tips of the ridges, where we apply a constant-heat-flux boundary condition. The subsequent hydrodynamic and thermal problems within the liquid are then analysed accounting for curvature of the liquid–gas interface (meniscus) using boundary perturbation, assuming a small deflection from flat. The effects of this surface deformation on both the effective hydrodynamic slip length and the Nusselt number are computed analytically in the form of eigenfunction expansions, reducing the problem to a set of dual series equations for the expansion coefficients which must, in general, be solved numerically. The Nusselt number quantifies the convective heat transfer, the results for which are completely captured in a single figure, presented as a function of channel geometry at each order in the perturbation. Asymptotic solutions for channel heights large compared with the ridge period are compared with numerical solutions of the dual series equations. The asymptotic slip length expressions are shown to consist of only two terms, with all other terms exponentially small. As a result, these expressions are accurate even for heights as low as half the ridge period, and hence are useful for engineering applications.

JFM classification

Convection: Convection Micro-/Nano-fluid dynamics: Microfluidics Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 811 , 25 January 2017 , pp. 315 - 349
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baier, T., Steffes, C. & Hardt, S. 2010 Thermocapillary flow on superhydrophobic surfaces. Phys. Rev. E 82 (3), 037301.Google Scholar
Bergman, T. L., Lavine, A. S., Incropera, F. P. & DeWitt, D. P. 2011 Fundamentals of Heat and Mass Transfer, 7th edn. Wiley.Google Scholar
Cheng, Y., Xu, J. & Sui, Y. 2015 Numerical study on drag reduction and heat transfer enhancement in microchannels with superhydrophobic surfaces for electronic cooling. Appl. Therm. Engng 88 (SI), 71–81.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.Google 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.Google Scholar
Davis, A. M. J. & Lauga, E. 2009 Geometric transition in friction for flow over a bubble mattress. Phys. Fluids 21 (1), 011701.Google Scholar
Enright, R., Eason, C., Dalton, T., Hodes, M., Salamon, T., Kolodner, P. & Krupenkin, T. 2006 Friction factors and Nusselt numbers in microchannels with superhydrophobic walls. In Proceedings of the 4th International Conference on Nanochannels, Microchannels and Minichannels, Parts A and B, pp. 599–609. ASME.Google Scholar
Enright, R., Hodes, M., Salamon, T. & Muzychka, Y. 2014 Isoflux Nusselt number and slip length formulae for superhydrophobic microchannels. Trans. ASME J. Heat Mass Transfer 136 (1), 012402.Google Scholar
Hodes, M., Kirk, T., Karamanis, G., Lam, L., MacLachlan, S. & Papageorgiou, D. 2015a Conformal map and asymptotic solutions for apparent slip lengths in the presence of thermocapillary stress. In Proceedings of First International ISHMT-ASTFE Conference (IHMTC 2015), p. 1254. ISHMT.Google Scholar
Hodes, M., Lam, L. S., Cowley, A., Enright, R. & MacLachlan, S. 2015b Effect of evaporation and condensation at menisci on apparent thermal slip. Trans. ASME J. Heat Mass Transfer 137 (7), 071502.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. Technol. 4 (1), 46–56.Google Scholar
Kays, W., Crawford, M. & Weigand, B. 2005 Convective Heat and Mass Transfer, 4th edn. McGraw-Hill.Google Scholar
Lam, L. S., Hodes, M. & Enright, R. 2015 Analysis of galinstan-based microgap cooling enhancement using structured surfaces. Trans. ASME J. Heat Mass Transfer 137 (9), 091003.Google Scholar
Lam, L. S., Hodes, M., Karamanis, G., Kirk, T. & MacLachlan, S. 2016 Effect of meniscus curvature on apparent thermal slip. Trans. ASME J. Heat Mass Transfer 138 (12), 122004.Google Scholar
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77.Google 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 Mass 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.Google Scholar
Maynes, D., Webb, B. W., Crockett, J. & Solovjov, V. 2013 Analysis of laminar slip-flow thermal transport in microchannels with transverse rib and cavity structured superhydrophobic walls at constant heat flux. Trans. ASME J. Heat Mass Transfer 135 (2), 021701.Google 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 Mass Transfer 130 (2), 022402.Google Scholar
Ou, J., Perot, B. & Rothstein, J. P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16 (12), 4635–4643.Google Scholar
Ou, J. & Rothstein, J. P. 2005 Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces. Phys. Fluids 17 (10), 103606.Google Scholar
Philip, J. R. 1972 Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (3), 353–372.Google Scholar
Priezjev, N. V., Darhuber, A. A. & Troian, S. M. 2005 Slip behavior in liquid films on surfaces of patterned wettability: comparison between continuum and molecular dynamics simulations. Phys. Rev. E 71 (4), 41608.Google ScholarPubMed
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
Teo, C. J. & Khoo, B. C. 2009 Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves. Microfluid. Nanofluidics 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. Nanofluidics 9 (2–3), 499–511.Google Scholar
Tuckerman, D. B. & Pease, R. F. W. 1981 High-performance heat sinking for VLSI. IEEE Electron. Device Lett. 2 (5), 126–129.Google Scholar
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.Google Scholar