Skip to main content Accessibility help
×
Home

Thermocapillary flow between grooved superhydrophobic surfaces: transverse temperature gradients

  • Ehud Yariv (a1) and Darren Crowdy (a2)

Abstract

We consider the thermocapillary motion of a liquid layer which is bounded between two superhydrophobic surfaces, each made up of a periodic array of highly conducting solid slats, with flat bubbles trapped in the grooves between them. Following the recent analysis of the longitudinal problem (Yariv, J. Fluid Mech., vol. 855, 2018, pp. 574–594), we address here the transverse problem, where the macroscopic temperature gradient that drives the flow is applied perpendicular to the grooves, with the goal of calculating the volumetric flux between the two surfaces. We focus upon the situation where the slats separating the grooves are long relative to the groove-array period, for which case the temperature in the solid portions of the superhydrophobic plane is piecewise uniform. This scenario, which was investigated numerically by Baier et al. (Phys. Rev. E, vol. 82 (3), 2010, 037301), allows for a surprising analogy between the harmonic conjugate of the temperature field in the present problem and the unidirectional velocity in a comparable longitudinal pressure-driven flow problem over an interchanged boundary. The main body of the paper is concerned with the limit of deep channels, where the problem reduces to the calculation of the heat transport and flow about a single surface and the associated ‘slip’ velocity at large distance from that surface. Making use of Lorentz’s reciprocity, we obtain that velocity as a simple quadrature, providing the analogue to the expression obtained by Baier et al. (2010) in the comparable longitudinal problem. The rest of the paper is devoted to the diametric limit of shallow channels, which is analysed using a Hele-Shaw approximation, and the singular limit of small solid fractions, where we find a logarithmic scaling of the flux with the solid fraction. The latter two limits do not commute.

Copyright

Corresponding author

Email address for correspondence: udi@technion.ac.il

References

Hide All
Baier, T., Steffes, C. & Hardt, S. 2010 Thermocapillary flow on superhydrophobic surfaces. Phys. Rev. E 82 (3), 037301.
Balasubramaniam, R. & Subramaniam, R. S. 1996 Thermocapillary bubble migration – thermal boundary layers for large Marangoni numbers. Intl J. Multiphase Flow 22 (3), 593612.
Balasubramaniam, R. & Subramanian, R. S. 2000 The migration of a drop in a uniform temperature gradient at large Marangoni numbers. Phys. Fluids 12, 733743.
Brown, J. W. & Churchill, R. V. 2003 Complex Variables and Applications. McGraw-Hill.
Choi, C.-H. & Kim, C.-J. 2006 Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface. Phys. Rev. Lett. 96 (6), 066001.
Crowdy, D. 2011 Frictional slip lengths and blockage coefficients. Phys. Fluids 23 (9), 091703.
Crowdy, D. 2017a Effect of shear thinning on superhydrophobic slip: perturbative corrections to the effective slip length. Phys. Rev. Fluids 2 (12), 124201.
Crowdy, D. G. 2017b Perturbation analysis of subphase gas and meniscus curvature effects for longitudinal flows over superhydrophobic surfaces. J. Fluid Mech. 822, 307326.
Feuillebois, F., Bazant, M. Z. & Vinogradova, O. I. 2009 Effective slip over superhydrophobic surfaces in thin channels. Phys. Rev. Lett. 102 (2), 026001.
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Hasimoto, H. 1958 On the flow of a viscous fluid past a thin screen at small Reynolds numbers. J. Phys. Soc. Japan 13 (6), 633639.
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 5577.
Lee, C., Choi, C.-H. & Kim, C.-J. 2008 Structured surfaces for a giant liquid slip. Phys. Rev. Lett. 101 (6), 064501.
Philip, J. R. 1972a Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (3), 353372.
Philip, J. R. 1972b Integral properties of flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. 23 (6), 960968.
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Quéré, D. 2005 Non-sticking drops. Rep. Prog. Phys. 68 (11), 24952532.
Quéré, D. 2008 Wetting and roughness. Annu. Rev. Mater. Res. 38 (1), 7199.
Schnitzer, O. & Yariv, E. 2017 Longitudinal pressure-driven flows between superhydrophobic grooved surfaces: large effective slip in the narrow-channel limit. Phys. Rev. Fluids 2 (7), 072101.
Subramanian, R. S. 1981 Slow migration of a gas bubble in a thermal gradient. AIChE J. 27, 646654.
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), 353382.
Yariv, E. 2017 Velocity amplification in pressure-driven flows between superhydrophobic gratings of small solid fraction. Soft Matt. 13, 62876292.
Yariv, E. 2018 Thermocapillary flow between longitudinally grooved superhydrophobic surfaces. J. Fluid Mech. 855, 574594.
Young, N. O., Goldstein, J. S. & Block, M. J. 1959 The motion of bubbles in a vertical temperature gradient. J. Fluid Mech. 6 (3), 350356.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed