# Pressure-driven plug flows between superhydrophobic surfaces of closely spaced circular bubbles

- 193 Downloads

## Abstract

Shear-driven flows over superhydrophobic surfaces formed of closely spaced circular bubbles are characterized by giant longitudinal slip lengths, viz., large compared with the periodicity (Schnitzer, Phys Rev Fluids 1(5):052101, 2016). This hints towards a strong superhydrophobic effect in the concomitant scenario of pressure-driven flow between two such surfaces, particularly for non-wide channels where bubble-to-bubble pitch and bubble radius are commensurate with channel width. We show here that such pressure-driven flows can be analyzed asymptotically and in closed form based on the smallness of the gaps separating the bubbles relative to the channel width (and bubble radius). We find that the flow adopts an unconventional plug profile away from the inter-bubble gaps, with the uniform velocity being asymptotically larger than the corresponding Poiseuille scale. For a given solid fraction and channel width, the net volumetric flux is maximized when the length of each semi-circular bubble-liquid interface is equal to the channel width. The plug flow identified herein cannot be obtained via a naive implementation of a Navier condition, which is indeed inapplicable for non-wide channels.

## Keywords

Lubrication approximation Singular perturbations Stokes flows## Notes

### Acknowledgements

Ehud Yariv was supported by the Israel Science Foundation (Grant No. 1081/16).

## References

- 1.Ou J, Perot B, Rothstein JP (2004) Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys Fluids 16:4635–4643CrossRefMATHGoogle Scholar
- 2.Ou J, Rothstein JP (2005) Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces. Phys Fluids 17:103606–11CrossRefMATHGoogle Scholar
- 3.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 Soft Matter 15:427–438CrossRefGoogle Scholar
- 4.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 19:123601CrossRefMATHGoogle Scholar
- 5.Davis AMJ, Lauga E (2010) Hydrodynamic friction of fakir-like superhydrophobic surfaces. J Fluid Mech 661:402–411CrossRefMATHGoogle Scholar
- 6.Lauga E, Stone HA (2003) Effective slip in pressure-driven stokes flow. J Fluid Mech 489:55–77MathSciNetCrossRefMATHGoogle Scholar
- 7.Philip JR (1972) Flows satisfying mixed no-slip and no-shear conditions. Z Angew Math Phys 23:353–372MathSciNetCrossRefMATHGoogle Scholar
- 8.Schnitzer O (2016) Singular effective slip length for longitudinal flow over a dense bubble mattress. Phys Rev Fluids 1(5):052101CrossRefGoogle Scholar
- 9.Karatay E, Haase AS, Visser CW, Sun C, Lohse D, Tsai PA, Lammertink RGH (2013) Control of slippage with tunable bubble mattresses. Proc Natl Acad Sci USA 110:8422–8426CrossRefGoogle Scholar
- 10.Schnitzer O (2017) Slip length for longitudinal shear flow over an arbitrary-protrusion-angle bubble mattress: the small-solid-fraction singularity. J Fluid Mech 820:580–603MathSciNetCrossRefMATHGoogle Scholar
- 11.Choi C-H, Kim C-J (2006) Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface. Phys Rev Lett 96:066001CrossRefGoogle Scholar
- 12.Lee C, Choi C-H, Kim C-J (2008) Structured surfaces for a giant liquid slip. Phys Rev Lett 101:064501CrossRefGoogle Scholar
- 13.Marshall JS (2017) Exact formulae for the effective slip length of a symmetric superhydrophobic channel with flat or weakly curved menisci. SIAM J Appl Math 77:1606–1630MathSciNetCrossRefMATHGoogle Scholar
- 14.Feuillebois F, Bazant MZ, Vinogradova OI (2009) Effective slip over superhydrophobic surfaces in thin channels. Phys Rev Lett 102:026001CrossRefGoogle Scholar
- 15.Schnitzer O, Yariv E (2017) Longitudinal pressure-driven flows between superhydrophobic grooved surfaces: large effective slip in the narrow-channel limit. Phys Rev Fluids 2:072101CrossRefGoogle Scholar
- 16.Yariv E (2017) Velocity amplification in pressure-driven flows between superhydrophobic gratings of small solid fraction. Soft Matter 13:6287–6292CrossRefGoogle Scholar
- 17.Sbragaglia M, Prosperetti A (2007) A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys Fluids 19:043603CrossRefMATHGoogle Scholar
- 18.Teo CJ, Khoo BC (2009) Analysis of stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves. Microfluid Nanofluid 7:353–382CrossRefGoogle Scholar
- 19.Belyaev AV, Vinogradova OI (2010) Hydrodynamic interaction with super-hydrophobic surfaces. Soft Matter 6:4563–4570CrossRefGoogle Scholar
- 20.Teo CJ, Khoo BC (2010) Flow past superhydrophobic surfaces containing longitudinal grooves: effects of interface curvature. Microfluid Nanofluid 9:499–511CrossRefGoogle Scholar
- 21.Teo CJ, Khoo BC (2014) Effects of interface curvature on poiseuille flow through microchannels and microtubes containing superhydrophobic surfaces with transverse grooves and ribs. Microfluid Nanofluid 17:891–905CrossRefGoogle Scholar
- 22.Hinch EJ (1991) Perturbation methods. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
- 23.Pozrikidis C (1992) Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar