Journal of Engineering Mathematics

, Volume 102, Issue 1, pp 65–87 | Cite as

Gas-cushioned droplet impacts with a thin layer of porous media

  • Peter D. Hicks
  • Richard Purvis


The pre-impact gas cushioning behaviour of a droplet approaching touchdown onto a thin layer of porous substrate is investigated. Although the model is applicable to droplet impacts with any porous substrate of limited height, a thin layer of porous medium is used as an idealized approximation of a regular array of pillars, which are frequently used to produced superhydrophobic- and superhydrophilic-textured surfaces. Bubble entrainment is predicted across a range of permeabilities and substrate heights, as a result of a gas pressure build-up in the viscous-gas squeeze film decelerating the droplet free-surface immediately below the centre of the droplet. For a droplet of water of radius 1 mm and impact approach speed 0.5 m s\(^{-1}\), the change from a flat rigid impermeable plate to a porous substrate of height \(5~\upmu \)m and permeability \(2.5~\upmu \)m\(^2\) reduces the initial horizontal extent of the trapped air pocket by \(48~\%\), as the porous substrate provides additional pathways through which the gas can escape. Further increases in either the substrate permeability or substrate height can entirely eliminate the formation of a trapped gas pocket in the initial touchdown phase, with the droplet then initially hitting the top surface of the porous media at a single point. Droplet impacts with a porous substrate are qualitatively compared to droplet impacts with a rough impermeable surface, which provides a second approximation for a textured surface. This indicates that only small pillars can be successfully modelled by the porous media approximation. The effect of surface tension on gas-cushioned droplet impacts with porous substrates is also investigated. In contrast to the numerical predictions of a droplet free-surface above flat plate, when a porous substrate is included, the droplet free-surface touches down in finite time. Mathematically, this is due to the regularization of the parabolic degeneracy associated with the small gas-film-height limit the gas squeeze film equation, by non-zero substrate permeability and height, and physically suggests that the level of surface roughness is a critical parameter in determining the initial touchdown characteristics.


Droplet impacts Gas entrainment Porous media 

Mathematics Subject Classification

76T10 76D09 76B45 



The authors are grateful to Dr. Manish Tiwari for introducing them to experiments involving droplet impacts with textured substrates. PDH is grateful for the use of the Maxwell High-Performance Computing Cluster of the University of Aberdeen IT Service. RP is grateful for the use of the High-Performance Computing Cluster supported by the Research and Specialist Computing Support service at the University of East Anglia.

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Engineering, Fraser Noble Building, King’s CollegeUniversity of AberdeenAberdeenUK
  2. 2.School of MathematicsUniversity of East AngliaNorwichUK

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