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

## Abstract

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.

## Keywords

Droplet impacts Gas entrainment Porous media## Mathematics Subject Classification

76T10 76D09 76B45## Notes

### Acknowledgments

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.

## Supplementary material

## References

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