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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
Article

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

  1. 1.
    van Dam DB, Le Clerc C (2004) Experimental study of the impact of an ink-jet printed droplet on a solid substrate. Phys Fluids 16(9):3403–3414ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Lembach AN, Tan HB, Roisman IV, Gambaryan-Roisman T, Zhang Y, Tropea C, Yarin AL (2010) Drop impact, spreading, splashing, and penetration into electrospun nanofiber mats. Langmuir 26(12):9516–9523CrossRefGoogle Scholar
  3. 3.
    Raza MA, van Swigchem J, Jansen HP, Zandvliet HJW, Poelsema B, Kooij ES (2014) Droplet impact on hydrophobic surfaces with hierarchical roughness. Surf Topogr 2(3):035002CrossRefGoogle Scholar
  4. 4.
    Tran T, Staat HJJ, Susarrey-Arce A, Foertsch TC, van Houselt A, Gardeniers HJGE, Prosperetti A, Lohse D, Sun C (2013) Droplet impact on superheated micro-structured surfaces. Soft Matter 9:3272–3282ADSCrossRefGoogle Scholar
  5. 5.
    Tsai P, van der Veen RCA, van de Raa M, Lohse D (2010) How micropatterns and air pressure affect splashing on surfaces. Langmuir 26(20):16090–16095CrossRefGoogle Scholar
  6. 6.
    Han D, Steckl AJ (2009) Superhydrophobic and oleophobic fibers by coaxial electrospinning. Langmuir 25(16):9454–9462CrossRefGoogle Scholar
  7. 7.
    Srikar R, Gambaryan-Roisman T, Steffes C, Stephan P, Tropea C, Yarin AL (2009) Nanofiber coating of surfaces for intensification of drop or spray impact cooling. Int J Heat Mass Transf 52(25–26):5814–5826CrossRefzbMATHGoogle Scholar
  8. 8.
    Delbos A, Lorenceau E, Pitois O (2010) Forced impregnation of a capillary tube with drop impact. J Colloid Interface Sci 341(1):171–177Google Scholar
  9. 9.
    Ding H, Theofanous TG (2012) The inertial regime of drop impact on an anisotropic porous substrate. J Fluid Mech 691:546–567ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Maitra T, Antonini C, Tiwari MK, Mularczyk A, Imeri Z, Schoch P, Poulikakos D (2014) Supercooled water drops impacting superhydrophobic textures. Langmuir 30(36):10855–10861CrossRefGoogle Scholar
  11. 11.
    Maitra T, Tiwari MK, Antonini C, Schoch P, Jung S, Eberle P, Poulikakos D (2014) On the nanoengineering of superhydrophobic and impalement resistant surface textures below the freezing temperature. Nano Lett 14(1):172–182ADSCrossRefGoogle Scholar
  12. 12.
    Chandra S, Avedisian CT (1991) On the collision of a droplet with a solid surface. Proc R Soc Lond A 432(1884):13–41ADSCrossRefGoogle Scholar
  13. 13.
    Thoroddsen ST, Etoh TG, Takehara K (2003) Air entrapment under an impacting drop. J Fluid Mech 478:125–134ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Thoroddsen ST, Etoh TG, Takehara K, Ootsuka N, Hatsuki Y (2005) The air bubble entrapped under a drop impacting on a solid surface. J Fluid Mech 545:203–212ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Driscoll MM, Nagel SR (2011) Ultrafast interference imaging of air in splashing dynamics. Phys Rev Lett 107(15):154502ADSCrossRefGoogle Scholar
  16. 16.
    Liu Y, Tan P, Xu L (2013) Compressible air entrapment in high-speed drop impacts on solid surfaces. J Fluid Mech 716:R9ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    de Ruiter J, van den Ende D, Mugele F (2015) Air cushioning in droplet impact. II. Experimental characterization of the air film evolution. Phys Fluids 27(1):012105ADSCrossRefGoogle Scholar
  18. 18.
    Tran T, de Maleprade H, Sun C, Lohse D (2013) Air entrainment during impact of droplets on liquid surfaces. J Fluid Mech 726:R6CrossRefzbMATHGoogle Scholar
  19. 19.
    van der Veen RCA, Tran T, Lohse D, Sun C (2012) Direct measurements of air layer profiles under impacting droplets using high-speed color interferometry. Phys Rev E 85:026315ADSCrossRefGoogle Scholar
  20. 20.
    Dell’Aversana P, Tontodonato V, Carotenuto L (1997) Suppression of coalescence and of wetting: the shape of the interstitial film. Phys Fluids 9(9):2475–2485ADSCrossRefGoogle Scholar
  21. 21.
    Cooker MJ (2012) A theory for the impact of a wave breaking onto a permeable barrier with jet generation. J Eng Math 79:1–12MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Iafrati A, Korobkin AA (2005) Self-similar solutions for porous/perforated wedge entry problem. In: Proceedings of the 20th international workshop on water waves and floating bodies, Longyearbyen, Norway, 29 May – 1 June 2005Google Scholar
  23. 23.
    Hicks PD, Purvis R (2010) Air cushioning and bubble entrapment in three-dimensional droplet impacts. J Fluid Mech 649:135–163ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Purvis R, Smith FT (2004) Air–water interactions near droplet impact. Eur J Appl Math 15:853–871MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Smith FT, Li L, Wu GX (2003) Air cushioning with a lubrication/inviscid balance. J Fluid Mech 482:291–318ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fitt AD, Howell PD, King JR, Please CP, Schwendeman DW (2002) Multiphase flow in a roll press nip. Eur J Appl Math 13(3):225–259Google Scholar
  27. 27.
    Knox DJ, Wilson SK, Duffy BR, McKee S (2015) Porous squeeze-film flow. IMA J Appl Math 80(2):376–409MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hicks PD, Purvis R (2011) Air cushioning in droplet impacts with liquid layers and other droplets. Phys Fluids 23(6):062104ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Mandre S, Mani M, Brenner MP (2009) Precursors to splashing of liquid droplets on a solid surface. Phys Rev Lett 102(13):134502ADSCrossRefGoogle Scholar
  30. 30.
    Hicks PD, Purvis R (2013) Liquid–solid impacts with compressible gas cushioning. J Fluid Mech 735:120–149ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Beavers GS, Joseph DD (1967) Boundary conditions at a naturally permeable wall. J Fluid Mech 30:197–207ADSCrossRefGoogle Scholar
  32. 32.
    Nield DA (2009) The Beavers–Joseph boundary condition and related matters: a historical and critical note. Transp Porous Med 78(3):537–540MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wilson SK (1991) A mathematical model for the initial stages of fluid impact in the presence of a cushioning fluid layer. J Eng Math 25(3):265–285MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Bouwhuis W, van der Veen RCA, Tran T, Keij DL, Winkels KG, Peters IR, van der Meer D, Sun C, Snoeijer JH, Lohse D (2012) Maximal air bubble entrainment at liquid-drop impact. Phys Rev Lett 109:264501ADSCrossRefGoogle Scholar
  35. 35.
    Alizadeh A, Bahadur V, Zhong S, Shang W, Li R, Ruud J, Yamada M, Ge L, Dhinojwala A, Sohal M (2012) Temperature dependent droplet impact dynamics on flat and textured surfaces. Appl Phys Lett 100(11):111601ADSCrossRefGoogle Scholar
  36. 36.
    Liu Y, Moevius L, Xu X, Qian T, Yeomans JM, Wang Z (2014) Pancake bouncing on superhydrophobic surfaces. Nat Phys 10:515–519CrossRefGoogle Scholar
  37. 37.
    van der Veen RCA, Hendrix MHW, Tran T, Sun C, Tsai PA, Lohse D (2014) How microstructures affect air film dynamics prior to drop impact. Soft Matter 10:3703–3707ADSCrossRefGoogle Scholar
  38. 38.
    Ellis AS, Smith FT, White AH (2011) Droplet impact onto a rough surface. Q J Mech Appl Math 64(2):107–139CrossRefzbMATHGoogle Scholar
  39. 39.
    Saffman PG (1971) On the boundary condition at the surface of a porous medium. Stud Appl Math 50:93–101CrossRefzbMATHGoogle Scholar
  40. 40.
    Hicks PD, Ermanyuk EV, Gavrilov NV, Purvis R (2012) Air trapping at impact of a rigid sphere onto a liquid. J Fluid Mech 695:310–320ADSCrossRefzbMATHGoogle Scholar
  41. 41.
    Le Bars M, Worster MG (2006) Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J Fluid Mech 550:149–173ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Nabhani M, El Khlifi M, Bou-saïd B (2010) A numerical simulation of viscous shear effects on porous squeeze-film using the Darcy–Brinkman model. Mech Ind 11:327–337Google Scholar
  43. 43.
    Jones IP (1973) Low Reynolds number flow past a porous spherical shell. Math Proc Camb Philos Soc 73:231–238ADSCrossRefzbMATHGoogle Scholar
  44. 44.
    King FW (2009) Hilbert transforms. Encyclopedia of mathematics and its applications, vol 1. Cambridge University Press, CambridgeGoogle Scholar

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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