Abstract
We examine the transient film flow under the action of gravity over solid substrates with three-dimensional topographical features. Our focus is placed on the coating of a periodic array of rectangular cuboid trenches. The Navier–Stokes equations are solved using the volume-of-fluid method, fully taking into account the flow in both the liquid and gas phases. Using this scheme, we are able to determine the different wetting patterns that may arise depending on parameters such as the various geometrical characteristics of the trench, the lateral distance between them, the substrate wettability and the liquid viscosity. We present flow maps that describe the conditions under which the liquid film may successfully coat the patterned substrate, resulting in the so-called Wenzel state, or air may become entrapped inside the topography of the substrate. In the latter case, we describe in detail the position and shape of the air inclusions, how they are formed and the conditions under which coating can approach the ideal Cassie–Baxter state. We investigate in detail the effect of the sidewalls, typically ignored when considering the case of ideal 2D trenches (i.e., trenches extending to infinity in the lateral direction), through the enhancement of the viscous resistance inside the trench and the effect of capillarity in the case of narrow trenches. We also examine the coating behavior for a wide range of liquids and show that successful coating is favored for liquids with moderate viscosities. Finally, we perform simulations for the coating of two successive trenches in the flow direction and show that in the case of 3D trenches, the differences between the coating of the first and subsequent trenches are not significant.
Similar content being viewed by others
References
Argyriadi K, Vlachogiannis M, Bontozoglou V (2006) Experimental study of inclined film flow along periodic corrugations: the effect of wall steepness. Phys Fluids 18:012102
Belyaev AV, Vinogradova OI (2010) Effective slip in pressure-driven flow past super-hydrophobic stripes. J Fluid Mech 652:489–499
Bhushan B, Jung YC, Koch K (2009) Micro-, nano- and hierarchical structures for superhydrophobicity, self-cleaning and low adhesion. Phil Trans R Soc A 367:1631–1672
Bodji MS, Kumar SV, Asthana A, Govardhan RN (2009) Underwater sustainability of the “Cassie” state of wetting. Langmuir 25:12120–12126
Bontozoglou V, Serifi K (2008) Falling film flow along steep two-dimensional topography: the effect of inertia. Int J Multiph Flow 34:734–747
Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface-tension. J Comp Phys 100:335–354
Busse A, Sandham ND, McHale G, Newton MI (2013) Change in drag, apparent slip and ptimum air layer thickness for laminar flow over an idealized superhydrophobic surface. J Fluid Mech 727:488–508
Byun D, Kim J, Ko HS, Park HC (2008) Direct measurement of slip flows in superhydrophobic microchannels with transverse grooves. Phys Fluids 20:113601
Cotin-Bizonne C, Barrat JL, Bocquet L, Charlaix E (2003) Low-friction flows of nanopatterned interfaces. Nat Mater 2:237–240
Craster RV, Matar OK (2009) Dynamics and stability of thin liquid films. Rev Mod Phys 81:1131–1198
Davies J, Maynes D, Webb BW, Woolford B (2006) Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs. Phys Fluids 18:087110
Decré MMJ, Baret JC (2003) Gravity-driven flows of viscous liquids over two-dimensional topographies. J Fluid Mech 487:147–166
Dilip D, Bodji MS, Govardhan RN (2015) Effect of absolute pressure on flow through a textured hydrophobic microchannel. Microfluid Nanofluid 19:1409–1427
Dong Z, Wu L, Li N, Ma J, Jlang L (2015) Manipulating overflow separation directions by wettability boundary positions. ACS Nano 9:6595–6602
Duez C, Ybert C, Clanet C, Bocquet L (2010) Wetting controls separation of inertial flows from solid surfaces. Phys Rev Lett 104:084503
Fraggedakis D, Kouris Ch, Dimakopoulos Y, Tsamopoulos J (2015) Flow of two immiscible fluids in a periodically constricted tube: transitions to stratified, segmented, Churn, spray or segregated flow. Phys Fluids 27:082102
Gao P, Feng JJ (2009) Enhanced slip on a patterned substrate due to depinning of the contact line. Phys Fluids 21:102102
Gaskell PH, Jimack PK, Sellier M, Thompson HM, Wilson CT (2004) Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography. J Fluid Mech 509:253–280
Goodwin R, Homsy GM (1991) Viscous flow down a slope in the vicinity of a contact line. Phys Fluids A 3(4):515–528
Gopala VR, van Wachem B (2008) Volume of fluid for immiscible-fluid and free-surface flows. Chem Eng J 141:204–221
Gramlich CM, Mazouchi A, Homsy GM (2004) Time-dependent free surface Stokes flow with a moving contact line. II. Flow over wedges and trenches. Phys Fluids 16:1660–1667
Grau G, Cen J, Kang H, Kitsomboonloha R, Scheideler WJ, Subramanian V (2016) Gravure-printed electronics: recent progress in tooling development, understanding of printing physics, and realization of printed devices. Flex Print Electron 1:023002
Hayes M, O’Brien SBG, Lammers JH (2000) Green, function for steady flow over a small two-dimensional topography. Phys Fluids 12:2845
Heining C, Aksel N (2009) Bottom reconstruction in thin-film flow over topography: steady solution and linear stability. Phys Fluids 21:083605
Higuera FJ, Medina A, Linan A (2008) Capillary rise of a liquid between two vertical plates making a small angle. Phys Fluids 20:102102
Huang C, Wang Z (2014) Planarization of high topography surfaces with deep holes and cavities using two-step polymer coating. Sens Actuators A 213:94–101
Huppert HE (1982) Flow and instability of a viscous current down a slope. Nature 300:427–429
Kalliadasis S, Bielarz C, Homsy GM (2000) Steady free surface thin film flows over topography. Phys Fluids 12:1889
Kalliadasis S, Homsy GM (2001) Stability of free-surfcae thin film flows over topography. J Fluid Mech 448:387–410
Karapetsas G, Chamakos NT, Papathanasiou AG (2016) Efficient modelling of droplet dynamics on complex surfaces. J Phys Condens Matter 28:085101
Kistler SF, Scriven LE (1994) The teapot effect: sheet-forming flows with deflection, wetting and hysteresis. J Fluid Mech 263:19–62
Kondic L, Diez J (2001) Pattern formation in the flow of thin films down an incline: constant flux configuration. Phys Fluids 13:3168
Lampropoulos NK, Dimakopoulos Y, Tsamopoulos J (2016) Transient flow of gravity-driven viscous films over substrates with rectangular topographical features. Microfluid Nanofluid 20:51
Lenz RD, Kumar S (2007) Steady two-layer flow in a topographically patterned channel. Phys Fluids 19:102103
Lv P, Xue Y, Shi Y, Lin H, Duan H (2014) Metastable states and wetting transition of submerged superhydrophobic structures. Phys Rev Lett 112:196101
Maynes D, Jeffs K, Woolford B, Webb BW (2007) Laminar flow in a microchannel with hydrophobic surface patterned microribs oriented parallel to the flow direction. Phys Fluids 19:093603
Mazloomi A, Moosavi A (2013) Thin liquid film flow over substrates with two topographical features. Phys Rev E 87:022409
Mazouchi A, Homsy GM (2001) Free surface Stokes flow over topography. Phys Fluids 13(10):2751–2761
Mazouchi A, Gramlich CM, Homsy GM (2004) Time-dependent free surface Stokes flow with a moving contact line. I. Flow over plane surfaces. Phys Fluids 16(5):1647–1659
Ou J, Rothstein JP (2005) Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces. Phys Fluids 17:103606
Park H, Park H, Kim J (2013) A numerical study of the effects of superhydrophobic surface on skin-friction drag in turbulent channel flow. Phys Fluids 25(11):110815
Pavlidis M, Dimakopoulos Y, Tsamopoulos J (2010) Steady viscoelastic film flow over 2D topography: I. The effect of viscoelastic properties under creeping flow. J Non Newt Fluid Mech 165:576–591
Pavlidis M, Karapetsas G, Dimakopoulos Y, Tsamopoulos J (2016) Steady viscoelastic film flow over 2D topography: II. The effect of capillarity, inertia and substrate geometry. J Non Newt Fluid Mech 234:201–214
Ponomarenko A, Quére D, Clanet C (2011) A universal law for capillary rise in corners. J Fluid Mech 666:146–154
Quére D (2005) Non-sticking drops. Rep Prog Phys 68:2495–2532
Rawlings C, Wolf H, Hedrick JL, Coady DJ, Duerig U, Knoll AW (2015) Accurate location and manipulation of nanoscaled objects buried under spin-coated films. ACS Nano 9:6188–6195
Rothstein JP (2010) Slip on superhydrophobic surfaces. Annu Rev Fluid Mech 42:89–109
Spaid MA, Homsy GM (1996) Stability of Newtonian and viscoelastic dynamic contact lines. Phys Fluids 8:460–478
Stillwagon LE, Larson RG (1990) Leveling of thin films over uneven substrates during spin coating. Phys Fluids 2:1937–1944
Teo CJ, Khoo BC (2010) Flow past superhydrophobic surfaces containing longitudinal grooves: effects of interface curvature. Microfluid Nanofluid 9:499–511
Troian SM, Herbolzheimer E, Safran SA, Joanny JF (1989) Fingering instabilities of driven spreading films. Europhys Lett 10:25–30
Tsai P, Peters AM, Pirat C, Wessling M, Lammertink RGH, Lohse D (2009) Quantifying effective slip length over micropatterned hydrophobic surfaces. Phys Fluids 21(11):112002
Veremieiev S, Thompson HM, Gaskell PH (2015) Free-surface film flow over topography: full three-dimensional finite element solutions. Comp Fluids 122:66–82
Wardle KE, Weller HG (2013) Hybrid Multiphase CFD solver for coupled dispersed/segregated flows in liquid–liquid extraction. Int J Chem Eng 2013:1–13
Xiang Y, Xue Y, Lv P, Li D, Duan H (2016) Influence of fluid flow on the stability and wetting transitions of submerged superhydrophobic surfaces. Soft Matter 12:4241–4246
Yin X, Kumar S (2006) Flow visualization of the liquid-emptying process in scaled-up gravure grooves and cells. Chem Eng Sci 61:1146–1156
Zhou C, Kumar S (2012) Two-dimensional two-layer channel flow near a step. Chem Eng Sci 81:38–45
Acknowledgements
This work has been supported financially by the General Secretariat of Research and Technology of Greece through the program “Excellence” (Grant No. 1918, entitled “FilCoMicrA”) in the framework “Education and Lifelong Learning” co-funded by the European Social Fund and National Resources (NL, YD and JT) and by the LIMMAT foundation under the grant MuSiComPS (GK). Part of the computations were performed on the “ARIS” National HPC Infrastructure of the Greek Research and Technology Network.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Karapetsas, G., Lampropoulos, N.K., Dimakopoulos, Y. et al. Transient flow of gravity-driven viscous films over 3D patterned substrates: conditions leading to Wenzel, Cassie and intermediate states. Microfluid Nanofluid 21, 17 (2017). https://doi.org/10.1007/s10404-017-1853-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10404-017-1853-3