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On the degree of wetting of a slit by a liquid film flowing along an inclined plane

Published online by Cambridge University Press:  02 May 2017

D. Pettas Open the ORCID record for D. Pettas [Opens in a new window] ,
G. Karapetsas ,
Y. Dimakopoulos  and
J. Tsamopoulos Open the ORCID record for J. Tsamopoulos [Opens in a new window]
D. Pettas
Affiliation:
Laboratory of Fluid Mechanics and Rheology, Department of Chemical Engineering, University of Patras, Patras 26500, Greece
G. Karapetsas
Affiliation:
Laboratory of Fluid Mechanics and Rheology, Department of Chemical Engineering, University of Patras, Patras 26500, Greece
Y. Dimakopoulos
Affiliation:
Laboratory of Fluid Mechanics and Rheology, Department of Chemical Engineering, University of Patras, Patras 26500, Greece
J. Tsamopoulos*
Affiliation:
Laboratory of Fluid Mechanics and Rheology, Department of Chemical Engineering, University of Patras, Patras 26500, Greece
*
†Email address for correspondence: [email protected]
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Abstract

Liquid film flow along an inclined plane featuring a slit, normal to the main direction of flow, creates a second gas–liquid interface connecting the two side walls of the slit. This inner interface forms two three-phase contact lines and supports a widely varying amount of liquid under different physical and geometrical conditions. The exact liquid configuration is determined by employing the Galerkin/finite element method to solve the two-dimensional Navier–Stokes equations at steady state. The interplay of inertia, viscous, gravity and capillary forces along with the substrate wettability and orientation with respect to gravity and the width of the slit determine the extent of liquid penetration and free-surface deformation. Finite wetting lengths are predicted in hydrophilic and hydrophobic substrates for inclination angles more or less than the vertical, respectively. Multiple steady solutions, connected by turning points forming a hysteresis loop, are revealed by pseudo-arclength continuation. Under these conditions, small changes in certain parameter values leads to an abrupt change in the wetting length and the deformation amplitude of the outer film surface. In hydrophilic substrates the wetting lengths exhibit a local minimum for small values of the Reynolds number and a very small range of Bond numbers; when inertia increases, they exhibit the hysteresis loop with the second limit point in a very short range of Weber numbers. Simple force balances determine the proper rescaling in each case, so that critical points in families of solutions for different liquids or contact angles collapse. The flow inside the slit is quite slow in general because of viscous dissipation and includes counter-rotating vortices often resembling those reported by Moffatt (J. Fluid Mech., vol. 18, 1964, pp. 1–18). In hydrophobic substrates, the wetting lengths decrease monotonically until the first limit point of the hysteresis loop, which occurs in a limited range of Bond numbers when the Kapitza number is less than 300 and in a limited range of Weber numbers otherwise. Here additional solution families are possible as well, where one or both contact points (Cassie state) coincide with the slit corners.

JFM classification

Materials Processing Flows: Coating Micro-/Nano-fluid dynamics: Microfluidics Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 820 , 10 June 2017 , pp. 5 - 41
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© 2017 Cambridge University Press 

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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.Google Scholar
Belyaev, A. V. & Vinogradova, O. I. 2010 Effective slip in pressure-driven flow past super-hydrophobic stripes. J. Fluid Mech. 652, 489–499.Google Scholar
Bhushan, B., Jung, Y. C. & Koch, K. 2009 Micro-, nano- and hierarchical structures for superhydrophobicity, self-cleaning and low adhesion. Phil. Trans. R. Soc. Lond. A 367, 1631–1672.Google Scholar
Bielarz, C. & Kalliadasis, S. 2003 Time-dependent free-surface thin film flows over topography. Phys. Fluids 15, 2512–2524.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.CrossRefGoogle Scholar
Bontozoglou, V. & Serifi, K. 2008 Falling film flow a ong steep two-dimensional topography: the effect of inertia. Intl J. Multiphase Flow 34, 734–747.Google Scholar
Busse, A., Sandham, N. D., Mchale, G. & Newton, M. I. 2013 Change in drag, apparent slip and optimum air layer thickness for laminar flow over an idealised superhydrophobic surface. J. Fluid Mech. 727, 488–508.Google Scholar
Byun, D., Kim, J., Ko, H. S. & Park, H. C. 2008 Direct measurement of slip flows in superhydrophobic microchannels with transverse grooves. Phys. Fluids 20, 113601.Google Scholar
Cottin-Bizonne, C., Barrat, J.-L., Bocquet, L. & Charlaix, E. 2003 Low-friction flows of liquid at nanopatterned interfaces. Nat. Mater. 2, 238–240.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.Google Scholar
Davies, J., Maynes, D., Webb, B. W. & Woolford, B. 2006 Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs. Phys. Fluids 18, 087110.Google Scholar
Dean, W. R. & Montagnon, P. E. 1949 On the steady motion of viscous liquid in a corner. Proc. Camb. Phil. Soc. 45, 389–394.Google Scholar
Decre, M. M. J. & Baret, J.-C. 2003 Gravity-driven flows of viscous liquids over two-dimensional topographies. J. Fluid Mech. 487, 147–166.Google Scholar
Dilip, D., Bobji, M. S. & Govardhan, R. N. 2015 Effect of absolute pressure on flow through a textured hydrophobic microchannel. Microfluid. Nanofluid. 19, 1409–1427.Google Scholar
Dimakopoulos, Y. & Tsamopoulos, J. 2003 A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations. J. Comput. Phys. 192, 494–522.Google Scholar
Dimakopoulos, Y. & Tsamopoulos, J. 2009 On the transient coating of a straight tube with a viscoelastic material. J. Non-Newtonian Fluid Mech. 159 (1–3), 95–114.Google Scholar
Duez, C., Ybert, C., Clanet, C. & Bocquet, L. 2010 Wetting controls separation of inertial flows from solid surfaces. Phys. Rev. Lett. 104, 084503.Google Scholar
Fraggedakis, D., Pavlidis, M., Dimakopoulos, Y. & Tsamopoulos, J. 2016 On the velocity discontinuity at a critical volume of a bubble rising in a viscoelastic fluid. J. Fluid Mech. 789, 310–346.Google Scholar
Gao, P. & Feng, J. J. 2009 Enhanced slip on a patterned substrate due to depinning of contact line. Phys. Fluids 21, 102102.Google Scholar
Gaskell, P. H., Jimack, P. K., Sellier, M., Thompson, H. M. & Wilson, M. C. T. 2004 Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography. J. Fluid Mech. 509, 253–280.Google Scholar
Georgiou, G., Schultz, W. & Olson, L. G. 1990 Singular finite elements for the sudden-expansion and the die-swell problems. Intl J. Numer. Meth. Fluids 10, 357–372.Google Scholar
Gibbs, J. W. 1906 The scientific papers by J. Willard Gibbs. Thermodynamics. vol. I. Longmans Green and Co. (Dover Reprint, 1961).Google Scholar
Gramlich, C. M., Mazouchi, A. & Homsy, G. M. 2004 Time-dependent free surface Stokes flow with a moving contact line. II. Flow over wedges and trenches. Phys. Fluids 16, 1660–1667.Google Scholar
Grau, G., Cen, J., Kang, H., Kitsomboonloha, R., Scheideler, W. J. & Subramanian, V. 2016 Gravure-printed electronics: recent progress in tooling development, understanding of printing physics, and realization of printed devices. Flexible Printed Electron. 1, 023002.Google Scholar
GUPTA2000 WSMP: Watson Sparse Matrix Package Part II – direct solution of general sparse systems.Google Scholar
Heining, C., Bontozoglou, V., Aksel, N. & Wierschem, A. 2009 Nonlinear resonance in viscous films on inclined wavy planes. Intl J. Multiphase Flow 35, 78–90.Google Scholar
Hodes, M., Lam, L., Cowley, A., Enright, R. & Maclachlan, S. 2015 Effect of evaporation and condensation at menisci on apparent thermal slip. Trans. ASME J. Heat Transfer 137, 071502.Google Scholar
Huang, C. & Wang, Z. 2014 Planarization of high topography surfaces with deep holes and cavities using two-step polymer coating. Sensors Actuators A 213, 94–101.Google Scholar
Iooss, G. & Joseph, D. D. 1990 Elementary Stability and Bifurcation Theory. Springer.Google Scholar
Kalliadasis, S., Bielarz, C. & Homsy, G. M. 2000 Steady free-surface thin film flows over topography. Phys. Fluids 12, 1889–1898.Google Scholar
Kalliadasis, S. & Homsy, G. M. 2001 Stability of free-surface thin-film flows over topography. J. Fluid Mech. 448, 387–410.Google Scholar
Karapetsas, G., Lampropoulos, N. K., Dimakopoulos, Y. & Tsamopoulos, J. 2017 Transient flow of gravity-driven viscous films over 3D patterned substrates: conditions leading to Wenzel, Cassie and intermediate states. Microfluid. Nanofluid. 21, 17; doi:10.1007/s10404-017-1853-3.Google Scholar
Kistler, S. & Scriven, L. E. 1994 The teapot effect: sheet-forming flows with deflection, wetting and hysteresis. J. Fluid Mech. 263, 19–62.Google Scholar
Lampropoulos, N. K., Dimakopoulos, Y. & Tsamopoulos, J. 2016 Transient flow of gravity-driven viscous films over substrates with rectangular topographical features. Microfluid. Nanofluid. 20, 51.Google Scholar
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.CrossRefGoogle ScholarPubMed
Marston, J. O., Thoroddsen, S. T., Thompson, J., Blyth, M. G., Henry, D. & Uddin, J. 2014 Experimental investigation of hysteresis in the break-up of liquid curtains. Chem. Engng Sci. 117, 248–263.Google Scholar
Maynes, D., Jeffs, K., Woolford, B. & Webb, B. W. 2007 Laminar flow in a microchannel with hydrophobic surface patterned microribs oriented parallel to the flow direction. Phys. Fluids 19, 093603.Google Scholar
Mazouchi, A., Gramlich, C. M. & Homsy, G. M. 2004 Time-dependent free surface Stokes flow with a moving contact line. I. Flow over plane surfaces. Phys. Fluids 16, 1647–1659.Google Scholar
Mazouchi, A. & Homsy, G. M. 2001 Free surface Stokes flow over topography. Phys. Fluids 13, 2751–2761.Google Scholar
Michael, D. H. 1958 The separation of a viscous liquid at a straight edge. Cambridge core. Mathematika 5 (1), 82–84.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.Google Scholar
Moulinet, S. & Bartolo, D. 2007 Life and death of a fakir droplet: impalement transitions on superhydrophobic surfaces. Eur. Phys. J. E 24, 251–260.Google Scholar
Nguyen, P. K. & Bontozoglou, V. 2011 Steady solutions of inertial film flow along strongly undulated substrates. Phys. Fluids 23, 052103.Google Scholar
Ou, J., Perot, B. & Rothstein, J. P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16, 4635–4643.Google Scholar
Ou, J. & Rothstein, J. P. 2005 Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces. Phys. Fluids 17, 103606.Google Scholar
Papaioannou, J., Giannousakis, A., Dimakopoulos, Y. & Tsamopoulos, J. 2014 Bubble deformation and growth inside viscoelastic filaments undergoing very large extensions. Indust. Engng Chem. Res. 53, 7548–7569.Google Scholar
Park, J., Park, J., Lim, H. & Kim, H.-Y. 2013 Shape of a large drop on a rough hydrophobic surface. Phys. Fluids 25, 022102.Google Scholar
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-Newtonian Fluid Mech. 165, 576–591.Google Scholar
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-Newtonian Fluid Mech. 234, 201–214.Google Scholar
Qianwen, C., Cui, H. & Zheyao, W. 2012 Benzocyclobutene polymer filling of high aspect-ratio annular trenches for fabrication of Through-Silicon-Vias (TSVs). Microelectronics Reliability 52, 2670–2676.Google Scholar
Quéré, D. 2008 Wetting and Roughness. Annu. Rev. Mater. Res. 38, 71–99.Google Scholar
Richardson, S. 1970 The die swell phenomenon. Rheol. Acta 9, 193–199.Google Scholar
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42, 89–109.Google Scholar
Saprykin, S., Koopmans, R. J. & Kalliadasis, S. 2007 Free-surface thin-film flows over topography: influence of inertia and viscoelasticity. J. Fluid Mech. 578, 271.Google Scholar
Seydel, R. 2010 Practical Bifurcation and Stability Analysis. Springer.Google Scholar
Stillwagon, L. E. & Larson, R. G. 1990 Leveling of thin films over uneven substrates during spin coating. Phys. Fluids A 2, 1937.Google Scholar
Teo, C. J. & Khoo, B. C. 2010 Flow past superhydrophobic surfaces containing longitudinal grooves: effects of interface curvature. Microfluid. Nanofluid. 9, 499–511.Google Scholar
Tsai, P., Peters, A. M., Pirat, C., Wessling, M., Lammertink, R. G. H. & Lohse, D. 2009 Quantifying effective slip length over micropatterned hydrophobic surfaces. Phys. Fluids 21, 112002.Google Scholar
Tsouka, S., Dimakopoulos, Y. & Tsamopoulos, J. 2016 Stress-gradient induced migration of polymers in thin films flowing over smoothly corrugated surfaces. J. Non-Newtonian Fluid Mech. 228, 79–95.Google Scholar
Wierschem, A. & Aksel, N. 2004 Influence of inertia on eddies created in films creeping over strongly undulated substrates. Phys. Fluids 16, 4566.Google Scholar
Xiang, Y., Xue, Y., Lv, P., Li, D. & Duan, H. 2016 Influence of fluid flow on the stability and wetting transition of submerged superhydrophobic surfaces. Soft Matt. 12, 4241–4246.Google Scholar
Yin, X. & Kumar, S. 2006 Two-dimensional simulations of flow near a cavity and a flexible solid boundary. Phys. Fluids 18, 063103.Google Scholar
Zacharioudaki, M., Kouris, C., Dimakopoulos, Y. & Tsamopoulos, J. 2007 A direct comparison between volume and surface tracking methods with a boundary-fitted coordinate transformation and third-order upwinding. J. Comput. Phys. 227, 1428–1469.Google Scholar