Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-27T17:14:25.410Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial drag-out problem: sheets and films on a rotating disc

Published online by Cambridge University Press:  03 December 2020

J. John Soundar Jerome Open the ORCID record for J. John Soundar Jerome [Opens in a new window] ,
Sébastien Thevenin ,
Mickaël Bourgoin Open the ORCID record for Mickaël Bourgoin [Opens in a new window]  and
Jean-Philippe Matas Open the ORCID record for Jean-Philippe Matas [Opens in a new window]
J. John Soundar Jerome*
Affiliation:
Univ Lyon, Université Claude Bernard Lyon 1, Laboratoire de Mécanique des Fluides et d'Acoustique, CNRS UMR-5509, Boulevard 11 novembre 1918, F-69622 Villeurbanne CEDEX, LYON, France
Sébastien Thevenin
Affiliation:
Univ Lyon, Université Claude Bernard Lyon 1, Laboratoire de Mécanique des Fluides et d'Acoustique, CNRS UMR-5509, Boulevard 11 novembre 1918, F-69622 Villeurbanne CEDEX, LYON, France
Mickaël Bourgoin
Affiliation:
Laboratoire de Physique, Universite Lyon, ENS de Lyon, Universite Lyon 1, CNRS, F-69342Lyon, France
Jean-Philippe Matas
Affiliation:
Univ Lyon, Université Claude Bernard Lyon 1, Laboratoire de Mécanique des Fluides et d'Acoustique, CNRS UMR-5509, Boulevard 11 novembre 1918, F-69622 Villeurbanne CEDEX, LYON, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The so-called Landau–Levich–Deryaguin problem treats the coating flow dynamics of a thin viscous liquid film entrained by a moving solid surface. In this context, we use a simple experimental set-up consisting of a partially immersed rotating disc in a liquid tank to study the role of inertia, and also curvature, on liquid entrainment. Using water and UCON$^{\textrm{TM} }$ mixtures, we point out a rich phenomenology in the presence of strong inertia: ejection of multiple liquid sheets on the emerging side of the disc, sheet fragmentation, ligament formation and atomization of the liquid flux entrained over the disc's rim. We focus our study on a single liquid sheet and the related average liquid flow rate entrained over a thin disc for various depth-to-radius ratio $h/R < 1$. We show that the liquid sheet is created via a ballistic mechanism as liquid is lifted out of the pool by the rotating disc. We then show that the flow rate in the entrained liquid film is controlled by both viscous and surface tension forces as in the classical Landau–Levich–Deryaguin problem, despite the three-dimensional, non-uniform and unsteady nature of the flow, and also despite the large values of the film thickness based flow Reynolds number. When the characteristic Froude and Weber numbers become significant, strong inertial effects influence the entrained liquid flux over the disc at large radius-to-immersion-depth ratio, namely via entrainment by the disc's lateral walls and via a contribution to the flow rate extracted from the three-dimensional liquid sheet itself, respectively.

JFM classification

Materials Processing Flows: Coating Interfacial Flows (free surface): Thin films Multiphase and Particle-laden Flows: Gas/liquid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , A7
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adachi, K., Tamura, T. & Nakamura, R. 1988 Coating flows in a nip region and various critical phenomena. AIChE J. 34 (3), 456–464.CrossRefGoogle Scholar
Ascanio, G., Carreau, P. J., Brito-De La Fuente, E. & Tanguy, P. A. 2004 Forward deformable roll coating at high speed with Newtonian fluids. Chem. Engng Res. Des. 82 (3), 390–397.CrossRefGoogle Scholar
Benilov, E. S., Chapman, S. J., Mcleod, J. B., Ockendon, J. R. & Zubkov, V. S. 2010 On liquid films on an inclined plate. J. Fluid Mech. 663, 53–69.CrossRefGoogle Scholar
Benjamin, D. F., Anderson, T. J. & Scriven, L. E. 1995 Multiple roll systems: steady-state operation. AIChE J. 41 (5), 1045–1060.CrossRefGoogle Scholar
Brown, P. P. & Lawler, D. F. 2003 Sphere drag and settling velocity revisited. J. Environ. Engng 129 (3), 222–231.CrossRefGoogle Scholar
Campana, D. M., Ubal, S., Giavedoni, M. D. & Saita, F. A. 2010 Numerical prediction of the film thickening due to surfactants in the Landau–Levich problem. Phys. Fluids 22 (3), 032103.CrossRefGoogle Scholar
Campana, D. M., Ubal, S., Giavedoni, M. D. & Saita, F. A. 2011 A deeper insight into the dip coating process in the presence of insoluble surfactants: a numerical analysis. Phys. Fluids 23 (5), 052102.CrossRefGoogle Scholar
Campanella, O. H. & Cerro, R. L. 1984 Viscous flow on the outside of a horizontal rotating cylinder: the roll coating regime with a single fluid. Chem. Engng Sci. 39 (10), 1443–1449.CrossRefGoogle Scholar
Cerro, R. L. & Scriven, L. E. 1980 Rapid free surface film flows. An integral approach. Ind. Engng Chem. Fundam. 19 (1), 40–50.CrossRefGoogle Scholar
Chalmers, B. 1937 Surface tension and viscosity phenomena in tinplate manufacture. Trans. Faraday Soc. 33, 1167–1176.CrossRefGoogle Scholar
Cohen, E. D. & Gutoff, E. B. 1992 Modern Coating and Drying Technology, vol. 1. VCH New York.Google Scholar
Coyle, D. J., Macosko, C. W. & Scriven, L. E. 1990 Stability of symmetric film-splitting between counter-rotating cylinders. J. Fluid Mech. 216, 437–458.CrossRefGoogle Scholar
Culick, F. E. C. 1960 Comments on a ruptured soap film. J. Appl. Phys. 31, 1128–1129.CrossRefGoogle Scholar
De Ryck, A. & Quéré, D. 1994 Quick forced spreading. Europhys. Lett. 25 (3), 187.CrossRefGoogle Scholar
De Ryck, A. & Quéré, D. 1996 Inertial coating of a fibre. J. Fluid Mech. 311, 219–237.CrossRefGoogle Scholar
De Ryck, A. & Quéré, D. 1998 a Fluid coating from a polymer solution. Langmuir 14 (7), 1911–1914.CrossRefGoogle Scholar
De Ryck, A. & Quéré, D. 1998 b Gravity and inertia effects in plate coating. J. Colloid Interface. Sci. 203 (2), 278–285.CrossRefGoogle Scholar
Deryaguin, B. V. 1943 Thickness of liquid layer adhering to walls of vessels on their emptying and the theory of photo-and motion-picture film coating. In Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS, vol. 39, pp. 13–16.Google Scholar
Deryaguin, B. V. & Levi, S. M. 1964 Film Coating Theory. Focal Press.Google Scholar
Dixit, H. N. & Homsy, G. M. 2013 The elastic Landau–Levich problem. J. Fluid Mech. 732, 5–28.CrossRefGoogle Scholar
Evans, P. L., Schwartz, L. W. & Roy, R. V. 2004 Steady and unsteady solutions for coating flow on a rotating horizontal cylinder: two-dimensional theoretical and numerical modeling. Phys. Fluids 16 (8), 2742–2756.CrossRefGoogle Scholar
Evans, P. L., Schwartz, L. W. & Roy, R. V. 2005 Three-dimensional solutions for coating flow on a rotating horizontal cylinder: theory and experiment. Phys. Fluids 17 (7), 072102.CrossRefGoogle Scholar
Filali, A., Khezzar, L. & Mitsoulis, E. 2013 Some experiences with the numerical simulation of Newtonian and Bingham fluids in dip coating. Comput. Fluids 82, 110–121.CrossRefGoogle Scholar
Gans, A., Dressaire, E., Colnet, B., Saingier, G., Bazant, M. Z. & Sauret, A. 2019 Dip-coating of suspensions. Soft Matter 15 (2), 252–261.CrossRefGoogle ScholarPubMed
Gaskell, P. H., Innes, G. E. & Savage, M. D. 1998 An experimental investigation of meniscus roll coating. J. Fluid Mech. 355, 17–44.CrossRefGoogle Scholar
Goucher, F. S. & Ward, H. 1922 The thickness of liquid films formed on solid surfaces under dynamic conditions. Phil. Mag. 44, 1002–1014.Google Scholar
Groenveld, P. 1970 a Dip-coating by withdrawal of liquid films. PhD thesis, TU Delft, the Netherlands.Google Scholar
Groenveld, P. 1970 b High capillary number withdrawal from viscous Newtonian liquids by flat plates. Chem. Engng Sci. 25 (1), 33–40.CrossRefGoogle Scholar
Groenveld, P. 1970 c Low capillary number withdrawal. Chem. Engng Sci. 25 (8), 1259–1266.CrossRefGoogle Scholar
Hasan, N. & Naser, J. 2009 Determining the thickness of liquid film in laminar condition on a rotating drum surface using ${CFD}$. Chem. Engng Sci. 64 (5), 919–924.CrossRefGoogle Scholar
Jeffreys, H. 1930 The draining of a vertical plate. Math. Proc. Camb. Phil. Soc. 26 (2), 204–205.CrossRefGoogle Scholar
Jin, B., Acrivos, A. & Münch, A. 2005 The drag-out problem in film coating. Phys. Fluids 17 (10), 103603.CrossRefGoogle Scholar
Jung, Y. D. & Ahn, K. H. 2013 Prediction of coating thickness in the convective assembly process. Langmuir 29 (51), 15762–15769.CrossRefGoogle ScholarPubMed
Kim, O. & Nam, J. 2017 Confinement effects in dip coating. J. Fluid Mech. 827, 1–30.CrossRefGoogle Scholar
Kizito, J. P., Kamotani, Y. & Ostrach, S. 1999 Experimental free coating flows at high capillary and Reynolds number. Exp. Fluids 27 (3), 235–243.Google Scholar
Kovac, J. P. & Balmer, R. T. 1980 Experimental studies of external hygrocysts. Trans. ASME: J. Fluids Engng 102 (2), 226–230.Google Scholar
Krechetnikov, R. & Homsy, G. M. 2005 Experimental study of substrate roughness and surfactant effects on the Landau–Levich law. Phys. Fluids 17 (10), 102108.CrossRefGoogle Scholar
Landau, L. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. URSS 17, 42.Google Scholar
Maleki, M., Reyssat, M., Restagno, F., Quéré, D. & Clanet, C. 2011 Landau–Levich menisci. J. Colloid Interface Sci. 354 (1), 359–363.CrossRefGoogle ScholarPubMed
Mayer, H. C. & Krechetnikov, R. 2012 Landau–Levich flow visualization: revealing the flow topology responsible for the film thickening phenomena. Phys. Fluids 24 (5), 052103.CrossRefGoogle Scholar
Melo, F. 1993 Localized states in a film-dragging experiment. Phys. Rev. E 48, 2704–2712.CrossRefGoogle Scholar
Middleman, S. 1978 Free coating of viscous and viscoelastic liquids onto a partially submerged rotating roll. Polym. Engng Sci. 18 (9), 734–737.CrossRefGoogle Scholar
Mill, C. C. & South, G. R. 1967 Formation of ribs on rotating rollers. J. Fluid Mech. 28 (3), 523–529.CrossRefGoogle Scholar
Moffatt, H. K. 1977 Behaviour of a viscous film on the outer surface of a rotating cylinder. J. Méc. 16 (5), 651–673.Google Scholar
Morey, F. C. 1940 Thickness of a liquid film adhering to surface slowly withdrawn from the liquid. J. Res. Natl Bur. Stand. 25, 385.CrossRefGoogle Scholar
Nasto, A., Brun, P.-T. & Hosoi, A. E. 2018 Viscous entrainment on hairy surfaces. Phys. Rev. Fluids 3, 024002.CrossRefGoogle Scholar
Nigam, K. D. P. & Esmail, M. N. 1980 Liquid flow over a rotating dip coater. Can. J. Chem. Engng 58 (5), 564–568.CrossRefGoogle Scholar
Owens, M. S., Vinjamur, M., Scriven, L. E. & Macosko, C. W. 2011 Misting of Newtonian liquids in forward roll coating. Ind. Engng Chem. Res. 50 (6), 3212–3219.CrossRefGoogle Scholar
Palma, S. & Lhuissier, H. 2019 Dip-coating with a particulate suspension. J. Fluid Mech. 869, R3.CrossRefGoogle Scholar
Pitts, E. & Greiller, J. 1961 The flow of thin liquid films between rollers. J. Fluid Mech. 11 (1), 33–50.CrossRefGoogle Scholar
Preziosi, L. & Joseph, D. D. 1988 The run-off condition for coating and rimming flows. J. Fluid Mech. 187, 99–113.CrossRefGoogle Scholar
Pukhnachev, V. V. 1977 Motion of a liquid film on the surface of a rotating cylinder in a gravitational field. J. Appl. Mech. Tech. Phys. 18 (3), 344–351.CrossRefGoogle Scholar
Quéré, D. 1999 Fluid coating on a fiber. Annu. Rev. Fluid Mech. 31 (1), 347–384.CrossRefGoogle Scholar
Rabaud, M. 1994 Dynamiques interfaciales dans l'instabilité de l'imprimeur. Ann. Phys. France 19, 659–690.CrossRefGoogle Scholar
Rubashkin, B. L. 1967 The problem of the thickness of a layer entrained by a rotating drum partially immersed in a liquid. J. Engng Phys. 13 (4), 305–307.Google Scholar
Ruschak, K. J. 1985 Coating flows. Annu. Rev. Fluid Mech. 17 (1), 65–89.CrossRefGoogle Scholar
Ruschak, K. J. & Scriven, L. E. 1976 Rimming flow of liquid in a rotating cylinder. J. Fluid Mech. 76 (1), 113–126.CrossRefGoogle Scholar
Savart, F. 1833 Mémoire sur le choc d'une veine liquide lancée contre un plan circulaire. Ann. Chim. Phys. 54 (56), 1833.Google Scholar
Savva, N. & Bush, J. W. M. 2009 Viscous sheet retraction. J. Fluid Mech. 31, 211–240.CrossRefGoogle Scholar
Schweizer, P. M. & Kistler, S. F. 2012 Liquid Film Coating: Scientific Principles and their Technological Implications. Springer Science & Business Media.Google Scholar
Scott, J. C. 1975 The preparation of water for surface-clean fluid mechanics. J. Fluid Mech. 69 (2), 339–351.CrossRefGoogle Scholar
Seiden, G. & Thomas, P. J. 2011 Complexity, segregation, and pattern formation in rotating-drum flows. Rev. Mod. Phys. 83 (4), 1323.CrossRefGoogle Scholar
Seiwert, J., Clanet, C. & Quéré, D. 2011 Coating of a textured solid. J. Fluid Mech. 669, 55–63.CrossRefGoogle Scholar
Snoeijer, J. H., Delon, G., Fermigier, M. & Andreotti, B. 2006 Avoided critical behavior in dynamically forced wetting. Phys. Rev. Lett. 96, 174504.CrossRefGoogle Scholar
Snoeijer, J. H., Ziegler, J., Andreotti, B., Fermigier, M. & Eggers, J. 2008 Thick films of viscous fluid coating a plate withdrawn from a liquid reservoir. Phys. Rev. Lett. 100, 244502.CrossRefGoogle Scholar
Soroka, A. J. & Tallmadge, J. A. 1971 A test of the inertial theory for plate withdrawal. AIChE J. 17 (2), 505–508.CrossRefGoogle Scholar
Spiers, R. P., Subbaraman, C. V. & Wilkinson, W. L. 1974 Free coating of a newtonian liquid onto a vertical surface. Chem. Engng Sci. 29 (2), 389–396.CrossRefGoogle Scholar
Tallmadge, J. A. 1971 A theory of entrainment for angular withdrawal of flat supports. AIChE J. 17 (1), 243–246.CrossRefGoogle Scholar
Taylor, G. I. 1959 The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets. Proc. R. Soc. Lond. A 253, 313–321.Google Scholar
Tharmalingam, S. & Wilkinson, W. L. 1978 The coating of newtonian liquids onto a rotating roll. Chem. Engng Sci. 33 (11), 1481–1487.CrossRefGoogle Scholar
Thoroddsen, S. T. & Mahadevan, L. 1997 Experimental study of coating flows in a partially-filled horizontally rotating cylinder. Exp. Fluids 23 (1), 1–13.CrossRefGoogle Scholar
Van Rossum, J. J. 1958 Viscous lifting and drainage of liquids. Appl. Sci. Res. A 7 (2-3), 121–144.CrossRefGoogle Scholar
Villermaux, E., Pistre, V. & Lhuissier, H. 2013 The viscous savart sheet. J. Fluid Mech. 730, 607–625.CrossRefGoogle Scholar
Weinstein, S. J. & Ruschak, K. J. 2001 Dip coating on a planar non-vertical substrate in the limit of negligible surface tension. Chem. Engng Sci. 56 (16), 4957–4969.CrossRefGoogle Scholar
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.CrossRefGoogle Scholar
White, D. A. & Tallmadge, J. A. 1965 Theory of drag out of liquids on flat plates. Chem. Engng Sci. 20 (1), 33–37.CrossRefGoogle Scholar
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16 (3), 209–221.CrossRefGoogle Scholar
Yih, C.-S. 1960 Instability of a rotating liquid film with a free surface. Proc. R. Soc. Lond. A 258 (1292), 63–89.Google Scholar

John Soundar Jerome et al. supplementary movie 1

A water sheet extending perpendicularly outward from the rim is ejected on the emerging side of a partially-submerged rotating wheel of radius 13.5 cm when depth-to-radius ratio of 0.2.

Download John Soundar Jerome et al. supplementary movie 1(Video)
Video 18.9 MB

John Soundar Jerome et al. supplementary movie 2

Holes nucleate, grow and disintegrate the water sheet on the emerging side of the wheel. This leads to ligaments and droplets ejected at the liquid rim.

Download John Soundar Jerome et al. supplementary movie 2(Video)
Video 39.9 MB

John Soundar Jerome et al. supplementary movie 3

Rim of the rotating disk for UCON (nearly 100 times more viscous than water) at 19 rpm and depth-to-radius ratio of 0.2.

Download John Soundar Jerome et al. supplementary movie 3(Video)
Video 15.4 MB

John Soundar Jerome et al. supplementary movie 4

A typical inertial liquid sheet for UCON (nearly 100 times more viscous than water) of a rotating wheel of radius 21 cm at 65 rpm and depth-to-radius ratio of 0.2.

Download John Soundar Jerome et al. supplementary movie 4(Video)
Video 7.6 MB

John Soundar Jerome et al. supplementary movie 5

Instantaneous visualization of sheet and entrained film thickness for a rotating wheel of diameter 42 cm and depth-to-radius ratio 0.43 at 3.5 m/s. The video shows relatively large variations of the film thickness, both locally and temporally.

Download John Soundar Jerome et al. supplementary movie 5(Video)
Video 32.3 MB