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Three-dimensional surfactant-covered flows of thin liquid films on rotating cylinders

Published online by Cambridge University Press:  03 April 2018

Weihua Li  and
Satish Kumar Open the ORCID record for Satish Kumar [Opens in a new window]
Weihua Li
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
Satish Kumar*
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
*
†Email address for correspondence: kumar030@umn.edu
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Abstract

The coating of discrete objects is an important but poorly understood step in the manufacturing of a broad variety of products. An important model problem is the flow of a thin liquid film on a rotating cylinder, where instabilities can arise and compromise coating uniformity. In this work, we use lubrication theory and flow visualization experiments to study the influence of surfactant on these flows. Two coupled evolution equations describing the variation of film thickness and concentration of insoluble surfactant as a function of time, the angular coordinate and the axial coordinate are solved numerically. The results show that surface-tension forces arising from both axial and angular variations in the angular curvature drive flows in the axial direction that tend to smooth out free-surface perturbations and lead to a stable speed window in which axial perturbations do not grow. The presence of surfactant leads to Marangoni stresses that can cause the stable speed window to disappear by driving flow that opposes the stabilizing flow. In addition, Marangoni stresses tend to reduce the spacing between droplets that form at low rotation rates, and reduce the growth rate of rings that form at high rotation rates. Flow visualization experiments yield observations that are qualitatively consistent with predictions from linear stability analysis and the simulation results. The visualizations also indicate that surfactants tend to suppress dripping, slow the development of free-surface perturbations, and reduce the shifting and merging of rings and droplets, allowing more time for solidifying coatings in practical applications.

JFM classification

Materials Processing Flows: Coating Interfacial Flows (free surface): Interfacial Flows (free surface) Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 844 , 10 June 2018 , pp. 61 - 91
Copyright
© 2018 Cambridge University Press 

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References

Badali, D., Chugunova, M., Pelinovsky, D. E. & Pollack, S. 2011 Regularized shock solutions in coating flows with small surface tension. Phys. Fluids 23 (9), 093103.10.1063/1.3635535Google Scholar
de Bruyn, J. R. 1997 Crossover between surface tension and gravity-driven instabilities of a thin fluid layer on a horizontal cylinder. Phys. Fluids 9 (6), 1599–1605.10.1063/1.869280Google Scholar
Burchard, A., Chugunova, M. & Stephens, B. K. 2012 Convergence to equilibrium for a thin-film equation on a cylindrical surface. Commun. Part. Diff. Equ. 37 (4), 585–609.10.1080/03605302.2011.648704Google Scholar
Campana, D. M. & Saita, F. A. 2006 Numerical analysis of the Rayleigh instability in capillary tubes: The influence of surfactant solubility. Phys. Fluids 18 (2), 022104.10.1063/1.2173969Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.10.1103/RevModPhys.81.1131Google 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), 2742–2756.10.1063/1.1758943Google 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), 072102.10.1063/1.1942523Google Scholar
Garg, D. & Dyer, P. N. 1990 Tungsten carbide erosion resistant coating for aerospace components. Mat. Res. Soc. Symp. Proc. 168, 213–220.10.1557/PROC-168-213Google Scholar
Kang, D., Nadim, A. & Chugunova, M. 2016 Dynamics and equilibria of thin viscous coating films on a rotating sphere. J. Fluid Mech. 791, 495–518.10.1017/jfm.2016.67Google Scholar
Kang, D., Nadim, A. & Chugunova, M. 2017 Marangoni effects on a thin liquid film coating a sphere with axial or radial thermal gradients. Phys. Fluids 29 (7), 072106.10.1063/1.4991580Google Scholar
Karabut, E. A. 2007 Two regimes of liquid film flow on a rotating cylinder. J. Appl. Mech. Tech. Phys. 48 (1), 55–64.10.1007/s10808-007-0008-9Google Scholar
Kelmanson, M. A. 1995 Theoretical and experimetal analyses of the maximum-suppotable fluid load on a rotating cylinder. J. Engng Maths 29 (3), 271–285.10.1007/BF00042858Google Scholar
Knox, W. J.1970. Photographic surfactant compositions. US patent no. 3514293.Google Scholar
Kovac, J. P. & Balmer, R. T. 1980 Experimental studies of external hygrocysts. Trans. ASME J. Fluids Engng 102 (2), 226–230.10.1115/1.3240653Google Scholar
Kramer, S. & Meikle, S.2002. Reduction of surface roughness during chemical mechanical planarization (CMP). US patent no. 2002018.Google Scholar
Li, W., Carvalho, M. S. & Kumar, S. 2017a Liquid-film coating on topographically patterned rotating cylinders. Phys. Rev. Fluids 2 (2), 024001.10.1103/PhysRevFluids.2.024001Google Scholar
Li, W., Carvalho, M. S. & Kumar, S. 2017b Viscous free-surface flows on rotating elliptical cylinders. Phys. Rev. Fluids 2 (9), 094005.10.1103/PhysRevFluids.2.094005Google Scholar
Li, W. & Kumar, S. 2015 Thin-film coating of surfactant-laden liquids on rotating cylinders. Phys. Fluids 27 (7), 072106.10.1063/1.4927222Google Scholar
Marmur, A. & Lelah, M. D. 1981 The spreading of aqueous surfactant solutions on glass. Chem. Engng Commun. 13 (1–3), 133–143.10.1080/00986448108910901Google Scholar
Mata, M. R. & Bertozzi, A. L. 2011 A numerical scheme for particle-laden thin film flow in two dimensions. J. Comput. Phys. 230 (16), 6334–6353.10.1016/j.jcp.2011.04.029Google Scholar
Matar, O. K. & Craster, R. V. 2001 Models for Marangoni drying. Phys. Fluids 13 (7), 1869–1883.10.1063/1.1378034Google Scholar
Moffatt, H. K. 1977 Behaviour of a viscous film on the outer surface of a rotating cylinder. J. Méc. 16, 651–673.Google Scholar
Narayanan, P., Llanos, G. H., Cook, D. & Leidner, J.2000. Device and process for coating stents. US patent no. 6723373.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931–980.10.1103/RevModPhys.69.931Google Scholar
Perizzolo, D., Lacefield, W. R. & Brunette, D. M. 2001 Interaction between topography and coating in the formation of bone nodules in culture for hydroxyapatite- and titanium-coated micromachined surfaces. J. Biomed. Mater. Res. 56 (4), 494–503.10.1002/1097-4636(20010915)56:4<494::AID-JBM1121>3.0.CO;2-X3.0.CO;2-X>Google Scholar
Preziosi, L. & Joseph, D. D. 1988 The run-off condition for coating and rimming flows. J. Fluid Mech. 187, 99–113.10.1017/S0022112088000357Google Scholar
Pukhnachev, V. V. 1978 Motion of a liquid film on the surface of a rotating cylinder in a gravitational field. J. Appl. Mech. Tech. Phys. 18 (3), 344–351.10.1007/BF00851656Google Scholar
Sahu, A. K. & Kumar, S. 2014 Thin-liquid-film flow on a topographically patterned rotating cylinder. Phys. Fluids 26 (4), 042102.10.1063/1.4869208Google Scholar
Schunk, P. R. & Scriven, L. E. 1997 Surfactant effects in coating processes. In Liquid Film Coating, pp. 495–536. Springer.10.1007/978-94-011-5342-3_14Google Scholar
Takagi, D. & Huppert, H. E. 2010 Flow and instability of thin films on a cylinder and sphere. J. Fluid Mech. 647, 221–238.10.1017/S0022112009993818Google Scholar
Warner, M. R. E., Craster, R. V. & Matar, O. K. 2002 Unstable van der Waals driven line rupture in Marangoni driven thin viscous films. Phys. Fluids 14 (5), 1642–1654.10.1063/1.1460878Google Scholar
Weidner, D. E. 2013 Suppression and reversal of drop formation on horizontal cylinders due to surfactant convection. Phys. Fluids 25 (8), 082110.10.1063/1.4818443Google Scholar
Witelski, T. P. & Bowen, M. 2003 ADI schemes for higher-order nonlinear diffusion equations. Appl. Numer. Maths 45 (2–3), 331–351.10.1016/S0168-9274(02)00194-0Google Scholar
Yih, C. S. & Kingman, J. F. C. 1960 Instability of a rotating liquid film with a free surface. Proc. R. Soc. Lond. A 258 (1292), 63–89.Google Scholar
Zhu, J., Chen, J. L., Lade, R. K., Suszynski, W. J. & Francis, L. F. 2015 Water-based coatings for 3D printed parts. J. Coat. Technol. Res. 12 (5), 889–897.10.1007/s11998-015-9710-3Google Scholar
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