Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-17T16:48:58.401Z Has data issue: false hasContentIssue false
  • English
  • Français

Film-splitting flows in forward roll coating

Published online by Cambridge University Press:  21 April 2006

D. J. Coyle ,
C. W. Macosko  and
L. E. Scriven
D. J. Coyle
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA Present address: General Electric Company, Corporate Research and Development, Schenectady, NY 12301, USA.
C. W. Macosko
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
L. E. Scriven
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A model based on the lubrication approximation is put forward for the general case of asymmetric forward roll coating of Newtonian liquids. Two more-rigorous theories are developed, one based on asymptotic expansions for small ratios of gap-to-roll diameter (H0/R), the second on Galerkin/finite-element solutions of the full Navier–Stokes equations over the relevant flow domain. The lubrication model is useful only as an approximation at high capillary numbers $(Ca\equiv \mu\overline{V}/\sigma) $. The asymptotic analysis is accurate when H0/R < 0.001 and Ca > 0.1. The ratio of the film thicknesses on the two rolls is predicted to equal the speed ratio to the 0.65 power, which is confirmed experimentally. The Galerkin/finite-element solutions give full details of the steady two-dimensional free-surface flows including complex recirculation patterns in the film-splitting region, and show how the film-splitting stagnation line becomes a static contact line in the limit as one roll surface becomes stationary.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 171 , October 1986 , pp. 183 - 207
Copyright
© 1986 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

Banks, W. H. & Mill C. C.1954 Some observations on the behaviour of liquids between rotating rollers Proc. R. Soc. Lond. A 223, 414.Google Scholar
Baumann T. M.1980 Roll coating in the presence of a stationary constraint. M.S. thesis, University of Massachusetts.
Benkreira H., Edwards, M. F. & Wilkinson W. L.1981a A semi-empirical model of the forward roll coating flow of Newtonian fluids. Chem. Engng Sci. 36, 423.Google Scholar
Benkreira H., Edwards, M. F. & Wilkinson W. L.1981b Roll coating of purely viscous liquids. Chem. Engng Sci. 36, 429.Google Scholar
Benkreira H., Edwards, M. F. & Wilkinson W. L.1982 Flux distribution in forward roll coating: a simple analysis. Chem. Engng J. 25, 211.Google Scholar
Bixler N. E.1982 Stability of a coating flow. Ph.D. thesis, University of Minnesota.
Cameron A.1966 Basic Lubrication Theory. Wiley.
Cole, J. A. & Hughes C. J.1956 Oil flow and film extent in complete journal bearings. IME Proceedings 170 (17), 499.Google Scholar
Coyle D. J.1984 The fluid mechanics of roll coating: steady flows, stability, and rheology. Ph.D. thesis, University of Minnesota.
Coyle D. J., Macosko, C. W. & Scriven L. E.1982 Computer simulation of nip flow in roll coating. In Computer Applications in Applied Polymer Science (ed. T. Provder), p. 251. ACS Symposium Series 197, American Chemical Society, Washington.
Coyle D. J., Macosko, C. W. & Scriven L. E.1986a Stability of symmetric film-splitting between counterrotating cylinders. J. Fluid Mech. (submitted).Google Scholar
Coyle D. J., Macosko, C. W. & Scriven L. E.1986b Film-splitting flows in forward roll coating of shear-thinning liquids. AIChE J. (submitted).Google Scholar
Coyne, J. C. & Elrod H. G.1970 Conditions for the rupture of a lubricating film. Part I: Theoretical model; Part II: New boundary conditions for Reynolds equations. Trans. ASME F: J. Lubric. Tech. 92, 451.Google Scholar
Dowson, D. & Taylor C. M.1979 Cavitation in bearings. Ann. Rev. Fluid Mech. 11, 35.Google Scholar
Floberg L.1961a Lubrication of two cylindrical surfaces considering cavitation. Trans. Chalmers Univ. Technol. 234, Goteborg.Google Scholar
Floberg L.1961b On hydrodynamic lubrication with special reference to cavitation in bearings. Dissertation, Chalmers Univ. Technol. 30, Goteborg.
Floberg L.1964 Cavitation in lubricating oil films. In: Cavitation in Real Liquids (ed. R. Davies). Elsevier.
Gartling D. K., Nickell, R. E. & Tanner R. I.1977 A finite element convergence study for accelerating flow problems. Intl J. Numer. Meth. Engng 11, 1155.Google Scholar
Gatcombe E. K.1945 Lubrication characteristics of involute spur gears—a theoretical investigation. Trans. ASME 67, 177.Google Scholar
Greener J.1979 Bounded coating flows of viscous and viscoelastic fluids. Ph.D. thesis, University of Massachusetts.
Greener, J. & Middleman S.1975 A theory of roll coating of viscous and viscoelastic fluids. Polymer Engng Sci. 15, 1.Google Scholar
Greener, J. & Middleman S.1979 Theoretical and experimental studies of the fluid dynamics of a two-roll coater. I & EC Fund. 18, 35.Google Scholar
Hintermaier, J. C. & White R. E.1965 The splitting of a water film between rotating rolls. Tappi 48, 617.Google Scholar
Hoffman, R. D. & Myers R. R.1962 The splitting of thin liquid films. Cavitation dynamics. Trans. Soc. Rheol. 6, 197.Google Scholar
Hood P.1976 Frontal solution program for unsymmetric matrices. Intl J. Numer. Meth. Engng 10, 379.Google Scholar
Hood P.1977 Correction. Intl J. Numer. Meth. Engng 11, 1055.Google Scholar
Hopkins M. R.1957 Viscous flow between rotating cylinders and a sheet moving between them. Br. J. Appl. Phys. 8, 443.Google Scholar
Kistler S. F.1983 The fluid mechanics of curtain coating and related viscous free surface flows. Ph.D. thesis, University of Minnesota.
Kistler, S. F. & Scriven L. E.1983 Coating flows. In: Computational Analysis of Polymer Processing (ed. J. R. A. Pearson & S. M. Richardson), p. 243. London and New York: Applied Science.
Kistler, S. F. & Scriven L. E.1984 Coating flow theory by finite element and asymptotic analysis of the Navier—Stokes system. Intl J. Numer. Meth. Fluids 4, 207.Google Scholar
Middleman S.1977 Fundamentals of Polymer Processing. McGraw-Hill.
Miller, J. C. & Myers R. R.1958 A photographic study of liquid flow in a roll nip. Trans. Soc. Rheol. 2, 77.Google Scholar
Myers, R. R. & Hoffman R. D.1961 The distribution of pressures in the roll application of Newtonian fluids. Trans. Soc. Rheol. 5, 317.Google Scholar
Myers R. R., Miller, J. C. & Zettlemoyer A. C.1959 The splitting of thin liquid films: kinematics. J. Colloid Sci. 14, 287.Google Scholar
Nickell R. E., Tanner, R. I. & Caswell B.1974 Solution of viscous incompressible jet and free surface flows using finite element methods. J. Fluid Mech. 65, 189.Google Scholar
Pitts, E. & Greiller J.1961 The flow of thin liquid films between rollers. J. Fluid Mech. 11, 33.Google Scholar
Reynolds O.1886 On the theory of lubrication and its application to Mr Beauchamp Tower's experiments including an experimental determination of the viscosity of olive oil Phil. Trans. R. Soc. Lond. A 177, 157.Google Scholar
Ruschak K. J.1980 A method for incorporating free boundaries with surface tension in finite element fluid flow simulators. Intl J. Numer. Meth. Engng 15, 639.Google Scholar
Ruschak K. J.1982 Boundary conditions at a liquid/air interface in lubrication flows. J. Fluid Mech. 119, 107.Google Scholar
Saito, H. & Scriven L. E.1981 Study of coating flow by the finite element method. J. Comp. Phys. 42, 53.Google Scholar
Savage M. D.1977 Cavitation in lubrication. Part 1: On boundary conditions and cavity—fluid interfaces. Part 2: Analysis of wavy interfaces. J. Fluid Mech. 80, 743.Google Scholar
Savage M. D.1982 Mathematical models for coating processes J. Fluid Mech. 117, 443.Google Scholar
Schneider G. B.1962 Analysis of forces causing flow in roll coaters. Trans. Soc. Rheol. 6, 209.Google Scholar
Silliman W. J.1979 Viscous film flows with contact lines. Ph.D. thesis, University of Minnesota, Minneapolis.
Silliman, W. J. & Scriven L. E.1980 Separating flow near a static contact line: slip at a wall and shape of a free surface. J. Comp. Phys. 34, 287.Google Scholar
Sullivan, T. M. & Middleman S.1979 Roll coating in the presence of a fixed constraining boundary. Chem. Engng Commun. 3 (6), 469.Google Scholar
Taylor C. M.1974 Film rupture for a lubricated cylinder lightly loaded against a plane. J. Mech. Engng Sci. 16, 225.Google Scholar
Taylor G. I.1963 Cavitation of a viscous fluid in narrow passages. J. Fluid Mech. 16, 595.Google Scholar
Van Dyke M.1975 Perturbation Methods in Fluid Mechanics. Stanford, CA: Parabolic.
Walters R. A.1980 The frontal method in hydrodynamics simulations. Comp. Fluids 8, 265.Google Scholar
Williamson A. S.1972 The tearing of an adhesive layer between flexible tapes pulled apart. J. Fluid Mech. 52, 639.Google Scholar