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TECHNICAL PAPERS

Laminar, Gravitationally Driven Flow of a Thin Film on a Curved Wall

[+] Author and Article Information
Kenneth J. Ruschak, Steven J. Weinstein

Manufacturing Research and Engineering Organization, Eastman Kodak Company, Rochester, NY 14652-3701

J. Fluids Eng 125(1), 10-17 (Jan 22, 2003) (8 pages) doi:10.1115/1.1522412 History: Received June 29, 2001; Revised June 20, 2002; Online January 22, 2003
Copyright © 2003 by ASME
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References

Kistler, S. F., and Schweizer, P. M., eds., 1997, Liquid Film Coating, Chapman & Hall, New York.
Ruschak,  K. J., and Weinstein,  S. J., 1999, “Viscous Thin-Film Flow Over a Round-Crested Weir,” ASME J. Fluids Eng., 121, pp. 673–677.
Ruschak,  K. J., and Weinstein,  S. J., 2000, “Thin-Film Flow at Moderate Reynolds Number,” ASME J. Fluids Eng., 122, pp. 774–778.
Ruschak,  K. J., and Weinstein,  S. J., 2001, “Developing Film Flow on an Inclined Plane With a Critical Point,” ASME J. Fluids Eng., 123, pp. 698–702.
Higuera,  F. J., 1994, “The Hydraulic Jump in Viscous Laminar Flow,” J. Fluid Mech., 274, pp. 69–92.
Levich, V. G., 1962, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, Chap. 12.
Schlichting, H., 1979, Boundary-Layer Theory, 7th Ed., McGraw-Hill, New York, pp. 157–158.
Atkinson,  B., and McKee,  R. L., 1964, “A Numerical Investigation of Non-uniform Film Flow,” Chem. Eng. Sci., 19, pp. 457–470.
Weinstein,  S. J., and Ruschak,  K. J., 1999, “On the Mathematical Structure of Thin Film Equations Containing a Critical Point,” Chem. Eng. Sci., 54(8), pp. 977–985.
Weinstein,  S. J., and Ruschak,  K. J., 2001, “Dip Coating on a Planar Non-vertical Substrate in the Limit of Negligible Surface Tension,” Chem. Eng. Sci., 56, pp. 4957–4969.
Hassan,  N. A., 1967, “Laminar Flow Along a Vertical Wall,” ASME J. Appl. Mech., 34, pp. 535–537.
Bohr,  T., Putkaradze,  V., and Watanabe,  S., 1997, “Averaging Theory for the Structure of Hydraulic Jumps and Separation in Laminar Free-Surface Flows,” Phys. Rev. Lett., 79, pp. 1038–1041.
Bohr,  T., Ellegaard,  C., Hansen,  A. E., and Haaning,  A., 1996, “Hydraulic Jumps, Flow Separation and Wave Breaking: An Experimental Study,” Physica B, 228, pp. 1–10.
Schwartz,  L. W., and Weidner,  D. E., 1995, “Modeling of Coating Flows on Curved Surfaces,” J. Eng. Math., 29, pp. 91–103.
Roy,  R. V., Roberts,  A. J., and Simpson,  M. E., 2002, “A Lubrication Model of Coating Flows Over a Curved Substrate in Space,” J. Fluid Mech., 454, pp. 235–261.
Berger,  R. C., and Carey,  G. F., 1998, “Free Surface Flow Over Curved Surfaces. Part I: Perturbation Analysis,” Int. J. Numer. Methods Fluids, 28, pp. 191–200.
Roy,  T. R., 1984, “On Laminar Thin-Film Flow Along a Vertical Wall,” ASME J. Appl. Mech., 51, pp. 691–692.
Alekseenko, S. V., Nakoryakov, V. E., and Pokusaev, B. G., 1994, Wave Flow of Liquid Films, Begell House, New York.
Ruschak,  K. J., 1978, “Flow of a Falling Film Into a Pool,” AIChE J., 24, pp. 705–710.
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Figures

Grahic Jump Location
Photograph of the standing wave marking the transition from supercritical to subcritical flow on a curved wall; the view is downward toward the standing wave from the side. Label 1, vertical wall; 2, curved wall; 3, standing wave with slope shadowed by the lighting; 4, wall inclined at 2 deg; 5, sidewalls. Areas beyond the sidewalls of the flow have been blackened to eliminate distractions in the surroundings. The computed film thickness profile for these conditions is plotted in Fig. 9.
Grahic Jump Location
Film thickness profiles for Re=20 and δ=0.0147 from the film equation with varying velocity profile and the Navier-Stokes equation on a downwardly curving wall. Only the position of the critical point from the Nusselt film equation is indicated because the profile is indistinguishable. The initially subcritical flow becomes supercritical as wall inclination increases.
Grahic Jump Location
Plot of the velocity profile parameter A for the conditions of Fig. 3. Upstream and downstream, A→0 as the velocity profile becomes fully developed.
Grahic Jump Location
The range of velocity profiles for the conditions of Fig. 3 from the film equation with varying velocity profile and from the Navier-Stokes equation. The two profiles from the film equation correspond to the extreme values of A. The range of velocity at each of 11 nodes is shown for the Navier-Stokes equation.
Grahic Jump Location
Film profiles for Re=20 and δ=0.0093 from the film equation with varying velocity profile and the Navier-Stokes equation on an upwardly curving wall. The initially supercritical flow becomes subcritical as wall inclination decreases.
Grahic Jump Location
An expanded view of film profiles for the conditions of Fig. 6 in the vicinity of the standing wave. The Nusselt film equation gives two sections that are discontinuous at the critical point; the section upstream of the critical point is indistinguishable and not shown
Grahic Jump Location
The range of velocity profiles for the conditions of Fig. 6 from the film equation with varying velocity profile and from the Navier-Stokes equation. The two profiles from the film equation correspond to the extreme values of A. The range of velocity at each of 11 nodes is shown for the Navier-Stokes equation.
Grahic Jump Location
Film profiles from the film equation with varying velocity profile and from the Navier-Stokes equation for flow on an upwardly curving wall. For the middle curves, Re=12,δ=0.021, and Bo=600, the conditions of Fig. 1; for the highest curves, Re=0.1; and for the lowest curves, Re=25.
Grahic Jump Location
Film thickness profile for Re=50 and δ=0.05 from the film equation with varying velocity profile for flow on a downwardly curving wall that is horizontal upstream and vertical downstream. The initially supercritical flow becomes subcritical as film thickness increases due to drag; the flow becomes supercritical again where the wall steepens.
Grahic Jump Location
Flow on a wall inclined at 20 deg from horizontal into a pool at Re=20 with and without surface tension

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