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The flow of a liquid film along a periodic wall

Published online by Cambridge University Press:  21 April 2006

C. Pozrikidis
C. Pozrikidis
Affiliation:
Department of Applied Mechanics and Engineering Sciences, B-010, University of California, San Diego, La Jolla, CA 92093, USA
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Abstract

The creeping flow of a liquid film along an inclined periodic wall of arbitrary geometry is considered. The problem is formulated using the boundary-integral method for Stokes flow. This method is extended to two-dimensional flows involving free surfaces, and is implemented in an iterative numerical procedure. Detailed calculations for flow along a sinusoidal wall are perfomed. The free-surface profile is studied as a function of flow rate, inclination angle, wave amplitude, and surface tension, and is compared with previous asymptotic solutions. The results include streamline patterns, velocity profiles and wall-shear-stress distributions, and establish criteria for flow reversal. For specified wall geometry, the asymptotic behaviour for very small flow rates is shown to be a strong function of surface tension. It is demonstrated that these results are valid in a qualitative sense for general wall geometries. The analogy between gravity-driven flow and the flow of a liquid layer on a rotating disk (spin coating) is also discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 188 , March 1988 , pp. 275 - 300
Copyright
© 1988 Cambridge University Press

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References

Chin, R. W., Abernathy, F. H. & Bertchy, J. R. 1986 Gravity and shear wave instability of free surface flows. Part 1. Numerical calculations. J. Fluid Mech. 168, 501–503.Google Scholar
Dassori, C. G., Deiber, J. A. & Cassano, A. E. 1984 Slow two-phase flow through a sinusoidal channel. Intl J. Multiphase Flow 10, 181–193.Google Scholar
Fulford, G. D. 1964 The flow of liquids in thin films. Adv. Chem. Engng 5, 151–236.Google Scholar
Geller, A. S., Lee, S. H. & Leal, L. G. 1986 The creeping motion of a spherical particle normal to a deformable interface. J. Fluid Mech. 169, 27–69.Google Scholar
Hasimoto, H. & Sano, O. 1980 Stokeslets and eddies in creeping flow. Ann. Rev. Fluid Mech. 12, 335–363.Google Scholar
Higdon, J. J. L. 1985 Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities. J. Fluid Mech. 159, 195–226.Google Scholar
Higgins, B. G. 1986 Film flow on a rotating disk. Phys. Fluids 29, 3522–3529.Google Scholar
Hirt, C. W., Cook, J. L. & Butler, T. D. 1970 A Lagrangian method for calculating the dynamics of an incompressible fluid with free surface. J. Comp. Phys. 5, 103–124.Google Scholar
Hirt, C. W. & Nichols, B. D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comp. Phys. 39, 201–225.Google Scholar
Kelmanson, M. A. 1983 Boundary integral equation solution of viscous flows with free surfaces. J. Engng Maths 17, 329–343.Google Scholar
Kistler, S. F. & Scriven, L. E. 1983 Coating Flows. In Computational Analysis of Polymer Processing (ed. J. R. A. Pearson & S. M. Richardson), chap. 8, pp. 243–299. Applied Science Publishers.
Krantz, W. B. & Goren, S. L. 1970 Finite-amplitude, long waves on liquid films flowing down a plane. Ind. Engng Chem. 9, 107–113.Google Scholar
Lee, S. H. & Leal, L. G. 1986 Low Reynolds number flow past cylindrical bodies of arbitrary cross-sectional shape. J. Fluid Mech. 164, 401–427.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.Google Scholar
Pierrehumbert, R. T. 1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 120–144.Google Scholar
Pozrikidis, C. 1987 Creeping flow in two-dimensional channels. J. Fluid Mech. 180, 495–514.Google Scholar
Rallison, J. M. 1984 The deformation of small viscous drops in viscous flows. Ann. Rev. Fluid. Mech. 16, 45–66.Google Scholar
Rallison, J. M. & Acrivos, A. A. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89, 191–200.Google Scholar
Ruschak, K. J. & Scriven, L. E. 1977 Developing flow on a vertical wall. J. Fluid Mech. 81, 305–316.Google Scholar
Ryskin, G. & Leal, L. G. 1984a Numerical solution of free-boundary problems in fluid mechanics. Part 1. The finite-difference technique. J. Fluid Mech. 148, 1–17.Google Scholar
Ryskin, G. & Leal, L. G. 1984b Numerical solution of free-boundary problems in fluid mechanics. Part 2. Buoyancy-driven motion of a gas bubble through a quiescent liquid. J. Fluid Mech. 148, 18–36.Google Scholar
Ryskin, G. & Leal, L. G. 1984c Numerical solution of free-boundary problems in fluid mechanics. Part 3. Bubble deformation in an axisymmetric straining flow. J. Fluid Mech. 148, 37–43.Google Scholar
Tougou, H. 1978 Long waves on a film flow of a viscous fluid down an inclined uneven wall. J. Phys. Soc. Japan, 44, 1014–1019.Google Scholar
Youngren, G. K. & Acrivos, A. 1976 On the shape of a gas bubble in a viscous extensional flow. J. Fluid Mech. 76, 433–442.Google Scholar
Wang, C. Y. 1981 Liquid film flowing slowly down a wavy incline. AIChE J. 27, 207–212.Google Scholar
Wang, C. Y. 1984 Thin film flowing down a curved surface. Z. Angew. Math. Phys. 35, 533–544.Google Scholar