Skip to main content Accessibility help
×
Home

Drop formation during coating of vertical fibres

  • Serafim Kalliadasis (a1) and Hsueh-Chia Chang (a1)

Abstract

When the coating film around a vertical fibre exceeds a critical thickness hc, the interfacial disturbances triggered by Rayleigh instability can undergo accelerated growth such that localized drops much larger in dimension than the film thickness appear. We associate the initial period of this strongly nonlinear drop formation phenomenon with a self-similar intermediate asymptotic blow-up solution to the long-wave evolution equation which describes how static capillary forces drain fluid into the drop. Below hc, we show that strongly nonlinear coupling between the mean flow and axial curvature produces a finite-amplitude solitary wave solution which prevents local finite-time blow up and hence disallows further growth into drops. We thus estimate hc by determining the existence of solitary wave solutions. This is accomplished by a matched asymptotic analysis which joins the capillary outer region of a large solitary wave to the thin-film inner region. Our estimate of hc = 1.68R3H–2, where R is the fibre radius and H is the capillary length H = (σ/ρg)½, is favourably compared to experimental data.

Copyright

References

Hide All
Aul, R. W. & Olbricht, W. L. 1960 Stability of a thin annular film in pressure-driven, low Reynolds-number flow through a capillary. J. Fluid Mech. 215, 585599.
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166188.
Chang, H.-C. 1986 Traveling waves on fluid interfaces - normal form analysis of the Kuramoto-Sivashinsky equation. Phys. Fluids 29, 31423147.
Chang, H.-C. 1989 Onset of nonlinear waves on falling films. Phys. Fluids A 1, 13141327.
Chang, H.-C., Demekhin, E. & Kopelevich, D. 1993a Laminarizing effects of dispersion in an active-dissipative nonlinear medium. Physica D D63, 299320.
Chang, H.-C., Demekhin, E. & Kopelevich, D. 1993b Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech. 250, 433480.
Cheng, M. & Chang, H.-C. 1992 Stability of axisymmetric waves on liquid films flowing down a vertical column to azimuthal and streamwise disturbances. Chem. Engng Commun. 118, 327346.
Elphick, C., Meron, E. & Spiegel, E. A. 1988 Spatiotemporal complexity in traveling patterns. Phys. Rev. Lett. 61, 496499.
Frenkel, A. L. 1992 Nonlinear theory of strongly undulating thin films flowing down vertical cylinders. Europhys. Lett. 18, 583588.
Frenkel, A. L., Babchin, A. J., Levich, B. G., Shlang, T. & Sivashinsky, G. I. 1987 Annular flows can keep unstable films from breakup: nonlinear saturation of capillary instability. J. Colloid Interface Sci. 115, 225233.
Glendinning, P. & Sparrow, C. 1984 Local and global behavior near homoclinic orbits. J. Stat. Phys. 35, 645696.
Hammond, P. S. 1983 Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech. 137, 363384.
Kawahara, T. & Toh, S. 1988 Pulse interaction in an unstable dissipative-dispersive nonlinear system. Phys. Fluids 31, 21032111.
Kerchman, V. I. & Frenkel, A. L. 1993 Interactions of coherent structures in a film flow: simulations of a highly nonlinear evolution equation. Theoret. Comput. Fluid Dyn. (to appear).
Lin, S. P. 1974 Finite-amplitude sideband instability of a viscous film. J. Fluid Mech. 63, 417429.
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291308.
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 2750.
Quéré, D. 1990 Thin films flowing on vertical fibers. Europhys. Lett. 13, 721726.
Ratulowski, J. & Chang, H.-C. 1989 Transport of gas bubbles in capillaries. Phys. Fluids A 1, 16421655.
Rosenau, P., Oron, A. & Hyman, J. M. 1992 Bounded and unbounded patterns of the Benney equation. Phys. Fluids A 4, 11021104.
Russo, M. J. & Steen, P. H. 1989 Shear stabilization of the capillary breakup of a cylindrical interface. Phys. Fluids A 1, 19261937.
Schwartz, L. W. 1989 Viscous flows down an inclined plane: instability and finger formation. Phys. Fluids. A 1, 443445.
Shlag, T. & Sivashinsky, G. I. 1982 Irregular flow of a liquid film down a vertical column. J. Phys. Paris 43, 459466.
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16, 209221.
Wilson, S. D. R. & Jones, A. F. 1983 The entry of a falling film into a pool and the air entrainment problem. J. Fluid Mech. 128, 219230.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Drop formation during coating of vertical fibres

  • Serafim Kalliadasis (a1) and Hsueh-Chia Chang (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed