Next: Introduction: Film flows and evolution equations

[Published in "Mathematical Modeling and Simulation in Hydrodynamic Stability", (ed. D. N. Riahi), World Scientific, Singapore, 1996. pp. 35-81]

DERIVATIONS AND SIMULATIONS OF EVOLUTION EQUATIONS
OF WAVY FILM FLOWS

ALEXANDER L. FRENKEL AND K. INDIRESHKUMAR
Department of Mathematics, University of Alabama
Tuscaloosa, Alabama 35487

Abstract:

Wavy flows of viscous films on solid surfaces are considered. The focus is on approximate decriptions which are hinged on a single evolution equation. Perturbative approaches to constructing such theories are discussed. For several film flows, evolution equations obtained--along with the validity conditions of those theories--with the multiparametric perturbation approach are reviewed. The results of their three-dimensional numerical simulations on extended spatial intervals are discussed. Some unresolved fundamental questions concerning such film-flow studies are posed and discussed.

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• Introduction: Film flows and evolution equations
• Perturbation approaches
  • An example of the less-formal MP approach at work: 2-D vertical film flow
  • The leading two orders of the fully formal MP approach: the strongly dispersive case
    • Power series
    • Leading-order system
    • The -correction to the leading order
  • Earlier perturbation approaches and the problem of validity conditions
    • Long-wave approach
    • Single-parameter approach
• Some results
  • Three-dimensional inclined-film flow
    • General evolution equation; impossibility of a single-equation description of large-amplitude regimes for large times
    • Evolution equation for small-amplitude regimes
    • Numerical studies of evolution equation
    • Unusual patterns on strange attractors for strongly dispersive falling films
    • Transient patterns: Qualitative agreement of simulations with experiments
  • Flow down a vertical fiber
    • Large-amplitude regimes
    • Small-amplitude evolution equation
  • Small-amplitude waves in core-annular flows
  • Vertical and horizontal core-annular flows with large-amplitude waves
• Some unresolved questions concerning
  foundations of the film flow research
  • On foundations of perturbation methods used
    in film flow studies.
  • On numerical simulations of evolution equations
• Summary
• Acknowledgments
• References
• About this document ...

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Alex Frenkel
Fri Nov 8 23:40:39 CST 1996