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Nonlinear evolution of waves on a vertically falling film

  • H.-C. Chang (a1), E. A. Demekhin (a2) and D. I. Kopelevich (a2)

Abstract

Wave formation on a falling film is an intriguing hydrodynamic phenomenon involving transitions among a rich variety of spatial and temporal structures. Immediately beyond an inception region, short, near-sinusoidal capillary waves are observed. Further downstream, long, near-solitary waves with large tear-drop humps preceded by short, front-running capillary waves appear. Both kinds of waves evolve slowly downstream such that over about ten wavelengths, they resemble stationary waves which propagate at constant speeds and shapes. We exploit this quasi-steady property here to study wave evolution and selection on a vertically falling film. All finite-amplitude stationary waves with the same average thickness as the Nusselt flat film are constructed numerically from a boundary-layer approximation of the equations of motion. As is consistent with earlier near-critical analyses, two travelling wave families are found, each parameterized by the wavelength or the speed. One family γ1 travels slower than infinitesimally small waves of the same wavelength while the other family γ2 and its hybrids travel faster. Stability analyses of these waves involving three-dimensional disturbances of arbitrary wavelength indicate that there exists a unique nearly sinusoidal wave on the slow family γ1 with wavenumber αs (or α2) that has the lowest growth rate. This wave is slightly shorter than the fastest growing linear mode with wavenumber αm and approaches the wave on γ1 with the highest flow rate at low Reynolds numbers. On the fast γ2 family, however, multiple bands of near-solitary waves bounded below by αf are found to be stable to two-dimensional disturbances. This multiplicity of stable bands can be interpreted as a result of favourable interaction among solitary-wave-like coherent structures to form a periodic train. (All waves are unstable to three-dimensional disturbances with small growth rates.) The suggested selection mechanism is consistent with literature data and our numerical experiments that indicate waves slow down immediately beyond inception as they approach the short capillary wave with wavenumber α2 of the slow γ1 family. They then approach the long stable waves on the γ2 family further downstream and hence accelerate and develop into the unique solitary wave shapes, before they succumb to the slowly evolving transverse disturbances.

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Nonlinear evolution of waves on a vertically falling film

  • H.-C. Chang (a1), E. A. Demekhin (a2) and D. I. Kopelevich (a2)

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