Quasi-Stationary Wave Evolution on a Falling Film

Instabilities of three-dimensional viscous falling films

Journal of Fluid Mechanics ◽

10.1017/s0022112092002489 ◽

1992 ◽

Vol 242 ◽

pp. 529-547 ◽

Cited By ~ 86

Author(s):

S. W. Joo ◽

S. H. Davis

Keyword(s):

Evolution Equation ◽

Three Dimensional ◽

Falling Film ◽

Two Dimensional ◽

Wave Evolution ◽

Strongly Nonlinear ◽

Long Wave ◽

Threshold Amplitude ◽

Permanent Wave ◽

Dimensional Wave

A long-wave evolution equation is used to study a falling film on a vertical plate. For certain wavenumbers there exists a two-dimensional strongly nonlinear permanent wave. A new secondary instability is identified in which the three-dimensional disturbance is spatially synchronous with the two-dimensional wave. The instability grows for sufficiently small cross-stream wavenumbers and does not require a threshold amplitude; the two-dimensional wave is always unstable. In addition, the nonlinear evolution of three-dimensional layers is studied by posing various initial-value problems and numerically integrating the long-wave evolution equation.

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Secondary and tertiary excitation of three-dimensional patterns on a falling film

Journal of Fluid Mechanics ◽

10.1017/s002211209400426x ◽

1994 ◽

Vol 270 ◽

pp. 251-276 ◽

Cited By ~ 20

Author(s):

H.-C. Chang ◽

M. Cheng ◽

E. A. Demekhin ◽

D. I. Kopelevich

Keyword(s):

Solitary Wave ◽

Three Dimensional ◽

Wave Structure ◽

Stationary Wave ◽

Wave Crest ◽

Checkerboard Pattern ◽

Secondary Transition ◽

Falling Film ◽

Two Dimensional ◽

Primary Instability

The primary instability of a falling film selectively amplifies two-dimensional noise down-stream over three-dimensional modes with transverse variation. If the initial three-dimensional noise is weak or if it has short wavelengths such that they are effectively damped by the capillary mechanism of the primary instability, our earlier study (Chang et al. 1993a) showed that the primary instability leads to a weakly nonlinear, nearly sinusoidal γ1 stationary wave which then undergoes a secondary transition to a strongly nonlinear γ2 wave with a solitary wave structure. We show here that the primary transition remains in the presence of significant three-dimensional noise but the secondary transition can be replaced by a selective excitation of oblique triad waves which can even include stable primary disturbances. The resulting secondary checkerboard pattern is associated with a subharmonic mode in the streamwise direction. If the initial transverse noise level is low, a secondary transition to a two-dimensional γ2 solitary wave is followed by a tertiary ‘phase instability’ dominated by transverse wave crest modulations.

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