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

Published online by Cambridge University Press:  26 April 2006

H.-C. Chang ,
M. Cheng ,
E. A. Demekhin  and
D. I. Kopelevich
H.-C. Chang
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
M. Cheng
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
E. A. Demekhin
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Permanent address: Department of Applied Mathematics, Krasnodar Polytechnical Institute. Krasnodar, 350072, Russia.
D. I. Kopelevich
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Permanent address: Department of Applied Mathematics, Krasnodar Polytechnical Institute. Krasnodar, 350072, Russia.
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Abstract

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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 270 , 10 July 1994 , pp. 251 - 276
Copyright
© 1994 Cambridge University Press

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