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

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 . 1993 a ) showed that the primary instability leads to a weakly nonlinear, nearly sinusoidal γ 1 stationary wave which then undergoes a secondary transition to a strongly nonlinear γ 2 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 γ 2 solitary wave is followed by a tertiary ‘phase instability’ dominated by transverse wave crest modulations.