Three-dimensional instabilities of film flows

Abstract

Two-dimensional (2-D) interfacial waves on flowing films are unstable with respect to both two-and three-dimensional instabilities. In this paper, several distinct three-dimensional instabilities that occur in different regions of the parameter space defined by the Reynolds number R and the frequency f of forced two-dimensional waves are discussed in detail. (a) A synchronous 3-D instability, in which spanwise deformations of adjacent wave fronts have the same transverse phase, appears over a wide range of frequency. These transverse modulations occur mainly along the troughs of the primary waves and eventually develop into sharp and nearly isolated depressions. The instability involves many higher harmonics of the fundamental 2-D waves. (b) A3-D surbharmonic instability occurs for frequencies close to the neutral curve f,(R). In this case, the transverse modulations are out of phase for successive wave fronts, and herringbone patterns result. It is shown that this weakly nonlinear instability is due to the resonant excitation of a triad of waves consisting of the fundamental two-dimensional wave and two oblique waves. The evolution of wavy films after the onset of either of these 3-D instabilities is complex. However, sufficiently far downstream, large-amplitude solitary waves absorb the smaller waves and become dominant. Q I995 American Institute of Ph.ysics.

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

“…Note that this result extends the one obtained for the pure hydrodynamic mode by Yih (1955) who extended the Squire's approach to free surface flows (Huerre & Rossi 1998); (iii) as a consequence of the previous point and as shown experimentally, it always exists a transition zone of purely two-dimensional flow when the film is periodically forced at the inlet (Liu et al 1995). Moreover, the secondary three-dimensional instability develops at a certain distance from the inlet that increases with decreasing Reynolds numbers (Alekseenko et al 1994).…”

Wave patterns in film flows: modelling and three-dimensional waves

“…The secondary instability is found to be not much selective, which explains the widespread presence of the synchronous instability observed in the experiments by Liu et al (1995) whereas Floquet analysis predicts a subharmonic scenario in most cases. Three-dimensional wave patterns are next computed assuming periodic boundary conditions.…”

“…More recent experimental results are presented for example in Nosoko et al (1996), Vlachogiannis & Bontozoglou (2001), Park & Nosoko (2003), Nosoko & Miyara (2004), or Argyriadi et al (2004). At Haverford, Gollub and coworkers have performed an extensive study of water-glycerin mixtures flowing down weakly inclined planes, see , , Liu & Gollub (1994), and Liu et al (1995). Controlling the entrance flow rate, they applied a periodic forcing at the inlet and observed the response of the film at given frequency.…”

“…The purpose of this paper is to propose an accurate modelling able to account for the experiments by Liu et al (1995) and ultimately get a unified theoretical understanding of the experimental data available in the literature. The separation of scales implied by the long-wave character of the instability allows one to define a small parameter , called the film parameter , basically measuring the slope of the interface in order of magnitude, and to apply Prandtl's simplification of the cross-stream momentum equation, usual in boundary layer theory, which helps one to get rid of the in-depth pressure distribution dominated here by surface tension and gravity.…”

“…Transition from 2D to 3D flows is shown to be strongly dependent on initial conditions. The herringbone patterns, the synchronously deformed fronts and the threedimensional solitary waves observed in experiments (Liu et al 1995;Park & Nosoko 2003;Alekseenko et al 1994) are recovered using our regularised model, which is found to be an excellent compromise between the complete model, which has seven equations, and the simplified model, which does not include the second-order inertia corrections. Those corrections are found to play a role in the selection of the type of secondary instability as well as of the spanwise wavelength of the emerging pattern.…”