Methods of deterministic chaos applied to the flow of thin wavy films

Abstract

The structure of thin, wavy falling films was studied to evaluate whether the random-appearing wave structure is a result of deterministic chaos or a purely stochastic process. The time-varying film thickness was obtained at different spatial locations near the point of wave inception for flow rates in the range of lntroduc t ionThe hydrodynamic behavior of thin wavy falling films has been a subject of intensive investigation for about forty years. These films are widely employed in equipment for heat transfer, mass transfer, and chemical reacting systems. In addition to the practical need for understanding the mechanics of this type of flow, there are challenging theoretical problems embedded in the task of modeling these wavy films. This combination has given rise to an extensive literature on this subject.The earliest work was based on the use of integral equations of the boundary layer type to solve the equations of motion (Kapitza and Kapitza, 1949;Shkadov, 1967). These approaches were based on the assumed existence of a periodic interface and produced first-order estimates for wavelength and velocity. A long series of papers including work by Benney (1966), Lin (1969), and Whitaker (1964), used linear stability analysis to find the wavelength and velocity of the fastest growing wave, again assuming a periodic perturbation. This initial periodic disturbance is thought to evolve into the more complex waveforms observed in experiments as a result of the nonlinear nature of the equations. Nonlinear stability analyses have also been pursued (Pumir, 1983). Recent work on the nonlinear But even a cursory examination of measured wave traces raises some doubts as to the usefulness of the idea of a smallamplitude periodic wave as the model for the initial phase of the wave motion or of isolated waves as a model for the developed ones. Figure Id shows a wave trace for a falling liquid film of water-glycerine solution taken with a conductivity probe mounted in a vertical pipe of 50.8 mm dia. as described below. The flow rate corresponds to a Reynolds number of 3.9. The probe was located 0.346 m below a carefully leveled, sharpedged overflow weir that served as the feed device. The film thickness data are shown after low-pass digital filtering at 25 Hz to remove noise. At this location the wave amplitude is less than 0.25% of the mean film thickness. At all positions closer to the feed the waves were so small that they could barely be detected even with the special circuitry used for this purpose. Note that while the period between successive waves is quite regular, the amplitude is very random. Kapitza and Kapitza (1949) in their classical study of waves on falling films found it necessary to pulse the feed to produce periodic waves. In the absence of pulsing they too reported that the waves were random in appearance. Thus one must question whether the

Chaotic behavior of fluidized beds based on pressure and voidage fluctuations

“…Time signals from natural phenomena are subject to noise which can significantly increase the correlation dimension in nonlinear time-sequence analysis (Lacy et al, 1991). In this study, a low-pass filter set at 20 Hz has been used in data sampling.…”

Instabilities of three-dimensional viscous falling films

“…Experimental studies of falling films have been done by, among others, Kapitza & Kapitza (1949), Krantz & Goren (1971), Portalski & Clegg (1972), Alekseenko, Nakoryakov & Pokusaev (1985), and Lacy, Sheintuch & Dukler (1991). As summarized by Alekseenko et al, in most of the experiments performed, twodimensional regular waves are observed only near the wave-inception line.…”

Azimuthal and streamwise disturbances in a fluid layer flowing down a rotating cylinder

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