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
The multiplicity patterns of olefin oxidation, catalyzed by a Pt wire controlled to maintain a preset average temperature, are traced by varying the reactant concentrations and are mapped in the concentration plane. The similar bifurcation maps of ethylene and propylene oxidations exhibit regions of tristability and isolated branches in the directions of olefin and oxygen concentrations.Isobutylene oxidation exhibits mushroom-shaped bistability. Several features indicate the occurrence of symmetry breaking to form a partially ignited wire. Comparison of these results with the characteristic bifurcation maps of inhomogeneous solutions, drawn in part 1 of this work, shows that these complex patterns are induced by coupling of reactant inhibition with strong exothermicity. This work also develops an efficient methodology for fast tracing of all the multiplicity features of reacting systems. Bifurcation diagrams are traced automatically with a microcomputer-governed experimental system, and the continuity of features in the second and third dimensions is studied off-line to decide on further experimentation.In the first part of this work (Sheintuch, 1989), we demonstrated that surprisingly complex multiplicity patterns can be induced by relatively simple kinetics in systems that attain stable inhomogeneous solutions. We reviewed previous observations of bistability in reactions catalyzed by an "isothermal" wire, i.e., controlled to maintain a preset (average) temperature, and showed that the patterns with counterclockwise hysteresis or with isolated branches can be accounted for by stationary thermal fronts in systems with simple kinetics or with reactant inhibition. The analysis was also extended to systems that show tristability in order to account for the observations reported in this part.This work presents a comparative study of multiplicity patterns and bifurcation maps in olefin oxidation on a Pt wire, using a thermochemic method. Regions of uniqueness, bistability, and tristability are mapped in the oxygen versus olefin concentration plane at several temperatures.
The conclusion from the RHF/4-31G*//RHF/4-31G* structure and frequency calculations is that 1-5 are stable. The structures 2-5 are more rigidly defined than 1, as measured by their lowest (nontorsional) vibrational frequency.The calculated NN bond lengths in 1-5 mimic analogous calculated NN bond lengths in smaller unstrained NH molecules in almost all cases.Assuming the calculated RHF/4-31G*//RHF/4-31G* energy differences accurately represent reality, all five N6 structures are highly metastable to decomposition into three N2 molecules. This
Stable asymmetric states may be induced in catalytic wires controlled to attain a preset average temperature by changing the current fed into it. Complex multiplicity patterns are found there even at a fixed average temperature. A single-valued rate expression induces two bistability domains in a uniform wire. Cooling effects at the wire edge may eliminate one of these domains. Apparent
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.