This paper presents a theoretical and experimental investigation into a novel class of water-wave motions in narrow open channels. The distinctive condition on these motions is that the lines of contact between the free surface and the sides of the channel are fixed, which condition bears crucially on the hydrodynamic effects of surface tension. Most of the account concerns travelling waves in channels of rectangular cross-section that are exactly brimful, but the relevance of this prototype to other, more usual situations is explained with reference to the phenomenon of contact-angle hysteresis.In § 2 a linearized theory is developed which poses an eigenvalue problem of unusual kind. Unlike the familiar and much simpler problem corresponding to mobile lines of contact at which the free surface remains horizontal, the new problem has no explicit solution and the edge conditions are not automatically compatible with the kinematic conditions at the solid boundaries. Treatment by functional analytic methods is necessary to verify that solutions exist having physically appropriate properties, but this approach gives a final bonus in securing comparatively easy estimates for some of these properties. A variational characterization of the eigenvalues is used to settle questions of existence and the ordering of possible wave modes, and finally to establish approximate formulae relating wavelength to frequency.In § 3 experiments are reported which were performed with clean water filling three channels made of Perspex. Over continuous ranges of frequency, delimited so that only the fundamental progressive-wave mode was generated, wavelengths were measured by an electronic technique. The measurements agree well with the theoretical predictions, diverging markedly from behaviour to be expected in the absence of edge constraints.Appendix A outlines a supplementary theoretical argument proving that the first eigenvalue of the problem treated in § 2 is always simple. Appendix B reviews three generalizations of the theory.
Surface-active material is present in most naturally occurring water samples, and it naturally diffuses steadily to free surfaces, where it both reduces the surface tension and gives the surface elastic properties which enable it to resist compression. When the water flows so that the surface layer is trapped and compressed against a fixed shallow-draught barrier the film material makes the surface incompressible, and flow beneath the barrier forms a viscous boundary layer under the film. The stresses associated with this boundary layer are found to distort the surface in the region of the leading edge of the film, giving rise to a phenomenon which is commonly observed in nature and which has been called the Reynolds ridge. This paper describes experimental work on the measurement of the ridge, and compares the results with a theoretical model due to Harper & Dixon. Good agreement is indicated.
This paper describes the development of a new technique for removing surface-active contaminants from water so that it may be used for surface-clean experiments in fluid mechanics. The removal of contamination from concentrated aqueous solutions of non-surface-active materials is also possible, allowing considerable variation of density and viscosity in these experiments. Instead of using the conventional distillation processes necessary for most work involving surface chemical phenomena, surface-active substances are removed by adsorption onto a dense current of small nitrogen gas bubbles rising in a long vertical column of the water. The efficiency of the technique was found to increase to a great extent when common salt was dissolved in the water, up to concentrations around 40 g/l, and experiments are described which allow the identification of several physical processes which aid the operation of the method. The adequate performance of the cleaning technique in the removal both of an ionic surface-active material and also of Gentian Violet dye is demonstrated. The method should permit the design of surface-clean experiments using much larger volumes of water than are possible when the water has to be distilled using conventional surface chemical techniques. A simple apparatus for the measurement of surface tension is also described.
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