The stability of a fluid film on the surface of a cone

Abstract

The flow is studied of a fluid film originating from a source at the apex of a right circular cone, and flowing under gravity over its outer surface. A modified Orr-Sommerfeld equation is derived for the flow. This and the corresponding boundary conditions contain terms introducing distance along the cone generator as a parameter. The system of equations is solved for small wave number to give e. stability criterion, which shows the flow to be unstable for all practical values of Reynolds number. Results of experiments on a 60 included angle cone are presented, giving the pattern of behaviour of the main film and waveform characteristics. Mean film thickness is expressed as a function of Reynolds number only and found to be less than that of an undisturbed laminar film. Wave amplitude is shown to reach a limiting value and thereafter decline. Wave number is shown to be linearly related to a non-dimensional parameter incorporating viscosity, surface tension and slope derived from one proposed by Berbente and Ruckonstein whose theory also matches well the limiting amplitudes measured. Wave speed and wavelength are also measured and discussed.