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Stability of thin, radially moving liquid sheets

An analysis of the stability of thin viscous liquid sheets, such as those emitted from industrial spraying nozzles, is presented. These sheets are in the form of a circular sector whose thickness reduces as the distance from the nozzle increases. The Kelvin-Helmholtz type of instability usually observed causes the breakup and atomization of the sheet, as required in most industrial spraying processes. Waviness, like that of a flapping flag, is produced and increasing amplitudes finally cause breakup. An analytical solution in the form of hypergeometric functions for the shape of the sheet and the waves is obtained. This solution includes, as special cases, analyses existing in the literature, in addition to establishing the possibility of a new type of instability dependent on the distance from the nozzle. Also, the classical type of instability, in which the waves increase with time, is examined and relations for unstable waves as a function of parameters such as the fluid viscosity, surface tension and sheet velocity are obtained. It is shown that there is no single wave that has a maximum growth rate, but that the wavenumber for maximum instability increases with the distance from the nozzle orifice.