On the temporal instability of a two-dimensional viscous liquid sheet

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Abstract
This paper reports a temporal instability analysis of a moving thin viscous liquid sheet in an inviscid gas medium. The results show that surface tension always opposes, while surrounding gas and relative velocity between the sheet and gas favour, the onset and development of instability. It is found that there exist two modes of instability for viscous liquid sheets – aerodynamic and viscosity-enhanced instability – in contrast to inviscid liquid sheets for which the only mode of instability is aerodynamic. It is also found that axisymmetrical disturbances control the instability process for small Weber numbers, while antisymmetrical disturbances dominate for large Weber numbers. For antisymmetrical disturbances, liquid viscosity, through the Ohnesorge number, enhances instability at small Weber numbers, while liquid viscosity reduces the growth rate and the dominant wavenumber at large Weber numbers. At the intermediate Weber-number range, Liquid viscosity has complicated effects due to the interaction of viscosity-enhanced and aerodynamic instabilities. In this range, the growth rate curve exhibits two local maxima, one corresponding to aerodynamic instability, for which liquid viscosity has a negligible effect, and the other due to viscosity-enhanced instability, which is influenced by the presence and variation of liquid viscosity. For axisymmetrical disturbances, liquid viscosity always reduces the growth rate and the dominant wavenumber, aerodynamic instability always prevails, and although the regime of viscosity-enhanced instability is always present, its growth rate curve does not possess a local maximum.