Absolute and convective instability of a liquid sheet

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Abstract
The linear stability of a viscous liquid sheet in the presence of ambient gas is investigated. It is shown that there are two independent modes of instability, sinuous and varicose. The large-time asymptotic amplitude of sinuous disturbances is found to be bounded but non-vanishing for all calculated values of Reynolds numbers and the gas-to-liquid density ratios when the Weber number is greater than one half. The Weber numberWeis defined as the ratio of the surface tension force to the inertia force per unit area of the gas–liquid interface. WhenWeis smaller than one half, the sinuous mode is stable if the gas-to-liquid density ratio is zero, otherwise it is convectively unstable. The varicose mode is always convectively unstable unless the density ratio,Q, is zero. Then it is asymptotically stable. The spatial growth rate of the varicose mode is smaller than that of the sinuous mode for the same flow parameters. The wavelength of the most amplified waves in both modes is found to scale with the product of the sheet thickness andQ/We. It is shown, by use of the energy equation, that the mechanism of instability is a capillary rupture whenWe[ges ] 0.5, and the convective instability is due to the interfacial pressure fluctuation whenWe< 0.5.

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