The influence of external gas on the stability of a liquid curtain developing in space

An inviscid model is developed to study the linear stability of a liquid curtain moving in a still gas. The growth of transverse waves, recognized to be responsible for instability, is analysed by resorting to localized perturbations, hence to wave packet solutions. As the gravity effect is fully considered, the basic flow is not invariant along continuous translations in the streamwise direction (spatially developing flow). However, the evolution length scale is large with respect to the curtain thickness and allows a multiple scale approach. The dispersion relations for both sinuous and varicose disturbances are derived to the zeroth order approximation: they exhibit a local character. For sinuous modes a critical local Weber, above which the flow is unstable, is found. The instability, due to the external gas, has a convective character, therefore it is not responsible for the curtain rupture. The range of unstable wavenumbers broadens as the gas-to-liquid density ratio increases. Previous related papers employing the spatial mode approach are revisited, because the growth in space is of physical significance only in the presence of growth in time.