Instability of a gravitational liquid sheet

In this paper the linear stability of a two-dimensional liquid sheet falling vertically under gravity is analyzed referring to wave packet solutions of the initial value problem. As the gravity effect is fully considered, the basic flow is nonuniform i.e. spatially developing. However, the length scale over which the inertia-gravity effects become significant is far larger than the sheet thickness, thus suggesting a multiple scale approach. This make for a local character (slightly non parallel flow) of the system response to any given perturbation. According to previous findings of the relevant literature, the model is developed within an inviscid approximation, as regards both the basic flow and the perturbed state. It is found that for sinuos disturbances the sheet undergoes a transition from locally absolute to convective instability at a certain location along the (vertical) stream-wise direction. This critical distance from the slit exit section increases with decreasing the flow Weber number (namely, for istance, the liquid flow rate per unit length), but is independent of the gas-to-liquid density ratio. In the region close to the slit an algebraic growth of disturbances is found, which exhibits an absolutely unstable character. However, if the region of locally absolute instability is sufficiently small, following Chomaz, Huerre and Redekopp (1998), it may be hypothesized that the system is globally stable. Beyond a critical size, self-sustained unstable global modes should arise. This agrees with the experimental evidence that the sheet breaks up as the flow rate is reduced, any other quantities being kept constant. Although it is believed that the liquid viscosity may act in removing the algebraic growth. The time after which this occurs could be not sufficient to avoid for possible nonlinear phenomena to appear and break-up the sheet.