Stability of radially expanding thin liquid sheets: a modified thinning model

The growth of instabilities and the subsequent break-up of liquid sheets have been widely studied to understand nozzle atomisation. Two different theoretical approaches have been proposed to explain the spatial growth of instabilities. The first is based on the aerodynamic interactions between the liquid sheet and the surrounding air, in which the sheet becomes unstable to infinitesimal perturbations, similar to the Kelvin–Helmholtz instability. In the absence of aerodynamic interactions, the theory predicts zero spatial growth. An alternate theory ignores the surrounding air and instead relies on the thinning of the liquid sheet to predict spatial growth. Here, we rederive the governing equations for the thinning theory and show that when a small vertical displacement (or perturbation) with zero slope is introduced at the centre of the liquid sheet, the perturbation is convected downstream with finite amplification, but the perturbation does not diverge at the edge of the sheet. On the other hand, a perturbation with a finite slope diverges. We compare the model predictions with displacement measurements over a range of Weber numbers and geometries to assess the model’s robustness. We demonstrate that low-frequency perturbations, in the absence of aerodynamic interactions, such as noise from the experimental set-up or impact waves generated at the point of impingement, can explain the spatial growth, particularly in the low-Weber-number regime. The theory predicts growth rates close to the measured values at low Weber numbers, where aerodynamic interactions are negligible.