Temporal instability of a liquid sheet in two gas streams of unequal velocities with acoustic oscillations

This study conducts a temporal instability analysis of liquid sheets subjected to two gas streams with unequal velocities, considering the influence of acoustic oscillations. Using Floquet theory, the dispersion relation is derived, and the effects of gas densities and velocities on instability are investigated. Two distinct unstable modes are identified: the para-sinuous mode, primarily dominated by the gas stream with larger aerodynamic forces, and the para-varicose mode, constrained by the gas stream with smaller aerodynamic forces. The difference between these modes and the purely varicose/sinuous modes is significant at small wavenumbers but diminishes at large wavenumbers. The introduction of acoustic oscillations leads to a parametric unstable region where para-varicose modes are more likely to occur, while para-sinuous modes are more sensitive to increases in oscillating amplitude. Additionally, an increase in the Ohnesorge number and time constant ratio, along with a decrease in the elasticity number, inhibits instability. The oscillating frequency plays a critical role in determining the location of the parametric unstable region, with higher frequencies generally inhibiting parametric instability while enhancing Kelvin–Helmholtz instability.