Threshold condition for spray formation by Faraday instability

Abstract A vertically vibrating liquid layer produces liquid ligaments that disintegrate to form a spray with drops of a controllable size. Previous experimental investigations of ultrasonic atomisation have shown that when such a spray forms, there exists a predominant surface-wave mode from which drops are generated with a mean diameter that follows Lang’s equation. In this paper, we determined this predominant surface-wave mode physically and, by utilising the coupled level-set and volume-of-fluid method, we numerically studied the threshold condition for spray formation based on a cell model of the predominant surface wavelength that excludes the effects of the container walls. We defined a condition whereby the broken drop holds a zero area-averaged vertical velocity in the laboratory reference frame as the criterion for the formation of a spray. The results of our calculations indicated that the onset of a spray occurs in the subharmonic unstable region for a threshold dimensionless forcing strength ${\it\beta}_{c}=({\it\rho}_{l}{\it\Delta}_{0}^{3}{\it\Omega}^{2})/{\it\sigma}\sim O(1)$ , where ${\it\rho}_{l}$ and ${\it\sigma}$ denote the liquid density and surface tension coefficient, respectively, ${\it\Delta}_{0}$ is the forcing displacement amplitude and ${\it\Omega}$ is the forcing angular frequency. Spray formation due to the Faraday instability can be considered as a process whereby the liquid layer absorbs energy from the inertial force, and releases it by producing drops that leave the surface of the liquid layer. We demonstrated that for a deep liquid layer, the threshold condition for the formation of a spray is determined only by the forcing strength, and is independent of the initial conditions of the liquid surface.