Abstract:Faraday instability on the droplet surface due to external periodic oscillation is widely used in ultrasonic atomization, spraying processing and other applications. The analysis of Faraday instability is of great significance to the study of the surface dynamics of vibrating droplet. In this paper, the Faraday instability problem is extended from radial vibration to vertical vibration, and the instability of inviscid droplet surface wave in vertical vibration is studied. The vertical vibration makes the droplet momentum equation a Mathieu equation with spatial correlation term and time periodic coefficient. The dispersion relations between the growth rate, the mode number and flow parameters of vertically vibrating droplet surface waves are obtained by using Floquet theory. The neutral stable boundary of vertically vibrating inviscid droplet under Faraday instability is obtained by solving an eigenvalue problem of surface deformation modes. The difference of droplet neutral stability boundary between vertical vibration and radial vibration is compared. The influence of elevation angle θon the neutral instability boundary is obtained by the approximate calculation under the assumption of large mode number. The results show that the difference between vertically vibrating droplets and radially vibrating droplets is obvious. The differences are as follows: in the case of harmonic, the unstable region of droplet surface wave becomes smaller, and the droplet will be more difficult to destabilize under external excitation; In the case of subharmonic, the neutral stable boundaries of the droplet surface wave coincide, and the droplet unstable wave will not appear subharmonic mode. Besides, for vertically vibrating droplets, the larger the elevation angle θ, the smaller the neutral instability region, and the easier it is for the droplet surface to remain stable under external excitation.