Differential and Variational Formalism for an Acoustically Levitating Drop

Abstract

Starting with a most general problem on interface waves between two ideal compressible fluids, treated here as an ullage gas and a liquid, respectively, and separating fast and slow time scales, differential and variational formalism for an acoustically levitating drop and its time-averaged shape (the drop vibroequilibrium) is developed. The drop vibroequilibria can differ from spherical shape; stable vibroequilibria are associated with local minima of the quasipotential energy whose analytical form is also derived in the present paper.

The Bateman-type variational formalism for an acoustically-driven drop

“…Th eorem 6. Under the smoothness assumption above, the zero fi rst variation of the action (5), (7) in fl uid domains, the kinematic boundary condition (8a) on the interface and the `vibrating box surface' condition (8b), the Bernoulli equations (14) (alternatively, the Euler equations (15)) in fl uid domains, the dynamic interface condition (20) on the interface as well as the vortex line conditions (10) and (12) provided by the defi nitions of pressure (5) and velocity fi elds (2).…”

“…Conclusions and discussion. By using the Clebsch potentials, the Bateman-type variational formulation from [1,2] can be generalised for barotropic fl uids to the case of rotational fl uid fl ows. Th e free-interface boundary value problem derived from the Bateman-type variational formulation not necessary has a unique solution.…”

“…Idea of the present paper comes from [1,2] whose objects are two oscillating compressible ideal barotropic fl uids when an acoustic vibrator is located in one of them to both generate a high-frequency acoustic fi eld and govern the interface motions. Physically, these two papers deal with an acoustical positioning of a large liquid mass (volume) in microgravity conditions and an acoustically-levitating drop, respectively.…”