Parametric instability of a liquid sheet

When a liquid sheet is subject to vertical vibration, parametric instabilities may occur and give rise to standing waves. The present work develops the linear theory of this phenomenon for inviscid, Newtonian and viscoelastic liquids. Floquet theory is applied to the governing equations to obtain recursion relations for the temporal modes of the deformations of the free surfaces. In contrast to the case where there is no vibration, we find that the symmetric and antisymmetric deformations of the liquid sheet are coupled to each other. For the inviscid case, the recursion relations are shown to be equivalent to a pair of coupled Mathieu equations. For the Newtonian and viscoelastic cases, the recursion relations are converted into an eigenvalue problem which is solved numerically to obtain the critical vibration amplitude needed to excite the instability, along with the corresponding critical wavenumber. The results display behaviour which is similar in several ways to that of the classic Faraday instability. The parametric instabilities studied here may be an important mechanism in the initial stages of foam destruction by ultrasound, as well as in several other practical applications.