Interfacial electrohydrodynamic waves under horizontal electric fields: Hamilton’s principle and multi-scale modeling

This paper is concerned with the multi-scale modeling of interfacial waves between two dielectric fluids under a horizontal electric field. First, we give a detailed proof of the Hamilton principle for this system. Next, based on the Hamiltonian structure and the analytical property of the Dirichlet-Neumann operator, the kinetic energy and electric potential energy in the Hamiltonian are expanded into the form of convergent series, and the order of truncation is determined. Finally, the reduced model is obtained by calculating the variational derivatives of the approximate total energy after truncation. The above process provides a systematic method for establishing nonlinear multi-scale models. Taking the case of “deep upper layer and shallow lower layer” as an example, we describe the whole modeling process in detail. Furthermore, the nonlinear coherent structure in the newly proposed model is computed using the modified Petviashvili iterative method. The asymptotic technique developed in this paper differs from previous work. Its advantage is that the derived reduced models naturally retain the energy conservation property; at the same time, this paper also extends the previous results to the three-dimensional situation.