A three-dimensional coupled Nitsche and level set method for electrohydrodynamic potential flows in moving domains

In this paper we present a new algorithm for computing three-dimensional electrohydrodynamic flow in moving domains which can undergo topological changes. We consider a non-viscous, irrotational, perfect conducting fluid and introduce a way to model the electrically charged flow with an embedded potential approach. To numerically solve the resulting system, we combine a level set method to track both the free boundary and the surface velocity potential with a Nitsche finite element method for solving the Laplace equations. This results in an algorithmic framework that does not require body-conforming meshes, works in three dimensions, and seamlessly tracks topological change. Assembling this coupled system requires care: while convergence and stability properties of Nitsche's methods have been well studied for static problems, they have rarely been considered for moving domains or for obtaining the gradients of the solution on the embedded boundary. We therefore investigate the performance of the symmetric and non-symmetric Nitsche formulations, as well as two different stabilization techniques. The global algorithm and in particular the coupling between the Nitsche solver and the level set method are also analyzed in detail. Finally we present numerical results for several time-dependent problems, each one designed to achieve a specific objective: (a) The oscillation of a perturbed sphere, which is used for convergence studies and the examination of the Nitsche methods; (b) The break-up of a two lobe droplet with axial symmetry, which tests the capability of the algorithm to go past flow singularities such as topological changes and preservation of an axi-symmetric flow, and compares results to previous axi-symmetric calculations; (c) The electrohydrodynamical deformation of a thin film and subsequent jet ejection, which will account for the presence of electrical forces in a non-axi-symmetric geometry.