Formation of root singularities on the free surface of a conducting fluid in an electric field

Abstract

The formation of singularities on a free surface of a conducting ideal fluid in a strong electric field is considered. It is found that the nonlinear equations of two-dimensional fluid motion can be solved in the small-angle approximation. This enables us to show that for almost arbitrary initial conditions the surface curvature becomes infinite in a finite time.Electrohydrodynamic instability of a free surface of a conducting fluid in an external electric field [1,2] plays an essential role in a general problem of the electric strength. The interaction of strong electric field with induced charges at the surface of the fluid (liquid metal for applications) leads to the avalanche-like growth of surface perturbations and, as a consequence, to the formation of regions with high energy concentration which destruction can be accompanied by intensive emissive processes.In this Letter we will show that the nonlinear equations of motion of a conducting fluid can be effectively solved in the approximation of small perturbations of the boundary. This allows us to study the nonlinear dynamics of the electrohydrodynamic instability and, in particular, the most physically meaningful singular solutions.Let us consider an irrotational motion of a conducting ideal fluid with a free surface, z = η(x, y, t), that occupies the region −∞ < z ≤ η(x, y, t), in an external uniform electric field E. We will assume the influence of gravitational and capillary forces to be negligibly small, which corresponds to the conditionwhere g is the acceleration of gravity, α is the surface tension coefficient, and ρ is the mass density. The potential of the electric field ϕ satisfies the Laplace equation, ∆ϕ = 0, with the following boundary conditions, ϕ → −Ez, z → ∞,

Formation of curvature singularities on the interface between dielectric liquids in a strong vertical electric field

“…According to Refs. [24,25], we have ∼ S 1/3 for the initial conditions (20) with S > 0 (see also Sec. VII).…”

“…Let us assume that the boundary is symmetric with respect to the point x = x c , where a singularity is formed. It follows from the expressions (16) and (24) that, in the vicinity of the singularity, the interface shape looks like…”

“…[24,25] for conducting liquids (e.g., liquid metals) and in Refs. [26,27] for dielectric liquids with conducting surfaces (e.g., liquid helium with the electroncharged surface [28]), the nonlinear evolution of the free surface can be effectively studied by analytical methods in the strong-field limit, where the surface motion is completely determined by electrostatic forces.…”

Formation of singularities on the surface of a liquid metal in a strong electric field

“…(The branch-point nature of this singularity agrees with the condition that the angles be small.) To answer this question, we will examine the evolution of the perturbation (18) according to the nonlinear equation (9).…”

“…Section IV is devoted to a study of the dynamics of one-dimensional surface perturbations. Integration of the model equations shows that it takes only a finite time for weak singularities of the root type to form in the system, i.e., singular points at which the curvature of surface is infinite (see also the Letter [9]). …”