Formation of singularities on the charged surface of a liquid-helium layer with a finite depth
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C PROPERTIES OF SOLIDS
Formation of Singularities on the Charged Surface of a Liquid-Helium Layer with a Finite Depth N. M. Zubarev Institute of Electrophysics, Ural Branch, Russian Academy of Sciences, ul. Amundsena 106, Yekaterinburg, 620016 Russia e-mail: [email protected] Received Mach 19, 2008
Abstract—The dynamics of the development of an instability of a charged surface of a liquid-helium layer with a finite depth is investigated. The equations describing the evolution of the free surface are derived with the use of conformal variables for the case in which the charge completely screens the electric field above the liquid. A model of the evolution of a spatially localized perturbation of a liquid-helium surface is proposed for the strongfield limit where the dynamics of the liquid is predominantly determined by the effect of electrostatic forces. This model describes the development of an instability of the initially planar boundary to the point of the formation of cuspidal dimples. The limit of an infinitely deep liquid is considered. The stability of the previously revealed liquid flow regime described by the Laplacian growth equations is proved without significant constraints on the surface geometry. PACS numbers: 67.30.hp, 68.03.-g, 47.65.-d DOI: 10.1134/S1063776108100154
1. INTRODUCTION The liquid-helium surface can be charged to rather high surface densities of a negative electric charge [1, 2]. The ability of electrons to freely move over the liquid-helium surface provides the equipotentiality of this surface on characteristic hydrodynamic times and scales. If the charge density is relatively low and the surface deformation is significant, the charge can appear to be insufficient to equalize the potential over the entire surface. This leads to the formation of socalled multiply charged dimples (see, for example, [3−5] and references therein). We will consider the opposite case in which the charge density is sufficiently high that the charges are redistributed over the deformed surface and completely screen the electric field above the surface. This situation was observed in the experiments described in [6, 7]. It was experimentally revealed that, on the liquid-helium surface, there appear dimples that are sharpened for a finite time. A similar scenario for the development of the instability is observed for a conducting liquid (liquid metal) in an electric fields [8–10]. The main difference is that the charge induced on the surface screens the electric field inside the liquid rather than the electric field above the liquid, as is the case in the experiments described in [6, 7]. This is responsible for the formation of cusps (rather than dimples), in particular, the so-called Taylor cones [11]. It should also be noted that the liquidhydrogen surface charged by ions is also characterized by an instability [12]. In a sufficiently strong electric field, cuspidal structures (geysers) resembling the Taylor cones are formed on the liquid-hydrogen surface. In
this paper, we will restrict our consideration to
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