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Register a new userEquilibrium Configurations of the Charged Surface of a Conducting Liquid at a Finite Interelectrode Distance
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The problem of determining equilibrium configurations of the free surface of a conducting liquid is considered with allowance for a finite interelectrode distance. The analogy is established between this electrostatic problem and that of finding the profile of a progressive capillary wave on the free surface of a liquid layer of a finite depth, which was solved by Kinnersley. This analogy allowed exact solutions to be obtained for the geometry of liquid electrodes, which expand the existing notions about the possible stationary states of the system.
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A wide class of exact solutions is obtained for the problem of finding the equilibrium configurations of charged jets of a conducting liquid; these configurations correspond to the finite-amplitude azimuthal deformations of the surface of a round jet. A critical value of the linear electric charge density is determined, for which the jet surface becomes self-intersecting, and the jet splits into two. It exceeds the density value required for the excitation of the linear azimuthal instability of the round jet. Hence, there exists a range of linear charge density values, where our solutions may be stable with respect to small azimuthal perturbations.
The dynamics of the development of instability of the free surface of liquid helium, which is charged by electrons localized above it, is studied. It is shown that, if the charge completely screens the electric field above the surface and its magnitude is much larger then the instability threshold, the asymptotic behavior of the system can be described by the well-known 3D Laplacian growth equations. The integrability of these equations in 2D geometry makes it possible to described the evolution of the surface up to the formation of singularities, viz., cuspidal point at which the electric field strength, the velocity of the liquid, and the curvature of its surface assume infinitely large values. The exact solutions obtained for the problem of the electrocapillary wave profile at the boundary of liquid helium indicate the tendency to a charge in the surface topology as a result of formation of charged bubbles.
A small drop of a heavier fluid may float on the surface of a lighter fluid supported by surface tension forces. In equilibrium, the drop assumes a radially symmetric shape with a circular triple-phase contact line. We show theoretically and experimentally that such a floating liquid drop with a sufficiently small volume has two distinct stable equilibrium shapes: one with a larger and one with a smaller radius of the triple-phase contact line. Next, we experimentally study the floatability of a less viscous water drop on the surface of a more viscous and less dense oil, subjected to a low frequency (Hz-order) vertical vibration. We find that in a certain range of amplitudes, vibration helps heavy liquid drops to stay afloat. The physical mechanism of the increased floatability is explained by the horizontal elongation of the drop driven by subharmonic Faraday waves. The average length of the triple-phase contact line increases as the drop elongates that leads to a larger average lifting force produced by the surface tension.
Applying the method of integral estimates to the analysis of three-wave processes we derive the sufficient criteria for the hard loss of stability of the charged plane surface of liquids with different physical properties. The influence of higher-order wave interactions on the instability dynamics is also discussed.
The nonlinear dynamics of charged-surface instability development was investigated for liquid helium far above the critical point. It is found that, if the surface charge completely screens the field above the surface, the equations of three-dimensional (3D) potential motion of a fluid are reduced to the well-known equations describing the 3D Laplacian growth process. The integrability of these equations in 2D geometry allows the analytic description of the free-surface evolution up to the formation of cuspidal singularities at the surface.
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