Development of strongly nonlinear structures at the charged boundary of a non-conducting liquid in an electric field

Abstract

Direct numerical simulation of the strongly nonlinear stages of instability development for a non-conducting liquid with a charged free surface in a normal electric field is performed. It is demonstrated that two main stages of the instability can be distinguished: an initial stage, during which dimples appear on the surface, and a developed stage, during which these dimples transform into expanding bubbles. The bubble size increases with increasing applied field, despite the fact that the scale corresponding to the dominant mode of the instability decreases.

Figures (4)

• Figure 1: Fig. 1. (Color online) Evolution of the free surface within the exact solution \ref{['eq8']} and \ref{['eq9']} for $\beta =1$ with the initial condition $A(0)=0.1$. The moment of collapse $t\approx 1.8076$. The inset shows the time dependencies at the point $x=0$ of the electrostatic pressure (blue line) and the capillary pressure (red line), which was not taken into account in the calculations.
• Figure 2: Fig. 2. (Color online) Evolution of the liquid surface obtained as a result of numerical solution of the model \ref{['eq6']} and \ref{['eq7']} for $\beta =1$ and the initial condition \ref{['eq10']} with $A=0.1$. Successive moments of time $t=0, \, 7.7,\, 10.0,\, 13.3$ are shown.
• Figure 3: Fig. 3. (Color online) Time dependencies of electrostatic pressure (blue line) and capillary pressure (red line) at point $x=0$ for $\beta =1$ and initial condition \ref{['eq10']} with $A=0.1$.
• Figure 4: Fig. 4. (Color online) Average radius of the formed bubble depending on the parameter $\beta$. The inset shows the shapes of the liquid surface at the moment of bubble detachment for $\beta ^{2} = 0.75, \, 0.8, \, 0.9, \, 1, \, 1.15, \,1.2$.