The stability of a horizontal fluid interface in a vertical electric field

The stability of the horizontal interface between conducting and non-conducting fluids under the influence of an initially uniform vertical electric field is discussed. To produce such a field when the conducting fluid is the heavier it is imagined that a large horizontal electrode is immersed in the non-conducting fluid. As the field increases the part of the interface below the electrode rises till at a voltage V, which depends on the interfacial tension, the height of the electrode above the interface and the density difference, the interface becomes unstable for vertical displacements Z which satisfy the equation (²{ x²} + ²{ y²}+k²) Z=0. The value of k consistent with the lowest value of V is found. When the electrode is situated above the interface at less than a certain distance the lowest value of V is attained when k = 0 so that the horizontal extent of an unstable crest is likely to be great. As the electrode height increases above this critical value k increases and the unstable crests become more closely spaced till an upper limiting value of k is obtained. Experiments made with several pairs of fluids verify these theoretical conclusions. In some cases sparking occurs before the potential V is reached, but in others, air at atmospheric pressure over water, for instace, the instability occurs first and the jet of water which results permits the passage of a spark which may inhibit further development of the instability. The physical condition which determines the sparking voltage to a fluid may therefore be very different from that which is operative between solid electrodes. This consideration might be relevant to the performance of power-line insulators in wet weather.