Dynamics of a drop under the influence of an inhomogeneous electric field

Abstract

The forced oscillations of an incompressible liquid drop under the influence of an alternating electric field are considered. The drop is sandwiched between two homogeneous plates and surrounded by an incompressible fluid of a different density. In equilibrium, the drop has the shape of a circular cylinder which is bounded axially by parallel solid plates. These plates have different surface properties. An external electric field acts as an external force which is a reason of the motion of the contact line. A modified Hocking boundary condition is used to describe the motion of the contact line: the velocity of the contact line is proportional to the deviation of the contact angle and the rate of fast relaxation processes, the frequency of which is proportional to the doubled frequency of the electric field. Using this equation allows one to qualitatively describe the experimental dependence of the contact angle as a function of stress, in contrast to the Young – Lippmann equation. The solution to the problem is presented as a Fourier series expansion in eigen functions of the Laplace operator. Graphs of the amplitude-frequency characteristics and evolution of the drop shape are constructed for various values of the problem parameters. In the case of identical surfaces, only odd modes of droplet oscillations are excited, while for different surfaces, even harmonics additionally appear. It was found that in a wide range of parameters the shape of the side surface of the drop is close to the description by an odd function. The oscillation amplitude of even modes is significant only near resonances at the frequencies of these modes. In the case of an inhomogeneous field, azimuthal oscillations are excited, which leads to the appearance of additional resonance peaks. Travel waves propagate along the lateral surface of the drop, which are caused by vibrations of the contact line and the contact angle. The dependences of the values of the contact angle on the upper and lower surfaces on the square root of the amplitude for different values of the frequency of the electric field, which qualitatively coincide with similar graphs of experimental data, are constructed. It is possible to determine the parameters of the Hocking.