To the Analytical Solution of the Problem of the Oscillations of a Drop on a Solid Substrate After Impact

The problem of nonforced oscillations of a liquid droplet is solved analytically. The droplet impacts, spreads on a solid substrate, rolls back, and finally comes to rest due to its excess potential energy. A mathematical model is proposed to describe the droplet boundary as a one-parameter family of surfaces. To implement the model, the Lagrange equations of the second kind are used. The oscillation frequency is the result of two factors: the capillary frequency factor and the geometric factor. A thorough analysis investigated the effects of several problem parameters, including surface tension, wetting angle, and the ratio of initial particle height to contact spot radius, on the oscillating frequency. Increased surface tension in a liquid droplet results in a proportional increase in oscillation frequency, which is directly proportional to the square root of the free surface energy. Projected calculations suggest that as the ratio of the initial particle height to the contact spot radius increases and the wetting edge angle increases, there is a decrease in the frequency of droplet free oscillations. These results qualitatively agree with known theoretical and experimental data.