Damped harmonic system modeling of post-impact drop-spread dynamics on a hydrophobic surface

Abstract

The post-impact spread, recoil, and shape oscillations of a droplet impinging on a dry horizontal hydrophobic substrate at low Weber numbers (We) (9.0 < We < 25) are modeled as the behavior of a second-order damped harmonic system. Experiments were conducted to capture the spread dynamics of droplets of six different liquids impinging on a Teflon substrate using a high speed digital visualization and image processing. The selected liquids cover a wide range of viscosities and surface tension coefficients, and their Ohnesorge and Capillary numbers vary by three orders of magnitude (0.002 ≤ Oh ≤ 1.57; 0.007 ≤ Ca ≤ 7.59). High-resolution photographic images of the post-impact spread-recoil process at different We are analyzed to obtain the temporal variations of the spread factor (ratio of liquid spread to droplet diameter) and the flatness factor (ratio of liquid height to droplet diameter). These are found to be represented by the damped harmonic response of a mass-spring-damper system, where the surface tension force acts as a spring and liquid viscosity provides the damping. Due to contact angle hysteresis, the frequency of oscillations for the transient flatness factor variation is slightly different from that for the spread factor variation. Semi-empirical correlations are developed for both the oscillation frequency and the damping factor as a function of drop Weber and Reynolds numbers. The predictions of temporal variations of the spread and flatness factors from these equations agree very well with experimental measurements on the hydrophobic Teflon substrate.