Onset of air-induced splash at low impact speeds for low-viscosity liquids

Recent experiments [Xu et al. PRL {\bf 94}, 184505 (2005)] revealed that the presence of air is essential for the splash formed after a low-viscosity liquid drop hits a dry, smooth solid. As the impact speed $U_0$ is increased from $2$ m/s to $8$ m/s, the threshold gas pressure, $P_T (U_0)$, below which the splash is suppressed, exhibits $2$ distinct trends. Above a critical impact speed $U_*$, $P_T$ decreases as $1/\sqrt{U_0}$. Below $U_*$, however, $P_T$ decreases much more rapidly with $U_0$. Here we show that a simple idea can account for both the different trend and the form of $P_T(U_0)$ below $U_*$. The idea is that, within the leading-edge of the thin liquid sheet ejected after impact, the flow dynamics is initially dominated by viscous effects. For a drop of radius $a$, surface tension $\sigma$, dynamic viscosity $\mu_L$, density $\rho_L$ falling in ambient gas with sound speed $C_g$, this idea gives the scaling law $U_* \sim (U^2_\rho U_\mu)^{1/3}$, where $U_\rho \sim \sqrt{\sigma/\rho_L a}$ is the capillary wave speed and $U_\mu \sim \sigma/\mu$ is the viscous decay speed for surface deformation. It also yields $P_T(U_0) \sim \sigma^2/(\mu_L a C_g U^2_0)$ for $U_0 \leq U_*$. The dependencies on $\mu_L$, $C_g$ and $a$ are all consistent with available measurements. In addition, our results suggest that, at fixed $U_0$, a different physical mechanism becomes relevant for splash formation when the liquid viscosity is increased above a cross-over value. The predicted cross-over value agrees with the measured value for $4$ m/s impact [Xu, PRE {\bf 75}, 056316 (2007)].