Experimental study on the bubble-induced gravity--capillary jet-like surface waves.

We experimentally study on the jet-like wave motion of a free surface caused by the motion of an electric-spark-generated underwater bubble near the free surface. The overall length scale of the bubble-induced free-surface jet is on the order of a few \textit{mm}, where both the gravity and the surface tension are important. Three different motions of the gravity--capillary jet-like surface waves are observed depending on the inception position of the bubble ($d)$ from the free surface, the maximum radius of the bubble ($R_{\mathrm{m}})$ and the maximum height of the gravity--capillary surface jet ($h)$ before pinch-off if any. When $d$/$R_{\mathrm{m}}$\textgreater 1.2, the surface jet shows a simple smooth hump (case 1). When 0.7\textless $d$/$R_{\mathrm{m}}$\textless 1.2, a single or multiple droplets are pinched off sequentially or simultaneously at the tip or from some points of the surface jet (case 2). Finally, when $d$/$R_{\mathrm{m}}$\textless 0.7, a series of squirting {\&} jetting phenomena are observed at the top of the surface jet (case 3). In particular, for cases 1 and 2 ($d$/$R_{\mathrm{m}}$\textgreater 0.7), we experimentally found the linearly proportional relationship between $h$/$R_{\mathrm{m\thinspace }}$and ($d$/$R_{\mathrm{m}})^{\mathrm{-4}}$. This proportional relationship is proven semi-analytically using a scaling argument and conservation of mass, momentum, and energy.