Gravity–capillary jet-like surface waves generated by an underwater bubble

Jet-like surface waves generated by an electric-spark-generated underwater bubble are experimentally studied. Three different motions of jet-like surface waves are observed depending on the inception position of the bubble (d: 0.28~7mm) below the free surface and the maximum radius of the bubble (R_m: 1.4~3.6mm). When d/R_m>1.3, the surface wave shows a simple smooth hump (case 1). When 1<d/R_m<1.3, a single droplet or multiple droplets are pinched off sequentially or simultaneously at the tip or from some points of the jet-like surface wave (case 2). Finally, when d/R_m<1, a series of squirting & jetting phenomena are observed at the top of the jet-like surface wave (case 3). For the case 1, a proportional relationship is found between ρgh/Δp and (d/R_m)^-4, where ρ is the density of the fluid, g is the gravitational acceleration, and Δp is the difference between the reference atmospheric pressure and the pressure inside a bubble. This proportional relationship is proved semi-analytically using a scaling argument, conservation of mass, momentum, and energy with the help of the Kelvin impulse theory.