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

Abstract

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 : 0.28-7 mm) below the free surface and the maximum radius of the bubble ( R-m : 1.5-3.6 mm). When d/R-m>1.3 , the surface wave shows a simple smooth hump (case 1). When 0.82 , 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 , a series of squirting and jetting phenomena are observed at the top of the jet-like surface wave (case 3). For case 1, a proportional relationship is found between rho gh/Delta p and (d/R-m)(-4.4) , where Delta p is the density of the fluid, rho is the gravitational acceleration and Delta p is the difference between the reference atmospheric pressure and the vapour pressure inside a bubble. This proportional relationship is explained semi-analytically using a scaling argument and conservation of momentum and energy, with the help of the Kelvin impulse theory. In addition, we solve the relevant axisymmetric Cauchy-Poisson problem where the initial condition is a jet-like surface wave near its maximum height. By comparing the analytical wave solution with the observed surface wave pattern, it is found that the resultant surface waves are indeed gravity-capillary waves where both the gravity and the surface tension are equally important.