Hole expansion from a bubble at a liquid surface

Abstract

For millimetre to micron sized bubbles, floating at the free surface of different low viscosity fluids with different surface tensions, and then collapsing, we study the ensuing expansion of the outer radius of the hole (ro) at the free surface, as well as its velocity of expansion (uo). Since the thin film cap of the bubble disintegrates before the hole in it reaches the static rim, the hole expansion at intermediate times occurs as if it initiates at the bubble’s static rim of radius Rr; the evolution of ro then results to be a strong function of gravity, since Rr depends strongly on the bubble radius R. A scaling analysis, which includes the increase in the tip radius due to mass accumulation and the resulting change in the retraction force, along with the gravity effects by considering the hole radius in excess of its initial static radius, re = ro − Rr, results in a novel scaling law re/R∼(t/tc)4/7, where tc=ρR3/σ is the capillary time scale; this scaling law is shown to capture the evolution of the hole radii in the present study. The dimensionless velocities of hole expansion, namely, the Weber numbers of hole expansion, Weo=ρuo2R/σ, scale as Weo∼(t/tc)−6/7, independent of gravity effects, matching the observations. We also show that these Weber numbers, which reduce with time, begin with a constant initial Weber number of 64, while the viscous limit of the present phenomena occurs when the bubble Ohnesorge number Oh=μ/σρR≃0.24.