Toward An Empirical Scaling For Bubble Coalescence Rates

Bubble coalescence in magma is a consequence of the thinning of the melt film that separates individual bubbles. New experiments on the drainage of such liquid films, to the point of spontaneous rupture, result in empirical equations for film drainage time as a consequence of gravitational and capillary forces. Application to magmatic systems gives estimates of bubble coalescence rates under conditions where bubble growth is negligible. We conducted laboratory experiments of air bubbles suspended in silicone oil and measured the time required for the liquid film that separates bubbles from the free surface to drain and rupture. The experiments spanned a wide range in Bond number (10-2 ≤ Bo ≤ 102). Independent of viscosity, our results show that for Bo < 0.25 film drainage was by capillary forces, whereas for Bo > 0.25 gravitational forces dominated. We obtain a scaling relation for film drainage time t = C ln(δo/δf) τ, where C is an empirical constant, δo is initial film thickness, δf is final film thickness. τ = ηR/σ is the characteristic capillary time scale (Bo<0.25, C=20),and τ = η/ΔρσR is the characteristic buoyancy time scale (Bo>0.25, C=5), where R is bubble radius, η is fluid viscosity, σ is surface tension and Δρ is density contrast between bubble and liquid. The predicted drainage times are 2-3 orders of magnitude shorter than previous estimates [1] and are consistent with film drainage times in silicate melts at 1≤ Bo ≤ 10. If our results may also be applicable to silicate melts at smaller Bond numbers, they indicate that bubble coalescence will occur on relatively short time scales, even in magmatic processes where bubble growth is negligible. [1] Proussevitch et al., J. Volcanol. Geotherm. Res., 59, 161-178 (1993). Figure 1. A single bubble rises in silicone oil (left), drains and ruptures (right) at the free surface. Figure 2. Experimental results. (a) Drainage time vs. bubble radius. (b) Dimensionless drainage time for Bo>0.25, and (c) for Bo<0.25.