Figure 4

Test of the coalescence condition, inequality (15). In panel (a) bubbles remain separated throughout the entire process of growth and deformation because, in addition to choosing a high \(\sigma _{\mathrm{SS}}\), the phase fields representing the two bubbles are not allowed to merge into a single one. In (b) and (c), however, this possibility is introduced via the coalescence condition, inequality (15). The only difference between (b) and (c) is the use of different initial bubble-densities. Otherwise, identical simulation parameters are employed. In (b) coalescence condition is satisfied at a time of \(t\approx 225\Delta t\) and the bubbles merge into a single one (see also panel d). In (c), the initial densities within the bubbles are lower than in (b) so that driving forces always remain below the coalescence-threshold. (d) Variation of the \(\Delta p^{\mathrm{film}}\) with time for the two cases (b) and (c). The horizontal lines marks the threshold pressure, \(\Pi _{\mathrm{disj}}\) which must be overcome for initiation of coalescence. In all the cases shown, the initial bubble radius is \(R(t=0)=30\,=5\eta\).