Single deformed bubbles rising through stagnant water of surfactant concentrations beyond CMC

Experiments were conducted on deformed bubbles in Triton X-100 aqueous solutions at concentrations exceeding the critical micelle concentration (CMC) to understand the bubble behavior in the so-called fully-contaminated state. The terminal velocity was significantly reduced at a concentration $C$ of 10 mmol/m³ ($<\mathrm{CMC}=240$ mmol/m³), and the velocity data were in agreement with the well-known slow velocity curve for contaminated systems. However, as indicated in $C$ beyond CMC, the terminal velocity experienced a further decrease, while the velocities at concentrations of $C=10,000$ and 20,000 mmol/m³ were the same, suggesting that at these concentrations the bubbles were completely contaminated. In the $Re$-$Eo$ plane, where the Eötvös number was calculated by the effective surface tension, ${\sigma }_{eq}$, for the equilibrium of adsorption and desorption, the bubble Reynolds numbers of contaminated bubbles agreed with those in a higher viscosity system with the effective Morton number for ${\sigma }_{eq}$. Therefore, bubbles were considered to be fully covered by surfactant, which causes the surface tension reduction and the Marangoni effect. The presence of surfactant decreased the shape deformation, and the aspect ratios for $C=10,000$ and 20,000 mmol/m³ were the same, again showing that the bubbles at these concentrations were fully contaminated. The Weber number defined for ${\sigma }_{eq}$ worked well to correlate the shapes of contaminated bubbles. A shape correlation for contaminated bubbles was developed, and an available drag correlation was extended by introducing the effects of bubble deformation to reproduce the fast and slow velocity curves of clean and contaminated bubbles that exhibit different characteristics in shape and path oscillations, in particular at $C$ beyond CMC.