Drag and interfacial vorticity of spherical bubble contaminated with soluble surfactant

Abstract

Numerical simulations of spherical bubbles contaminated with soluble surfactant were carried out to investigate the surfactant effects on the drag coefficient, $C_D$, and the interfacial vorticity, $\omega$, produced at the bubble interface. The different surface contamination regimes are considered in both the diffusion-dominant case and advection-dominant case, for different ambient contamination conditions controlled by varying the Marangoni, Langmuir and Hatta numbers, $Ma$, $La$ and $Ha$. The combinations, ${\Pi }_M=LaMa$ and ${\Pi }_H=Ha/La$, of the dimensionless groups were found to play dominant roles in the surfactant effects on $C_D$ and $\omega$ in both cases. Four different regimes for the dependence of the drag force and vorticity distribution as a function of the above dimensionless group were identified. In the diffusion-dominant case the vorticity is well correlated with a weighting average for those of clean and fully-contaminated bubbles, and a linear relation between $C_D$ and the maximum vorticity holds as in the case with clean bubbles. The characteristics of $C_D$ in the advection-dominant case are more complicated, but they have been classified into four regimes in terms of ${\Pi }_M$ and ${\Pi }_H$. A simple correlation of the stagnant-cap angle expressed in terms of ${\Pi }_M$ was also obtained. This study thus revealed the surfactant effects on $C_D$ and $\omega$ and the drag-vorticity relations in detail at the first time for the different regimes of surface contamination.