Stress field in the vicinity of a bubble/sphere moving in a dilute surfactant solution

In this study, we experimentally investigate the stress field around a gradually contaminated bubble as it moves straight ahead in a dilute surfactant solution with an intermediate Reynolds number ( $20 \lt {{\textit{Re}}} \lt 220$ ) and high Péclet number. Additionally, we investigate the stress field around a falling sphere unaffected by surface contamination. A newly developed polarisation measurement technique, highly sensitive to the stress field in the vicinity of the bubble or the sphere, was employed in these experiments. We first validated this method by measuring the flow around a solid sphere sedimenting in a quiescent liquid at a terminal velocity. The measured stress field was compared with established numerical results for ${{\textit{Re}}} = 120$ . A quantitative agreement with the numerical results validated this technique for our purpose. The results demonstrated the ability to determine the boundary layer. Subsequently we measured a bubble rising in a quiescent surfactant solution. The drag force on the bubble, calculated from its rise velocity, was set to transiently vary from that of a clean bubble to a solid sphere within the measurement area. With the intermediate drag force between clean bubble and solid sphere, the stress field in the vicinity of the bubble front was observed to be similar to that of a clean bubble, and the structure near the rear was similar to that of a solid sphere. Between the front and rear of the bubble, the phase retardation exhibited a discontinuity around the cap angle at which the boundary conditions transitioned from no slip to slip, indicating an abrupt change in the flow structure. A reconstruction of the axisymmetric stress field from the phase retardation and azimuth obtained from polarisation measurements experimentally revealed that stress spikes occur around the cap angle. The cap angle (stress jump position) shifted as the drag on the bubble increased owing to surfactant accumulation on its surface. Remarkably, the measured cap angle as a function of the normalised drag coefficient quantitatively agreed with the numerical results at intermediate ${{\textit{Re}}} = 100$ of Cuenot et al. (1997 J. Fluid Mech. 339 , 25–53), exhibiting only a slight deviation from the curve predicted by the stagnant cap model at low ${\textit{Re}}$ (creeping flow) proposed by Sadhal & Johnson (1983 J. Fluid Mech. 126 , 237–250).