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

TL;DR

This work addresses how surfactant-induced Marangoni stresses alter the stress field and boundary conditions around a bubble moving in a dilute surfactant solution at intermediate Reynolds numbers. It introduces a polarization-based, integrated photoelasticity method to measure near-interface stresses and validates it against numerical simulations for a solid sphere, then applies it to a rising bubble where surface contamination progresses from clean to fully contaminated, causing a cap-angle-driven transition in boundary conditions. The study reconstructs axisymmetric stress fields, identifies localized stresses near the cap angle, and demonstrates that the cap angle versus normalized drag aligns with the stagnant-cap model and with finite-Re numerical results, thereby linking interfacial chemistry with hydrodynamic stress structures. The method provides a powerful tool for probing surfactant effects in dispersion flows and holds potential for extending to more complex bubble dynamics and non-Newtonian fluids.

Abstract

In this study, we experimentally investigate the stress field around a bubble rising in a dilute surfactant solution (20 < Re < 220, high Peclet numbers) whose surface gradually becomes contaminated, and compare it with that around a sphere free from surface contamination. We employ a newly developed polarization measurement technique, highly sensitive to stress fields near interfaces. First, we validate this method by measuring the flow around a solid sphere settling at Re = 120 and comparing results with numerical predictions, confirming its accuracy. We then measure the stress field around a bubble whose drag force transitions from that of a clean interface to that of a rigid interface within the observation region. The stress near the bubble's front resembles that of a clean bubble, while the rear behaves like a solid sphere. Between these regions, a discontinuous phase retardation near the cap angle indicates a transition from slip to no-slip boundary conditions. Axisymmetric stress reconstruction reveals localized stress spike at the cap angle, which shifts as surfactant accumulates and increases the drag. Remarkably, the measured cap angle versus normalized drag coefficient agrees well with numerical simulations at Re = 100 (Cuenot et al. 1997) and shows only a slight deviation from the creeping-flow stagnant cap model (Sadhal and Johnson 1983). This work demonstrates that polarization-based stress field measurements effectively capture the interplay between surface contamination and hydrodynamics at intermediate Reynolds numbers.

Figures (10)

• Figure 1: The schematic of the experimental setup. (a) Overview. (b) The schematic diagram of measurement principle.
• Figure 2: Flow surrounding a solid sphere at $Re$ = 120. (a) Experimental phase retardation. (b) Experimental azimuth. (c) Numerical phase retardation. (d) Numerical azimuth. (e) Phase retardation near the surface. (f) Azimuth near the surface. The dashed line of (a) is the region of $r = 1.05R$ to $1.30R$.
• Figure 3: (Left) Schematic of the flow field around a solid sphere at $Re = 120$. Key features of the flow are labeled: (a) mushroom-shaped region of vanishing vorticity ($\omega \sim 0$), (b) region where the normal stress gradient is negligible ($\partial u_y / \partial x \sim 0$), and (c) regions in the wake where (c-i) the tangential stress gradient at the surface vanishes ($\partial u_t / \partial n \sim 0$) and (c-ii) normal stress gradient is nearly zero ($\partial u_y / \partial y \sim 0$). (Right) 3D visualization of the flow structure around the sphere, showing the boundary layer, wake, and standing eddy formation. Experimental results related to this flow field are shown in figure \ref{['fig:solid']}.
• Figure 4: Numerical results of flow around a solid sphere for various Reynolds numbers ($Re = 0.1, 1, 10, 100, 300$) and analytical results for two limiting cases (Stokes flow and potential flow). (a) Phase retardation $\Delta$ and (b) Azimuth $\phi$ distribution. These results illustrate the evolution of stress fields and their dependence on $Re$ and flow regimes.
• Figure 5: Drag coefficient of the bubble for various rising heights. The colors show the bubble sizes in each experiment. (i)–(iv) correspond to the figure \ref{['fig:main']}.