J Surfactant Effects on the Buoyancy-Driven Motion of a Drop

A more common source of Marangoni effects in systems of interest to chemical engineers is surfactants, as discussed in Chap. 2. This is particularly pertinent to the motion of gas bubbles (or drops) in water, or in any liquid that has a large surface tension (the surface tension of a pure air-water interface is approximately 70 dyn/cm). Experiments on the motion of gas bubbles in water at low Reynolds numbers show the perplexing result illustrated in Fig. 7-18. For bubbles larger than about 1 mm millimeter in diameter, the translation velocity is approximately equal to the predicted value for a spherical bubble with zero shear stress at the interface, that is, 

However, for smaller bubbles the translation velocity decreases until it is approximately two-thirds of this value that is, the small bubbles move through the liquid with a velocity that is consistent with a no-slip boundary condition at the interface (hence, yielding Stokes result (miiaU for the drag) rather than a ffee-surface condition [leading to the value 4n/xaU, according to (7-206)]. 

A qualitative explanation for this observation was discovered many years ago by Frumkin and Levich,30 who carried out experiments on buoyancy-driven motion of gas bubbles in 

If the concentration of surfactant is uniform at the interface, the effect is a uniform decrease in surface tension, which may generally change the bubble shape (because of an increase in the capillary number) but otherwise has no influence on the bubble dynamics or on the hydrodynamic drag. The fact is, however, that the surfactant concentration at 

However, such a gradient of y modifies the boundary condition at the bubble (or drop) surface, from a zero-shear-stress condition (tr9 = 0) to the condition (2-141), which becomes 

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J. Surfactant Effects on the Buoyancy-Driven Motion of a Drop... 

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