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Surfactant effect on rising bubbles - a third order theory. (English) Zbl 1289.76031
Summary: We study the surfactant effects on the motion of an air-bubble risin in a vertical capillary tube of small radius $$R$$, filled with a viscous fluid and sealed at one end. The thickness $$b$$ of the thin film behind the bubble is small comparated with $$R$$. We give a theoretical estimate of $$b$$, by using an expansion of order $$O(b/R)^3$$ of some functions appearing in the mathematical model. An upper bound of the rising velocity $$U$$ is given in terms of $$b$$ and other physical data of the problem. The obtained estimates are providing the thinning and the delay effect due to the surfactant presence of the bubble interface: $$b$$ and $$U$$ are smaller, compared with the ”clean” case. Our results are consistent with previous experimental and numerical data. In the case of horizontal (infinite) capillary tubes we have the opposite effect of surfactant, proved theoretically by Daripa and Paşa.
##### MSC:
 76D45 Capillarity (surface tension) for incompressible viscous fluids
##### Keywords:
capillarity; surface tension; Stokes equations