The effect of surfactant on long bubbles rising in vertical capillary tubes

In this letter we investigate the effect of interfacial surfactant on the motion of an air bubble rising in a vertical capillary tube filled with a viscous fluid and sealed at one end. A thin layer of liquid, with almost constant thickness b, exists between the bubble interface and the tube wall. The fluid displaced by the front meniscus flows down through this layer because the tube is sealed far up at the top. The steady rising velocity U of the bubble is related to the thickness b. An upper bound for U is obtained in terms of b and other physical data of the problem, which is in good agreement with previous experimental results. It is proved here analytically that the presence of surfactant on the bubble interface causes a thinning and a delay effect: the thickness of the liquid layer behind the bubble and the rise velocity of the bubble are smaller than those for the 'clean' case. Exactly the opposite effect of surfactant in the horizontal case has been derived analytically by Daripa and Pasa (2010 J. Stat. Mech. L02002) and numerically by Ratulowski and Chang (1990 J. Fluid Mech. 210 303). These effects of interfacial surfactant are consistent with previous experimental and numerical results.