Influence of soluble surfactants on the motion of a finite bubble in a capillary tube

Abstract

The steady motion of a long inviscid bubble displacing a viscous fluid in a capillary tube was first analyzed by Bretherton [J. Fluid Mech. 10, 166 (1961)] in the limit of small capillary number (Ca). While his theoretical prediction for the wetting film thickness left behind the moving front of the bubble showed good agreement with experimental observations for a moderately small Ca, it underpredicted the film thickness when Ca was smaller than about 10−4. In an attempt to resolve this discrepancy, several investigators studied the Marangoni effects of surface contaminants and predicted that the film thickness could be increased by a maximum factor of 42/3 if the surfactant transport is bulk-diffusion controlled. This prediction, however, is limited to the case of a semi-infinitely long bubble. The present study examines the motion of a finite length bubble in the presence of a small amount of surfactants. Both front and rear ends of the bubble are analyzed in the limit of small capillary number. The results indicate that due to the accumulation of the surfactant at the rear end of the bubble, the film thickening effect of the surfactant occurs only when the bubble length is larger than a certain critical value. It is also shown that the total pressure drop to drive the finite bubble through the capillary tube increases linearly with the length of the bubble as long as the bubble length is smaller than the critical value. When the bubble is longer than the critical value, the pressure drop does not increase any longer but remains to be constant. It appears that these results are capable of explaining the experimental observations of Schwartz and co-workers [J. Fluid Mech. 127, 259 (1986)] in which the film-thickening behavior at low Ca was shown to be dependent on the bubble length.