The effect of surfactant on the motion of long bubbles in horizontal capillary tubes

In experiments involving slow steady motion of a long finite bubble of fluid with very small viscosity in a capillary tube filled with a liquid of viscosity mu, a thin film of liquid of uniform thickness adheres to the tube walls between the front and the rear menisci of the bubble. Taking the contact angle of the liquid at the walls as zero and neglecting the gravitational effects, Bretherton (1961) obtained the film thickness as proportional to Ca(2/3) where Ca is the capillary number. In the range Ca < 5 x 10(-3), however, the thickness obtained experimentally was significantly larger than the theoretical values. Bretherton speculated that the presence of small traces of surfactant at the bubble-liquid interface may explain this thickening phenomenon and could be the source of this discrepancy, but he did not provide a proof. In this article we give a theoretical proof for the above thickening phenomenon using perturbation theory and a lubrication approximation of the flow equations. We consider the case of a small amount of surfactant at the bubble interface with a small variation of concentration. The main new result is a lower bound on the film thickness in terms of the Marangoni number M and G for small capillary number. A comparison with the 'clean' case proves the thickening effect of the interfacial surfactant.