Marangoni effects of adsorption—desorption controlled surfactants on the leading end of an infinitely long bubble in a capillary

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Abstract
The leading end of an infinitely long gas bubble which displaces a viscous surfactant solution in a capillary tube is studied. The surfactant is present at elevated concentration and has sorption controlled mass transfer. The displaced fluid wets the wall, forming a continuous liquid film between the bubble and the capillary wall. Both the thickness of this film and the additional pressure required to aspirate the bubble depend upon the Marangoni stresses caused by non-uniform surfactant adsorption along the interface. The equations governing this flow are solved at asymptotically small capillary number for the case where the balances of momentum and mass transfer are coupled to leading order. As the Marangoni effect is increased over several orders of magnitude, the additional pressure and the wetting-layer thickness increase above the stress-free interface values found by Bretherton (1961) and approach an upper bound of 42/3 times the Bretherton values. Accompanying changes in the surface tension, Marangoni stress and surface velocity profiles as this upper bound is approached are described. Finally, surface viscosities that are intrinsic to the interfacial region are incorporated in the analysis. When small departures from surface equilibrium states are considered, the terms representing surface viscous effects have the same functional form as the Marangoni stresses and result in thicker films and higher additional pressures.

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