On bubbles with small immobile adsorbed films rising in liquids at low Reynolds numbers

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Abstract
Surface-active impurities may collect as a stationary film on the lowest part of a bubble rising in liquid while the remainder of the surface moves freely. Numerical approximations for the motion are available if the Reynolds number is low, but they fail for small films. We give the steady-state asymptotic solution for that case, and obtain the perturbation of the drag coefficient from its value for a completely free surface. It depends on the amount by which the surface tension is reduced at the rear stagnation point. This reduction has usually been taken to be the maximum possible for the particular impurity; we consider also the case where dilution is so great that that maximum cannot be reached because the impurity would then be diffusing off the surface at the rear faster than onto it elsewhere.

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