“ Marginal pinching ” in soap films

– We discuss the behaviour of a thin soap film facing a frame element: the pressure in the Plateau border around the frame is lower than the film pressure, and the film thins out over a certain distance λ(t), due to the formation of a well-localized pinched region of thickness h(t) and extension w(t). We construct a hydrodynamic theory for this thinning process, assuming a constant surface tension: Marangoni effects are probably important only at late stages, where instabilitites set in. We find λ(t) ∼ t 1/4 , and for the pinch dimensions, h(t) ∼ t −1/2 and w(t) ∼ t −1/4. These results may play a useful role for the discussion of later instabilitites leading to a global film thinning and drainage, as first discussed by K. Mysels under the name " marginal regeneration ". Early experiments by K. Mysels and coworkers [1] showed that a vertical soap film, suspended on a frame, (and made with " mobile " surfactant) thins out by nucleation and growth of black, thin spots near the Plateau borders. They called this process " marginal regeneration ". There are in fact (at least) two steps in marginal regeneration: a) The pressure in the Plateau border is lower than the pressure in the film. This thins out the film near the border, and leads to a " pinch ". b) The pinched state must have an intrinsic instability leading to the black spots. The two steps are very different: the pinch can occur at constant surface tension—i.e. without any Marangoni effect. On the other hand, the later instabilities are probably triggered by Marangoni flows, as pointed out by a number of authors [2–5]. Our aim in the present note is restricted to the first step, i.e. the description of the pinched state with its dynamics. Pinching has already been studied in connection with the elimination of dimples in the coalescence of drops [6, 7] or in the drainage of thin films [8, 9]. However, in these problems, the extension of the dimpled zone is prescribed and fixes one of the spatial scales involved. Our case is different: we consider a semi-infinite film (of initial thickness e 0) facing a straight Plateau border. At t = 0 the film begins to pinch, and is perturbed over a certain distance λ(t) increasing with time (as we shall see λ(t) ∼ t 1/4). Also the width w(T) of …