Soap bubble

Abstract: In this paper we provide a complete classification of the local structure of singularities in a wide class of two-dimensional surfaces in R3 collected under the adjective (M, i, a) minimal by Almgren [A3] (see I(8)). The results, Theorems II. 4, IV. 5, IV. 8, are that the singular set of an (M, i, a) minimal set consists of H6lder continuously differentiable curves along which three sheets of the surface meet (Holder continuously) at equal (120?) angles, together with isolated points at which four such curves meet bringing together six sheets of the surface (H6lder continuously) at equal anglesin fact, in a neighborhood of each singular point, the surface is H6lder continuously diffeomorphic to either the surface Y of Figure 1 or the surface T of Figure 2 (both of which are defined in I(11)). The results apply to (idealizations of) many actual surfaces which are governed by surface tension, such as soap films as in Figure 4 and compound soap bubbles as in Figure 3 (and therefore to aggregates of some kinds of biological and metallurgical cells) (Corollary IV. 9 (i), (ii)), and thus are a proof of a result deduced experimentally by Plateau over 100 years ago [P]. They also apply to surfaces which minimize integrals which equal the area integral times some Holder continuous function on R3. A necessary first step in classifying singularities is to determine all possible area minimizing cones (Proposition II. 3). (In 1864 Lamarle claimed to make such a determination but his analysis of the technically most difficult case Figure 12 (p. 503)-was wrong.) Also included is a proof that the surface T of Figure 2 is in fact area minimizing (Theorem IV. 6); it seems to require the full force of Theorem IV.5 and I have never seen it proved elsewhere. The methods of this paper are

Abstract: It is obvious that for a complete comprehension of the phenomena connected with thunderstorms we must understand the behaviour of free water drops in strong electric fields such as exist in these storms. While no direct experiments have been recorded, there are several papers which give results bearing on the problem. Zeleny made very careful observations on water drops suspended from capillary tubes and showed how filaments of various shapes are formed, but his results are not directly applicable to free drops. Experiments on the breaking of soap bubbles have been made by Wilson and Taylor and by the writer, which showed that for a given bubble there is a definite field strength at which the bubble becomes unstable. Allowing for the difference in surface tensions, deductions were made of the fields required to produce instability in water drops.

Abstract: We have measured the surface dilational elastic moduli of bubbles immersed in water and soap bubbles in air. The short time response was obtained by submitting the bubbles to a rapid expansion after which the surface tension evolution was monitored, using either image analysis or pressure measurements. It was possible with this method to measure directly the Gibbs elasticity. The longer time response was obtained by submitting the bubbles to low frequency oscillations. Experiments were performed with solutions of non-ionic surfactants, C12E6, C12G2, their 1:1 mixture, Pluronic F-68 and 127 and the surface elastic moduli were compared with the stability of foams made with these surfactants. The foams evolve with time, first by Ostwald ripening, controlled by the low frequency elasticity, and then by bubbles coalescence, controlled by the high frequency elasticity.