The Apollonian decay of beer foam bubble size distribution and the lattices of young diagrams and their correlated mixing functions

Abstract

We present different methods to characterise the decay of beer foam by measuring the foam heights and recording foam images as a function of time. It turns out that the foam decay does not follow a simple exponential law but a higher-order equation V(t)=a−bt−ct2.5, which can be explained as a superposition of two processes, that is, drainage and bubble rearrangement. The reorganisation of bubbles leads to the structure of an Apollonian gasket with a fractal dimension of D≈1.3058. Starting from foam images, we study the temporal development of bubble size distributions and give a model for the evolution towards the equilibrium state based upon the idea of Ernst Ruch to describe irreversible processes by lattices of Young diagrams. These lattices generally involve a partial order, but one can force a total order by mapping the diagrams onto the interval [0,1] using ordering functions such as the Shannon entropy. Several entropy-like and nonentropy-like mixing functions are discussed in comparison with the Young order, each of them giving a special prejudice for understanding the process of structure formation during beer foam decay.