Instabilities in Guinness

The settling processes observable in a freshly poured pint of stout demonstrates two unusual phenomena. Firstly, the dispersion of bubbles appears to be sinking. Previous work has shown, using computational fluid dynamics simulations, that this can be explained by the small size of stout bubbles and the tilted shape of the glass. Secondly, the sinking bubbles form into a regular pattern of descending waves. Previous work “Waves in Guinness” compared these waves to disturbances which occur in industrial vertical bubbly column processes. Based on such technology, a one-dimensional analytical model of a glass of settling stout beer was built. Such industrial flows are prone to flow instability, where a uniform dispersion of rising bubbles transitions to slug flow. Large bubbles form across the column width separating the flow into slugs of liquid. While the mechanism of this phenomenon is not yet fully understood, a body of literature exists which asserts that the physics of slug instability are similar to those causing roll waves, those seen flowing down roads of shallow incline after heavy rain. Extending upon this, the work “Waves in Guinness” was able to predict the waves seen in stout beer. I show here that these two observed phenomena can be predicted by a single two-dimensional analytical model. This model confirms the previously proposed mechanism of sinking bubbles, and in a coupled manner, also predicts an evolving flow field within the glass, and shows that the characteristics of this flow field are responsible for the descending waves. This model therefore predicts both sinking bubbles and waves of bubbles. The equations of dispersed two phase flow were applied to a long slab, and nondimensionalised. Asymptotic reduction in the steady flow case allowed an analytic solution to be determined confirming the existence of sinking bubbles. A stability analysis of this steady flow field lead to an equation similar to the Orr-Sommerfeld equation and revealed an instability similar in character to the observed waves. Mathematical models of industrial problems give greater insight into their underlying physics. This can be utilised to both improve and better control such processes. Stout beer is one example of a more general group of two-phase separation processes. And while the sinking bubbles and wave patterns form part of the settling process that leads to a desirable creamy head, their occurrence in industrial scale processes are typically problematic.