Given a finite graph, the arboreal gas is the measure on forests (subgraphs without cycles) in which each edge is weighted by a parameter β greater than 0. Equivalently this model is bond percolation conditioned to be a forest, the independent sets of the graphic matroid, or the q→0 limit of the random cluster representation of the q-state Potts model. Our results rely on the fact that this model is also the graphical representation of the nonlinear sigma model with target space the fermionic hyperbolic plane H^{0|2}, whose symmetry group is the supergroup OSp(1|2).

The main question we are interested in is whether the arboreal gas percolates, i.e., whether for a given β the forest has a connected component that includes a positive fraction of the total edges of the graph. We show that in two dimensions a Mermin-Wagner theorem associated with the OSp(1|2) symmetry of the nonlinear sigma model implies that the arboreal gas does not percolate for any β greater than 0. On the other hand, in three and higher dimensions, we show that percolation occurs for large β by proving that the OSp(1|2) symmetry of the non-linear sigma model is spontaneously broken. We also show that the broken symmetry is accompanied by massless fluctuations (Goldstone mode). This result is achieved by a renormalisation group analysis combined with Ward identities from the internal symmetry of the sigma model.

A Hele-Shaw Newton's cradle: Circular bubbles in a Hele-Shaw channel. (Daniel Booth)

We present a model for the motion of approximately circular bubbles in a Hele-Shaw cell. The bubble velocity is determined by a balance between the hydrodynamic pressures from the external flow and the drag due to the thin films above and below the bubble. We find that the qualitative behaviour depends on a dimensionless parameter and is found to agree well with experimental observations.  Furthermore, we show how the effects of interaction with cell boundaries and/or other bubbles also depend on the value of this dimensionless parameter For example, in a train of three identical bubbles travelling along the centre line, the middle bubble either catches up with the one in front or is caught by the one behind, forming what we term a Hele-Shaw Newton's cradle.
 

Reciprocity in Fluids (Matthew Cotton)

Reciprocity is a useful, and often underused, way to calculate integrated quantities when a to solution to a related problem is known. In the remaining time, I will overview these ideas and give some example use cases

Elkem is developing a new method for categorising the reactivity between silicon carbide (SiC) powder and silicon monoxide gas (SiO(g)). Experiments have been designed which pass SiO gas through a powdered bed of SiC inside of a heated crucible, resulting in a reaction between the two. The SiO gas is produced via a secondary reaction outside of the SiC bed. Both reactions require specific temperature and pressure constraints to occur. Therefore, we would like to mathematically model the temperature distribution and gas flow within the experimental set-up to provide insight into how we can control the process.

Complexities arise from:

• Endothermic reactions causing heat sinks
• Competing reactions beyond the two we desire
• Dynamically changing properties of the bed, such as permeability