Supplementary material from "Dipole and pairwise models for the motion of bubbles in a Hele-Shaw cell"

We consider the motion of an arbitrary number of approximately circular bubbles in a Hele-Shaw cell. Each bubble is assumed to be large enough that it is flattened by the cell walls into a pancake-like shape, but small enough to remain approximately circular in plan view. Numerical solutions of the full Hele-Shaw problem become computationally expensive when there is a large number of bubbles. It is therefore common in the literature when modelling a large number of bubbles to assume that each bubble acts like a dipole. Here, we provide the theoretical basis for this approach through the use of matched asymptotic expansions, in the limit where the bubbles are all far apart. We find that this method qualitatively reproduces the behaviour of the full model at a much reduced computational cost, provided the bubbles remain well separated. We also derive a pairwise interaction model by summing over the contributions due to each possible bubble pair. This improved model has computational complexity comparable to that of the dipole model but remains valid in situations in which two bubbles become close.