# Industrial Mathematics Group

### Bubbles in Hele-Shaw flows

Microbubbles have vast applications ranging from cancer detection to targeted drug delivery and the food industry. These bubbles are generally produced in a Hele-Shaw channel. A Hele-Shaw channel is made up of two parallel plates separated by a small distance, in microfluidics this gap is usually around 10-100 μm. Due to this small gap height when bubbles are pushed through the channel they are squashed into a pancake-like shape. It is the study of these bubbles we address in our paper.

In our work we found the dynamics of the bubble depend solely on a single parameter δ, with the bubbles having opposite behaviour depending on if δ<1 or δ>1. For a single bubble we found that it will travel faster than the outer fluid flow if δ>1, but slower if δ<1. Working with collaborators in Princeton we have found good experimental agreement with our model. The model reproduces the surprising classical Taylor-Saffman result that a circular bubble without surface tension will travel at twice the outer fluid velocity.

We found that depending if δ<1 or δ>1 we get striking changes to the qualitative behaviour. The addition of a wall will slow down the bubble for δ>1, but speed it up for δ<1. If we have two bubbles in along the centreline of the channel this will travel faster than a single bubble for δ>1, but again slower for δ<1.

Even more interesting behaviour occurs when additional bubbles are included. In a system of three identical bubbles along the centreline we get Newton's cradle-like dynamics. In the case of δ>1 if we start with 2 bubbles together behind a third bubble, the pair will be faster and catch up with the third. Once they are all close an exchange will occur where the middle bubble now becomes a pair with the front bubble and these then move away together. Whereas this happens the opposite way around for δ<1, where now if we start with a bubble pair ahead of a single bubble, this pair will move slower than the single bubble causing an exchange in the opposite direction as before. Including even more bubbles we can create a cascading effect of bubble pair exchanges.

In a system of two bubbles of different sizes we observe a rollover effect, this is where the bubbles will “do-si-do” around each other. This phenomenon has been seen in experiments done by our collaborators at Princeton. Furthermore, if the bubbles are the same size, and not aligned with the flow direction they will drift away from the centreline, and the direction of this drift changes depending on whether δ<1 or δ>1.