  A circular Hele-Shaw cell bounded by a volumetrically confined elastic solid
can act as a fluidic fuse: during radially outward fluid flow, the solid
deforms in response to the viscous pressure field such that the gap expands
near the inlet (at the centre) and contracts near the outlet (around the rim).
If the flow rate exceeds a critical value, the gap at the outlet can close
completely, interrupting/choking the flow. Here, we consider the injection of
gas into such a soft-walled Hele-Shaw cell filled with viscous liquid. Our
theoretical model and numerical simulations for axisymmetric flow driven by the
injection of an expanding gas bubble show that the bubble increases the
critical flow rate of choking via two mechanisms. Firstly, as the interface
approaches the rim, it reduces the length over which the viscous pressure
gradient deforms the solid, which increases the critical flow rate above which
choking occurs. Secondly, compression of the gas reduces the outlet flow rate
relative to the inlet flow rate. As a consequence, for large injection rates, a
near-choking regime is established in which the outlet flow rate becomes
independent of the injection rate and instead depends only on the instantaneous
position of the interface. Our travelling-wave model for the advancement of the
bubble front will enable future reduced-order modelling of non-axisymmetric
problems, such as viscous fingering.
