Response of a two-dimensional liquid foam to air injection: Influence of surfactants, critical velocities and branched fracture

Experiments where air is injected into a foam confined in a Hele-Shaw cell are convenient to study the rheology of foams far from the quasistatic regime, and their limit of stability. At low overpressure, the injected air forms a ductile crack, whereas at high overpressure, it breaks the foam like a brittle material. We present new results in this configuration, complementary with previous studies. We show that air injection is slowed down for surfactants giving incompressible interfaces instead of mobile ones. The injection rate is quantitatively captured by a simple model balancing the air overpressure with known foam/wall friction laws for incompressible interfaces. We also revisit the critical velocity criteria for the injected air proposed by Arif et al. [1]. The upper bound of velocity in the ductile regime, based on the resistance of soap films against wall friction, is shown to hold much better for mobile than for incompressible interfaces. The propagation speed of shear waves is confirmed to be a good lower bound for the velocity in the brittle regime, provided the motion of all liquid within the foam is accounted for. Finally, a short description of branching in the fragile regime is given.