Air invasion into three-dimensional foam induces viscous fingering instabilities.

We conducted an experimental investigation to examine the immiscible radial displacement flows of air invading three-dimensional foam in a Hele-Shaw cell. Our study successfully identified three distinct flow regimes. In the initial regime, characterized by relatively low fingertip velocities, the foam underwent a slow displacement through plug flow. During this process, the three-phase contact lines slipped at the cell walls. Notably, we discovered that the air injection pressure exhibited a proportional relationship with the power of the fingertip velocity. This relationship demonstrated excellent agreement with a power law, where the exponent was determined to be 2/3. Transitioning to the second regime, we observed relatively high velocities, resulting in the displacement of the foam as a plug within single layers of foam bubbles. The movement of these bubbles near the cell walls was notably slower. Similar to the first regime, the behavior in this regime also adhered to a power law. In the third regime, which manifested at higher air injection pressures, the development of air fingers occurred through narrow channels. These channels had the potential to isolate the air fingers as they underwent a process of "healing." Furthermore, our results unveiled a significant finding that the width of the air fingers exhibited a continuous scaling with the air injection pressure, irrespective of the flow regimes being observed.