Dynamics of Hele-Shaw flow in Multi-connected Regions

Abstract
The Hele-Shaw problem holds significant importance in fluid mechanics, both theoretically and practically. In this talk, we explore flow dynamics and pattern formation within a radial, multi-layered Hele-Shaw flow that features a fluid injection source, with non-concentric layers adding complexity to the system. Employing boundary integral simulations, we capture the long-term behavior of the flow and reveal striking patterns along various interfaces.

We investigate both constant and time-dependent flux scenarios. Under a constant flux, our simulations show that the Saffman-Taylor-like instability is highly sensitive to the cell's geometric configuration and the fluids' viscosities. By carefully positioning the inner interfaces we can effectively control finger development on the outer interface. Furthermore, when the flux is scaled to the radius of the outer interface, the instability is completely suppressed leading to self-similar growth. Our study also shows that interactions among multiple interfaces can trigger an earlier onset of self-similarity.