| Abstract: |
| Many applications of multiphase flows in Engineering, Physics and Life Sciences involve the motion of fluids and thei mutual interplay at interfaces separating immiscible components. Complex mechanisms already appear in simple experiments, such as breakup and coalescence of drops, mixing in a driven cavity and thermocapillary flow, revealing remarkable similarities with global-impact issues, like tumor growth dynamics. Understanding and modelling the transition occurring when interfaces merge and reconnect is still a major challenge in Fluid dynamics.
The Hele-Shaw flow is a paradigm to approximate fluid flows characterized by predominance of viscuos forces as opposed to the inertial forces. Such approximation is justified in the so-called Hele-Shaw cell, namely the fluids are confined between two flat plates separated by an infinitesimally small gap. The Cahn-Hilliard-Hele-Shaw (CHHS) system is a diffuse interface model describing the motion of two globally immiscible, incompresible and viscous fluids in a Hele-Shaw cell. This model couples a Darcy`s law, which governs the volume-averaged fluid velocity, with a convective Cahn-Hilliard equation for the difference of the fluid concentrations (order parameter). In this talk I will present some recent results for the CHHS system with logarithmic potential, concerning uniqueness of physical weak solutions, the well-posedness of strong solutions as well as their further regularity properties. |
|