This lecture is devoted to the study of some two-fluid flows in the context of the Diffuse Interface Method also called the Phase Field Method.
The traditional approach to two-fluids flows leads to some free boundary value problems which are very nice to study, but the theory is limited by the possible appearance of singularities. The Diffuse Interface method which developed recently tends to replace the sharp interface by a thin layer interface making the problems more regular. These models are fully physically grounded since their derivation is based on the principles of thermodynamics and statistical mechanics. The theory has been very effective for the modeling and numerical simulation of classical examples such as transition in alloys, droplet interactions in fluids and solidification, as well as novel applications in biology and engineering, such as tumor growth, lithium battery, architecture of nanomaterials and image processing.
This lecture will focus on the Navier-Stokes-Cahn-Hilliard equations.
Work done in collaboration with Andrea Giorgini.