Fluid dynamics, from the non-local to the local Navier-Stokes-Korteweg system (ANR INFAMIE)

This project addresses the mathematical analysis of non-classical fluid systems formed by several components (granular media, suspension flows) in a dense regime. We are particularly interested in saturation and non-locality phenomena in such physical systems. One observes indeed the activation of nonlocality phenomena in relation with the creation of chains of contacts between the microscopic components of the medium when it is sufficiently dense. From the mathematical standpoint, saturation and non-locality effects raise difficult theoretical and numerical issues, these issues being linked to the multi-scale nature (in time and space) of the phenomena, and the co-existence of different dynamics within the flow (saturated or free). We are interested in this project in modeling issues (justification of constitutive laws and models), as well as theoretical issues (existence of solutions, singular limit). More precisely, we intend to study a fluid system involving an order parameter representing the non-locality effects, and to establish the rigorous convergence of the constructed solutions towards solutions of a capillarity fluid system, this for a relatively large family of capillarity laws.