6h-minicourse at the Summer School on PDEs and Randomness at MPI Leipzig, Germany
Abstract:
This minicourse is devoted to the large-scale behavior of suspensions of rigid particles in fluids. These systems are ubiquitous both in nature and in practical applications, and are known to give rise to complex rheological behaviors, in particular non-Newtonian effects. The rigorous micro-macro derivation of effective models for these systems has attracted considerable interest in recent years and will be the focus of this minicourse. Different aspects will be discussed, starting with the definition of the effective viscosity of suspensions by means of homogenization theory, Einstein's celebrated formula for dilute suspensions, as well as its corrections, and finally the derivation of semi-dilute mean-field models capturing some non-Newtonian behaviors. Along the way, this will lead us to discuss qualitative stochastic homogenization, cluster expansion techniques, renormalization of long-range interactions, as well as rigorous mean-field limit techniques.
Slides:
Lecture 1: Homogenization and effective viscosity
Lecture 2: Einstein's viscosity formula
Lecture 3: Semi-dilute cluster expansions
Lecture 4: Back to dunamics: derivation of dilute mean-field models