Mathematical analysis of force networks in granular and suspension flow*

We will discuss interaction networks that spontaneously form in particulate-based systems. These networks, most commonly known as `force chains' in granular systems, are weighted, 

dynamic structures, which are by now known to be of fundamental importance for the purpose of revealing the underlying physical causes of a number of physical phenomena involved in statics and dynamics of particulate-basedsystems. The presentation will focus on applications of algebraic topology, and in particular of persistent homology (PH) to analysis of such networks found in both simulations and experiments. PH allows for a simplified representation of complex interaction field in both two and three spatial dimensions in terms of persistent diagrams (PDs) that are essentially point clouds. These point clouds could be compared in a meaningful manner, meaning that they allow for the analysis of both static and dynamic properties of the underlying systems. It is important to point out that such representation is robust with respect to small perturbations, which is crucial in particular when applying the method to the analysis of experimental data.  In this context, we also point out that PDs allow for extraction of the properties of interaction networks even if contact forces are not well resolved.  In the second part of the talk, we will focus on two case studies: interaction networks in suspensions with a varied particle-particle interactions, and in dry granular systems  experiencing stick-slip, intermittent type of dynamics.  In the case of suspensions, we will show that the interaction networks are closely related to the rheology, and in the context of stick-slip dynamics, we will discuss potential of the considered approach to explain and possibly even predict system's behavior.