Granular roll waves on a μ (J) rheology model: A dynamical systems perspective

Abstract

This paper focuses on the formation and the unique characteristics of granular roll waves generated by utilizing the depth-averaged dynamic model recently introduced by Fei et al. [Appl. Math. Modell. 119, 763–781 (2023)], which employs μ(J) rheology. Notably, the symmetrical shape of the generated roll waves deviates from the typical pattern observed in both dry granular and water roll waves, which demonstrate a gradually rising flank followed by a sudden drop. Through a dynamical systems analysis and the associated phase space diagrams, which provide a comprehensive geometrical overview, we reveal the mathematical properties associated with the stable-uniform flow and the emergence of the granular roll waves. We then delve into the correlation between the shape of these roll waves and the nullclines of the dynamical system, whose analytical expression is presented. Additionally, we highlight the qualitative similarities and differences between the aforementioned model for liquid-immersed flowing granular matter and its well-established counterpart for dry granular flows based on the μ(I) rheology.