A comparison of macroscopic models describing the collective response of sedimenting rod-like particles in shear flows

Highlights: • Presentation of a kinetic multi-scale model, describing the sedimentation of rod-like particles. • Derivation of macroscopic models, which describe the collective behavior of the system. • A linear stability analysis, which predicts instability and a wavelength selection mechanism. • Numerical simulations, which compare the macroscopic models with the kinetic model. We consider a kinetic model, which describes the sedimentation of rod-like particles in dilute suspensions under the influence of gravity, presented in Helzel and Tzavaras (submitted for publication). Here we restrict our considerations to shear flow and consider a simplified situation, where the particle orientation is restricted to the plane spanned by the direction of shear and the direction of gravity. For this simplified kinetic model we carry out a linear stability analysis and we derive two different nonlinear macroscopic models which describe the formation of clusters of higher particle density. One of these macroscopic models is based on a diffusive scaling, the other one is based on a so-called quasi-dynamic approximation. Numerical computations, which compare the predictions of the macroscopic models with the kinetic model, complete our presentation.