We consider a kinetic-fluid model, derived previously by Helzel and Tzavaras [Multiscale Model. Simul., 15 (2017), pp. 500–536], which models the sedimentation process in dilute suspensions of rod-like particles. The kinetic equation describes the evolution of a density distribution function for the suspended particles as a function of space, orientation, and time. It has the form of a drift-diffusion equation on the sphere (𝑆2) for each point in physical space. In addition, the density distribution of the rod-like particles is translated and diffused in physical space in a way that depends on the particle orientation as well as the flow velocity. The fluid is modeled by a Navier–Stokes equation with a buoyancy term and additional stress tensor that is imposed by the suspended particles. To approximate this high-dimensional system, moment equations are derived, approximating the high-dimensional scalar kinetic equation by a lower-dimensional system of hyperbolic PDEs which only depends on the spatial variables as well as on time. The derivation of the system of moment equations is based on an expansion of the density distribution function using spherical harmonic basis functions. Numerical simulations validate all approximations introduced in this paper. This paper extends previous work by Dahm and Helzel [Multiscale Model. Simul., 20 (2022), pp. 1002–1039], where a simplified version of the kinetic equation was considered that restricts the orientation of the rod-like particles to 𝑆1 instead of 𝑆2.
Link: https://epubs.siam.org/doi/10.1137/25M1741169