Theoretische Modellierung granularer Stroeme in duennen Roehren mit Langevin-Gleichungen

TL;DR

This paper models granular flow in narrow vertical pipes using Langevin equations, deriving hydrodynamic equations, analyzing stability, and verifying results through numerical simulations, with applications to traffic flow modeling.

Contribution

It extends Langevin equation formalism to inhomogeneous granular flows and traffic, including stability analysis and numerical verification.

Findings

Identified critical densities for particle clustering.

Derived hydrodynamic equations from Langevin and Boltzmann equations.

Validated homogeneous solutions and critical densities via simulations.

Abstract

This is the final version of the author's diploma thesis written at the Humboldt University of Berlin in 1995. The topic is the flow of granular material in narrow vertical pipes, driven by the gravity, that is described by Langevin equations. Neglecting the interactions, we can solve the resulting Fokker-Planck equation for the homogeneous case. The consideration of inelastic collisions leads to a Boltzmann equation. Assuming local equilibrium, the hydrodynamic equations lead to the extension of the Langevin equation formalism for the inhomogeneous case. For certain parameter ranges, our formalism can also be used to describe traffic flows. We applied stability analyses to the hydrodynamic equations and found critical densities for the occurrence of particle clustering. We used numerical simulations of the Langevin equations to verify our homogeneous solution as well as the critical densities.