# Non-Fickian macroscopic model of axial diffusion of granular materials in a long cylindrical tumbler

## TL;DR

This paper introduces a non-Fickian macroscopic model for axial diffusion in granular materials within a cylindrical tumbler, accounting for particle collision-induced diffusion in a localized surface layer, with exact solutions and comparisons to classical models.

## Contribution

It presents a novel non-Fickian diffusion model for granular flow in tumblers, linking parameters to measurable quantities and deriving exact solutions.

## Key findings

- The model captures diffusion with memory effects.
- Exact solutions are derived for finite-width pulse initial conditions.
- The non-Fickian model differs from classical diffusion in granular flows.

## Abstract

We provide new concepts for understanding transport phenomena in flows of granular materials by using a non-Fickian macroscopic model of axial diffusion of a granular material in a finite cylindrical tumbler. The model accounts for diffusion induced by particle collisions only in a thin surface flowing layer due to localization of shear within the cross-section of the drum. All model parameters are related to measurable quantities in a granular flow. It is shown that the proposed model is a member of the general class of linear constitutive relations with memory. An exact solution for the spreading of a finite-width pulse initial condition under the proposed non-Fickian model is derived and compared to the solution of a Fickian model (i.e., the ``classical'' diffusion equation).