Numerical Studies of
kinetic theory of granular flows (KTGF) on the viscosity of granular fluid

Y. (Yupeng) Xu¹, J.T. (Johan)
Padding¹*, M.A. (Martin) van der Hoef², J.A.M. (Hans)
Kuipers¹

¹ Department
of Chemical Engineering and Chemistry, Eindhoven University of Technology, 5600
MB, Eindhoven, The Netherlands

² Department
of Science and Technology, University of Twente, 7500 AE, Enschede, The
Netherlands

Corresponding
author隆炉s e-mail: j.t.padding@tue.nl

In the past few decades, the two-fluid
models (TFM) based on the kinetic theory of granular flows (KTGF) have experienced
a rapid development and are used to investigate different types of gas-solid
two-phase flows. In the KTGF, collisional particle interactions in a granular
medium

are modelled following the Chapman-Enskog
approach for dense gases.

The kinetic theory of gases uses the model
of an ideal gas to relate temperature to the average kinetic energy of the
atoms in a container of gas. In this simple case, there is no energy dissipation
during particle-particle collisions and the motion of particles is considered
as the only source for the momentum and kinetic energy transfer in the system.
Based on the temperature, macroscopic properties such as the particulate
pressure and shear viscosity can be calculated. For the dense gas system, the
collisions between particles will also contribute to the transfer of momentum
and kinetic energy. Enskog first studied the effect of the particle-particle
collisions and obtained the standard Enskog theory (SET).

When it comes to the granular flow, the transport coefficients
are completely determined by the excess compressibility. Therefore, the radial
distribution function is of fundamental importance in the kinetic theory of
dense granular flow. Various expressions have been proposed in literature based
on the virial coefficient and simulation data.

In our simulations, the viscosity of the
granular flow is tested using a time-driven soft-sphere model with increasingly
high stiffness. The granular temperature of the granular fluid is kept constant
with the novel Lowe-Andersen thermostat. The radial distribution function is
measured and compared with the existing expressions. The viscosity of the
system is obtained by measuring the integral of the time correlation functions
of stress, known as the Green-Kubo method. A large difference is observed at
high solid fraction between the simulated results and the existing
correlations, which can be explained by the increasing importance of multibody
contacts among the particles.

Figure 1:  Comparison
between the simulation results and existing correlations of the viscosity for a
system with the granular temperature 0.001 m²/s² and d_p
= 2 mm