# Towards a better understanding of granular flows

VICENTEGARZÓ$\dagger$ Departamento de Física, Instituto Universitario de
Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, Avda. de
Elvas s/n, 06006 Badajoz (Spain)

###### Abstract

Understanding the transport of particles immersed in a carrier fluid (bedload
transport) is still an exciting challenge. Among the different types of gas-
solid flows, when the dynamics of solid particles is essentially dominated by
collisions between them, kinetic theory can be considered as a reliable tool
to derive continuum approaches from a fundamental point of view. In a recent
paper, Chassagne et al. (2023) have proposed a two-fluid model based on
modifications to a classical kinetic theory model Garzó & Dufty (1999). First,
in contrast to the classical model, the model proposed by Chassagne _et al._
takes into account the interparticle friction not only in the radial
distribution function but also through an effective restitution coefficient in
the rate of dissipation term of granular temperature. As a second
modification, at the top of the bed where the volume fraction is quite small,
the model accounts for the saltation regime in the continuum framework. The
theoretical results derived from the model agree with discrete simulations for
moderate and high densities and they are also consistent with experiments.
Thus, the model proposed by Chassagne et al. (2023) helps to a better
understanding on the combined impact of friction and inelasticity on the
macroscopic properties of granular flows.

Key words: particle-fluid flow, kinetic theory, granular media

$\dagger$Email address for correspondence<EMAIL_ADDRESS>
##  1 Introduction

It is well known that granular materials are ubiquitous in nature and play an
important role in many industrial processes, such as those involving
pharmaceutical, agricultural, or construction products among others. The fact
that these materials are widely used in industry is likely one of the main
reasons explaining why their understanding has attracted the attention of
physicists and engineers in the past few years. Apart from their practical
interest, their study is interesting by itself since they behave differently
depending on the external conditions to which they are subjected. For
instance, grains contained in a jar will behave as a solid, liquid, or gas
depending on how they are shaken. Even when the motion of grains is quite
similar to the random motion of atoms or molecules of a molecular fluid (rapid
flow conditions), granular flows differ from conventional fluid flows since
the size of grains is macroscopic and hence, their collisions are inelastic.

In the dilute regime, rapid flow conditions can be achieved when the energy
dissipated by collisions is compensated for by the energy supplied to the
system by external excitations. In this regime, kinetic theory (which can be
considered as a mesoscopic description intermediate between statistical
mechanics and hydrodynamics), conveniently adapted to account for dissipative
dynamics, has been employed in the past few years as the starting point to
derive hydrodynamic equations with explicit forms of the transport
coefficients. Usually, a simple model has been considered in the granular
literature: a gas of _smooth_ hard spheres where the inelasticity in
collisions is accounted for via a (positive) constant coefficient of normal
restitution $e\leq 1$. The _inelastic_ versions of the classical Boltzmann and
Enskog kinetic equations for dilute and moderate dense gases, respectively,
have been proposed to determine the dynamic properties of granular gases Garzó
(2019). However, technical difficulties in the analysis of the above equations
entailed approximations that limited the applicability of the first attempts
Lun et al. (1984); Jenkins & Richman (1985) to weakly dissipative granular
flows ($e\lesssim 1$). Years later, Garzó & Dufty (1999) (GD-theory) solved
the Enskog equation by means of the Chapman–Enskog method and obtained
expressions for the Navier–Stokes transport coefficients for the whole range
of values of the coefficient of restitution $e$.

Although grains in nature are usually surrounded by a fluid like water or air,
all the above models Lun et al. (1984); Jenkins & Richman (1985); Garzó &
Dufty (1999) neglect the effect of the interstitial fluid on the dynamics of
grains. Needless to say, understanding the flow of solid particles immersed in
one or more fluid phases is a very intricate problem. Among the different
types of multiphase flows, a particularly interesting set corresponds to the
so-called particle-laden suspensions, in which small, immiscible, and
typically dilute particles are immersed in a carrier fluid (for instance, fine
aerosol particles in air) Subramaniam (2020). When the dynamics of grains is
essentially dominated by collisions among them, kinetic theory is again a
convenient tool to describe this type of flows. Due to complexity of the
problem, a coarse-grained approach is usually adopted, where the effect of the
background fluid on grains is accounted for through an effective fluid-solid
force. In some situations Tsao & Koch (1995), only a viscous drag force
(proportional to the instantaneous particle velocity) is considered, while
other more realistic models Garzó et al. (2012); Gómez González & Garzó (2019)
also incorporate a stochastic Langevin-like term defined in terms of the
background temperature $T_{\text{b}}$.

On the other hand, for dense granular flows, kinetic theory has been so far
incapable of describing this type of regime. These type of flows are generally
described with the so-called $\mu(I)$ rheology Forterre & Pouliquen (2008),
where a phenomenological law (obtained by fitting experimental and discrete
simulation data) relates the stress rate $\mu$ with the inertial number $I$.
Although this phenomenological theory gives good predictions in the dense
regime, it fails for dilute systems.

A recent paper by Chassagne et al. (2023) proposes a frictional-collisional
two-fluid model based on kinetic theory for modeling bedload transport. The
proposed model (which employs the GD frictionless kinetic theory as a
baseline) covers dilute and dense regimes and exhibits a good agreement with
DEM simulations and experiments Ni & Capart (2018).

Figure 1: Plot of the (reduced) coefficients $\eta^{*}(e)/\eta^{*}(1)$,
$\kappa^{*}(e)/\kappa^{*}(1)$, and $\kappa_{\phi}^{*}(e)$ versus the
coefficient of restitution $e$ for $\mu^{p}=0.8$ and
$T_{\text{b}}^{*}=T_{\text{b}}/md^{2}\gamma^{2}=1$. In the case of the
coefficients $\eta^{*}(e)/\eta^{*}(1)$ and $\kappa^{*}(e)/\kappa^{*}(1)$ (left
and middle panels), the lines (a), (b), and (c) correspond to the results
obtained from the CBC-theory, the GG-theory, and the GD-theory, respectively,
for $\phi=0.2$. In the case of the coefficient $\kappa_{\phi}^{*}$ (right
panel), the solid (dash-dotted) lines refer to the results obtained from the
GG-theory (a) and the GD-theory (b) for $\phi=0.1$ ($\phi=0.2$).

##  2 Overview

To mitigate the discrepancies observed between the theoretical predictions of
the frictionless GD-theory with those obtained in DEM simulations, Chassagne
et al. (2023) (CBC-theory) corrects the above kinetic theory in the following
ways. First, since the GD-theory neglects the effect of the surrounding fluid,
a drag force term between the fluid and particle phases is introduced. This
drag force leads to a dissipation term in the balance equation for the
granular temperature due to fluctuating motion particle. Second, given that
simulations clearly show that interparticle friction adds another source of
energy dissipation different from that of inelasticity, an _effective_
restitution coefficient $e_{\text{eff}}$ Jenkins & Zhang (2002) is introduced
in the expression of the rate of dissipation of granular temperature. The
effective coefficient $e_{\text{eff}}(\mu^{p})$ is smaller than the (constant)
coefficient of restitution $e$ and depends on the friction coefficient
$\mu^{p}$. In addition, since interparticle friction also affects the
geometrical packing structure of the granular flow, they modify the usual form
of the radial distribution function $g_{0}$ (which takes into account spatial
correlations) by adding a $\mu^{p}$-dependent term adjusted to reproduce the
discrete simulations. A third modification to the GD-theory refers to the
saltation regime observed at the top of the bed, where the volume fraction is
small. Since this regime is essentially dominated by the fluid drag force, it
cannot be reproduced by the GD-theory. Although in most of the previous works
saltation is treated as a boundary condition, Chassagne et al. (2023) model
saltation in a continuum framework. While the saltation contribution
$\eta^{\text{salt}}$ to the shear viscosity $\eta$ was obtained in previous
works Jenkins et al. (2010), Chassagne et al. (2023) propose that the
saltation contribution $\kappa^{\text{salt}}$ to the thermal conductivity
$\kappa$ is simply given by $\kappa^{\text{salt}}=\eta^{\text{salt}}/\sigma$,
where $\sigma=0.5$ is an empirical constant found by fitting to the DEM
simulations.

To put the results in a proper context, it is interesting to compare the
predictions for the Navier–Stokes transport coefficients obtained from the GD-
theory, the CBC-theory and the GG-theory Gómez González & Garzó (2019). While
the former theory is for a _dry_ (no fluid phase) granular gas, the two latter
take into account the impact of the fluid phase on grains. To perform a clean
comparison between the CBC and GG theories, the fluid drag force coefficient
$K$ appearing in Chassagne et al. (2023) may be related to the Stokes drift
coefficient $\gamma$ of the GG-theory by the relation
$\gamma=(1-\phi)K/\rho^{p}$, where $\phi$ is the solid volume fraction,
$\rho^{p}=m/(\frac{\pi}{6}d^{3})$ is the particle density, $m$ is the particle
mass, and $d$ is the diameter of the sphere. In terms of the granular
temperature $T$, the transport coefficients can be written as
$\eta=d\rho^{p}\sqrt{T/m}\eta^{*}$,
$\kappa=d\rho^{p}\sqrt{T/m^{3}}\kappa^{*}$, and
$\kappa_{\phi}=T^{3/2}/(d^{2}\phi\sqrt{m})\kappa_{\phi}^{*}$. Here,
$\kappa_{\phi}$ is the transport coefficient linking the heat flux with the
solid volume fraction gradient. This contribution (which vanishes for elastic
collisions) to the heat flux was neglected by Chassagne et al. (2023). Figure
1 shows the dependence of the dimensionless coefficients
$\eta^{*}(e)/\eta^{*}(1)$, $\kappa^{*}(e)/\kappa^{*}(1)$ and
$\kappa_{\phi}^{*}(e)$ on $e$, as given by the three theories. For this
moderate value of the solid volume fraction ($\phi=0.2$), we observe that the
predictions of the (frictionless) GG-theory for the coefficients $\eta^{*}$
and $\kappa^{*}$ agree well (especially in the case of the shear viscosity)
with those obtained from the CBC-theory, even for quite strong inelasticities
($e\gtrsim 0.5$). This is likely due to the fact that, while the effect of the
friction coefficient $\mu^{p}$ on $g_{0}$ is relatively small, the impact of
the fluid phase on transport is quite important. This latter aspect is not
considered in the GD-theory and hence, it exhibits differences with the CBC-
theory which are much more significant than those found with the GG-theory.
Another interesting conclusion of the GG-theory is the negligible impact of
the coefficient $\kappa_{\phi}^{*}$ on the heat transport (see the right panel
of Fig. 1); this result clearly differs from the one obtained in the GD-model
(where $\kappa_{\phi}^{*}$ can be even larger than the thermal conductivity
coefficient $\kappa^{*}$ in a frictionless dry granular fluid) but agrees with
the simulation data of Chassagne et al. (2023).

One of the weaknesses of kinetic theory is its inability to predict dense
granular flows. However, the results derived from the CBC-theory (based on the
combination of kinetic theory with a frictional model) reproduces the $\mu(I)$
rheology in the dense regime. Although recent attempts to model dense granular
flows with kinetic theory have been based on the introduction of a correlation
length in the dissipation term, the results obtained by Chassagne et al.
(2023) suggest that dense granular flows are essentially dominated by the
competition between elastic-frictional and kinetic-collisional stresses rather
than the development of velocity correlations (breakdown of molecular chaos
hypothesis) for high densities.

Finally, a comparison with experiments Ni & Capart (2018) of the two-fluid
model proposed by Chassagne et al. (2023), when turbulence is negligible, has
also shown a good agreement. In this sense, the present model can be taken as
a starting point to adapt it to the study of more complex configurations.

##  3 Future

The paper by Chassagne et al. (2023) shows the potentiality of kinetic theory
to accurately describe bedload transport. Their results suggest that a
complicated extension of the GD frictionless kinetic theory to account for the
combined effect of the coefficients of friction and normal restitution on the
transport coefficients may not be necessary to capture the behavior of
granular flows for dilute and dense regimes. As a potential extension of the
CBC-theory to account for the effect of interparticle friction on transport in
a more rigorous way one could consider a granular gas of inelastic rough hard
spheres, where a constant coefficient of tangential restitution characterizes
the ratio between the magnitude of the tangential component of the relative
velocity after and before collision. In this context, as a first step, one
should extend the results derived by Kremer et al. (2014) to moderate
densities and consider this theory as a baseline to account for the saltation
regime in the continuum framework, as made by Chassagne et al. (2023).

Fundings. Financial support from Grant PID2020-112936GB-I00 and from Grants
IB20079 funded by Junta de Extremadura (Spain) and by ERDF A way of making
Europe is acknowledged.

Declaration of interests. The author reports no conflict of interest.

Author ORCIDs.
Vicente Garzó: https://orcid.org/0000-0001-6531-9328

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