# Rheology of granular particles immersed in a molecular gas under uniform
shear flow

Rubén Gómez González111Electronic address<EMAIL_ADDRESS>Departamento de
Física, Universidad de Extremadura, E-06006 Badajoz, Spain  Moisés García
Chamorro111Electronic address<EMAIL_ADDRESS>Departamento de Física,
Universidad de Extremadura, E-06006 Badajoz, Spain  Vicente Garzó222Electronic
address<EMAIL_ADDRESS>URL:
https://fisteor.cms.unex.es/investigadores/vicente-garzo-puertos/ Departamento
de Física and Instituto de Computación Científica Avanzada (ICCAEx),
Universidad de Extremadura, E-06006 Badajoz, Spain

###### Abstract

Non-Newtonian transport properties of a dilute gas of inelastic hard spheres
immersed in a molecular gas are determined. We assume that the granular gas is
sufficiently rarefied and hence, the state of the molecular gas is not
disturbed by the presence of the solid particles. In this situation, one can
treat the molecular gas as a bath (or thermostat) of elastic hard spheres at a
given temperature. Moreover, in spite of the fact that the number density of
grains is quite small, we take into account their inelastic collisions among
themselves in its kinetic equation. The system (granular gas plus a bath of
elastic hard spheres) is subjected to a simple (or uniform) shear flow (USF).
In the low-density regime, the rheological properties of the granular gas are
determined by solving the Boltzmann kinetic equation by means of Grad’s moment
method. These properties turn out to be highly nonlinear functions of the
shear rate and the remaining parameters of the system. Our results show that
the kinetic granular temperature and the non-Newtonian viscosity present a
discontinuous shear thickening (DST) effect for sufficiently high values of
the mass ratio $m/m_{g}$ ($m$ and $m_{g}$ being the mass of grains and gas
particles, respectively). This effect becomes more pronounced as the mass
ratio $m/m_{g}$ increases. In particular, in the Brownian limit ($m_{g}/m\to
0$) the expressions of the non-Newtonian transport properties derived here are
consistent with those previously obtained by considering a coarse-grained
approach where the effect of gas phase on grains is through an effective
force. Theoretical results are compared against computer simulations in the
Brownian limit showing an excellent agreement.

##  I Introduction

Understanding the flow of solid particles immersed in an interstitial fluid is
likely one of the most challenging open problems in granular flows. Among the
different types of multiphase flows, the so-called particle-laden suspensions
Subramaniam (2020) (where dilute particles flow in a carrier fluid)
constitutes one of the most studied problems in the granular literature. When
collisions (which are generally assumed to be nearly instantaneous) play a
relevant role is such suspensions, the kinetic theory conveniently adapted to
account for the dissipative dynamics of grains can be considered as an useful
tool to analyze the behavior of these sort of granular suspensions.

However, due to the technical difficulties embodied in the description of two
or more phases, a coarse-grained approach is usually adopted Koch (1990);
Gidaspow (1994); Jackson (2000); Fullmer and Hrenya (2017). In this approach,
the effect of the surrounding gas on solid particles is taken into account
through an effective fluid-solid force. While in some works Gidaspow (1994);
Zhao and Wang (2021); Chassagne  _et al._ (2023) the gas–solid force is simply
proportional to the relative mean flow velocity between the solid and gas
phases, in other works the force also incorporates a term proportional to the
velocity particle (Stokes linear drag law) Tsao and Koch (1995); Sangani  _et
al._ (1996); Wylie  _et al._ (2009a); Heussinger (2013); Chamorro  _et al._
(2015); Saha and Alam (2017); Alam  _et al._ (2019); Saha and Alam (2020) plus
a Langevin-like stochastic term Garzó  _et al._ (2012). While the Stokes drag
force term accounts for the energy dissipated by grains due to their friction
on the viscous gas, the stochastic term takes into account the energy
transferred to grains due to their “interactions” with the background
particles. When both terms (viscous drag force plus Langevin-like term) are
considered, a Fokker–Planck term is incorporated in the corresponding kinetic
equation.

For small spatial gradients, the above suspension model Garzó  _et al._ (2012)
has been solved by means of the Chapman–Enskog method Chapman and Cowling
(1970) to determine the Navier–Stokes transport coefficients in terms of the
coefficient of restitution and the parameters of the system Garzó  _et al._
(2012, 2013a, 2013b); Gómez González and Garzó (2023). The knowledge of the
transport coefficients has allowed to assess the impact of the surrounding gas
on the dynamic properties of grains. Beyond the Navier–Stokes regime, this
sort of effective models have been also employed to obtain the non-Newtonian
transport properties in (uniform) _sheared_ granular suspensions Tsao and Koch
(1995); Sangani  _et al._ (1996); Heussinger (2013); Seto  _et al._ (2013);
Kawasaki  _et al._ (2014); Chamorro  _et al._ (2015); Saha and Alam (2017);
Hayakawa  _et al._ (2017, 2020); Hayakawa and Takada (2019); Saha and Alam
(2020); Takada  _et al._ (2020, 2023). Theoretical and computational results
have clearly shown the existence in sheared gas-solid flows of the so-called
DST, namely, the non-Newtonian shear viscosity of the suspension drastically
increases with increasing the shear rate.

Although the above suspension models yield good agreement with computer
simulations Capecelatro and Desjardins (2013); Capecelatro  _et al._ (2015);
Garzó  _et al._ (2016); Fullmer  _et al._ (2017), it would be convenient to
consider a suspension model that considers the real collisions between grains
and particles of the molecular gas. A possibility would be to start from a set
of two coupled kinetic equations for the one-particle velocity distribution
functions of the solid and gas phases. However, the determination of transport
properties of the solid particles from this suspension model would be in
general a quite intricate task since in particular the different phases evolve
over quite different spatial and temporal scales.

To overcome the above difficulties, a suspension model inspired in a paper of
Biben _et al._ Biben  _et al._ (2002) has been recently proposed Gómez
González and Garzó (2022). In this model, one assumes first that the
concentration of the granular particles (or “granular gas”) is much smaller
than that of the molecular gas and so, one can assume that the state of the
latter is not perturbed by the presence of grains. This assumption can be
justified in the case of particle-laden suspensions where _dilute_ particles
are immersed in a carried fluid (for instance, fine aerosol particles in air).
Under these conditions, the background (molecular) gas can be treated as a
bath or _thermostat_ at a certain temperature $T_{g}$. Needless to say, this
suspension model (granular particles immersed in a molecular gas) can be
considered as the _tracer_ limit of a binary mixture; the grains of course are
the defect species of the mixture. In addition, although the number density of
grains is very small, one has to take into account not only the _elastic_
collisions between solid and gas particles in the kinetic equation of grains,
but also the _inelastic_ collisions among grains themselves. Thus, in contrast
to the coarse-grained approaches Koch (1990); Gidaspow (1994); Jackson (2000);
Garzó  _et al._ (2012); Fullmer and Hrenya (2016), two new input parameters
are introduced in this suspension model: the diameter $\sigma_{g}/\sigma$ and
mass $m_{g}/m$ ratios. Here, $\sigma_{g}$ and $m_{g}$ are the diameter and
mass of the particles of the molecular gas, respectively, while $\sigma$ and
$m$ are the diameter and mass of the solid particles, respectively. In fact,
as shown in Ref. Gómez González and Garzó (2022), the expressions of the
Navier–Stokes transport coefficients derived from this collisional model
reduce to those previously obtained from the coarse-grained approach Gómez
González and Garzó (2019a) when the grains are much heavier than the particles
of the molecular gas (Brownian limit $m_{g}/m\to 0$).

The goal of this paper is to determine the non-Newtonian transport properties
of a granular suspension under uniform shear flow (USF). Here, in contrast to
previous attempts Tsao and Koch (1995); Sangani  _et al._ (1996); Heussinger
(2013); Seto  _et al._ (2013); Kawasaki  _et al._ (2014); Chamorro  _et al._
(2015); Saha and Alam (2017); Hayakawa  _et al._ (2017); Hayakawa and Takada
(2019); Saha and Alam (2020); Takada  _et al._ (2020, 2023) carried out by
starting from the Stokes and/or Langevin suspension models, we will start from
the collisional model proposed in Ref. Gómez González and Garzó (2022) where
the collisions between the solid and molecular gas particles are taken into
account. In the case of a binary mixture constituted by a granular gas
thermostated by a molecular gas of elastic hard spheres, the USF state is
characterized by constant number densities for solid and gas particles, a
uniform temperature, and a (common) linear velocity profile
$U_{g,x}=U_{x}=ay$, where $a$ is the constant shear rate. Here,
$\mathbf{U}_{g}$ and $\mathbf{U}$ are the mean flow velocities of the
molecular and granular gases, respectively.

In the low-density regime, the distribution function
$f_{g}(\mathbf{r},\mathbf{v};t)$ of the molecular gas verifies the nonlinear
Boltzmann equation for hard spheres. Since we are interested in a steady
state, an external thermostat force must be introduced to compensate for the
viscous heating effect produced by the shear field in the molecular gas. Here,
as usual in non-equilibrium molecular dynamics simulations Evans and Morriss
(1990), we introduce a “drag” force of the form
$\mathbf{F}_{g}=-m_{g}\xi\mathbf{V}$, where $m_{g}$ is the mass of a particle
of the molecular gas and $\mathbf{V}=\mathbf{v}-\mathbf{U}$ is the peculiar
velocity. This sort of thermostat is usually referred to as the Gaussian
thermostat. The quantity $\xi$ is a function of the shear rate adjusted as to
keep the temperature $T_{g}$ constant. It is important to remark that only in
the especial case of Maxwell molecules Garzó and Santos (2003), there is an
exact equivalence between the results derived with and without the Gaussian
thermostat Dufty  _et al._ (1986). Regarding the distribution function
$f(\mathbf{r},\mathbf{v};t)$ of the granular gas, its time evolution involves
the Boltzmann $J[f,f]$ and Boltzmann–Lorentz $J_{g}[f,f_{g}]$ collision
operators. While the first nonlinear operator $J[f,f]$ takes into account the
rate of change of $f$ due to the inelastic collisions between grains
themselves, the linear operator $J_{g}[f,f_{g}]$ accounts for the rate of
change of $f$ due to the elastic collisions between particles of the molecular
gas and grains.

As in previous works Hayakawa  _et al._ (2017, 2020); Hayakawa and Takada
(2019); Takada  _et al._ (2020) on sheared granular suspensions, the
rheological properties of both the granular and molecular gases are
approximately obtained by means of Grad’s moment method Grad (1949). In spite
of the simplicity of Grad’s method, it is important to remark that the
rheological properties obtained from this method agrees in general quite well
with computer simulations Hayakawa  _et al._ (2017); Hayakawa and Takada
(2019); Takada  _et al._ (2020, 2023) in the Brownian limit ($m_{g}/m\to 0$),
even for high values of the shear rate. Therefore, the conventional Grad’s
moment method can be still considered as a powerful method to describe the
rheology of gas-solid suspensions. In particular, theory and simulations have
clearly shown the discontinuous transition of the kinetic temperature and
nonlinear shear viscosity for dilute suspensions Hayakawa and Takada (2019);
Gómez González and Garzó (2019b). Moreover, as the density increases, there is
a transition from the DST (observed in dilute gases) to the continuous shear
thickening (CST) for denser systems Hayakawa  _et al._ (2017); Takada  _et
al._ (2020).

As mentioned before, one of the added values of the present results is that
they provide an explicit dependence of the rheological properties on the mass
and diameter ratios, apart from their dependence on the remaining parameters
of the system (shear rate, coefficient of restitution for inelastic grain-
grain collisions, and background temperature). Our results clearly show how
the DST effect increases with increasing the mass ratio $m/m_{g}$ (or,
analogously, the diameter ratio $\sigma/\sigma_{g}$). In particular, for
sufficiently large values of $m/m_{g}$, the results derived here reduce to
those previously obtained Hayakawa and Takada (2019); Gómez González and Garzó
(2019b) in the Brownian limit from the Fokker–Planck operator. This agreement
justifies the use of the conventional Langevin equation to analyze non-
Newtonian transport in dilute granular suspensions, even for high values of
the shear rate Pelargonio and Zaccone (2023).

The plan of the paper is as follows. In Sec. II, we present the Boltzmann
kinetic equation for a granular gas in contact with a bath of elastic hard
spheres. The balance equations for the densities of mass, momentum, and energy
of the granular and molecular gases are also displayed. Section III deals with
the rheological properties of the molecular gas. As expected, a shear thinning
effect for the nonlinear shear viscosity is observed from both Grad’s solution
and DSMC results. The rheological properties of the granular gas are
determined in Sec. IV. In dimensionless form, they are given in terms of the
mass ratio $m/m_{g}$, the diameter ratio $\sigma/\sigma_{g}$, the coefficient
of restitution $\alpha$, the (reduced) bath temperature $T_{g}^{*}$, and the
(reduced) shear rate $a^{*}=a/\gamma$. Here, $\gamma$ is the drift coefficient
characterizing the friction of solid particles on the viscous gas. The
Brownian limiting case ($m_{g}/m\to 0$) is considered in Sec. V where we show
the consistency of the present results with those derived before Hayakawa  _et
al._ (2017); Hayakawa and Takada (2019); Gómez González and Garzó (2019b);
Takada  _et al._ (2020) by employing a coarse-grained approach. The shear-rate
dependence of the rheological properties of the granular gas is illustrated in
Sec. VI. Here, we relate the diameter and mass ratios, fix the background
temperature and consider different values of both the coefficient of
restitution and the mass ratio. Our results highlight the existence of a DST
effect as the mass ratio $m/m_{g}$ increases; the temperature ratio
$\theta=T/T_{g}$ and the non-Newtonian shear viscosity $\eta^{*}$
discontinuosly increases/decreases (at a certain value of $a^{*})$ as the
(reduced) shear rate $a^{*}$ gradually increases/decreases. This effect is
more pronounced as $m/m_{g}$ becomes more large. Theoretical results obtained
from Grad’s method are compared with computer simulations in the Brownian
limit showing an excellent agreement between both approaches. We close the
paper in Sec. VII with a brief summary and some concluding remarks on the
results reported here.

##  II Granular gas thermostated by a bath of elastic hard spheres. Boltzmann
kinetic theory

Let us consider a gas of inelastic hard disks ($d=2$) or spheres ($d=3$) of
mass $m$ and diameter $\sigma$. Inelasticity of collisions is characterized
through a constant (positive) coefficient of normal restitution $\alpha\leq
1$. We assume that the granular gas is immersed in gas of elastic hard disks
or spheres (molecular gas) of mass $m_{g}$ and diameter $\sigma_{g}$.
Collisions between granular particles and particles of the molecular gas are
elastic. As mentioned in Sec. I, we want to consider a situation where the
number density of the granular gas $n$ is much smaller than that of the
molecular gas $n_{g}$. In this case, the granular gas is sufficiently rarefied
so that the state of the molecular gas is not affected by the presence of the
solid (grains) particles. This means that the molecular gas may be considered
as a thermostat at the temperature $T_{g}$. In addition, although the
concentration of granular particles is small, the collisions among grains
themselves are also accounted for in the corresponding kinetic equation for
the one-particle velocity distribution function $f(\mathbf{r},\mathbf{v};t)$
of solid particles.

###  II.1 Molecular gas

Under the above conditions, in the low-density regime, the kinetic equation of
the one-particle velocity distribution function
$f_{g}(\mathbf{r},\mathbf{v};t)$ of the molecular gas is the Boltzmann kinetic
equation Cercignani (1988); Garzó and Santos (2003):

$\frac{\partial f_{g}}{\partial t}+\mathbf{v}\cdot\nabla
f_{g}+\frac{\partial}{\partial\mathbf{v}}\cdot\left(\frac{\mathbf{F}_{g}}{m_{g}}f_{g}\right)=J_{g}[f_{g},f_{g}],$
(1)

where the Boltzmann collision operator $J_{g}[f_{g},f_{g}]$ is

$\displaystyle J_{g}[\mathbf{v}_{1}|f_{g},f_{g}]$ $\displaystyle=$
$\displaystyle\sigma_{g}^{d-1}\int d\mathbf{v}_{2}\int
d\widehat{\boldsymbol{\sigma}}\Theta(\widehat{{\boldsymbol{\sigma}}}_{g}\cdot{\mathbf{g}}_{12})(\widehat{\boldsymbol{\sigma}}\cdot{\mathbf{g}}_{12})$
(2)
$\displaystyle\times\left[f_{g}({\mathbf{v}}_{1}^{\prime\prime})f_{g}({\mathbf{v}}_{2}^{\prime\prime})-f_{g}({\mathbf{v}}_{1})f_{g}({\mathbf{v}}_{2})\right].$

In Eq. (2), $\mathbf{g}_{12}=\mathbf{v}_{1}-\mathbf{v}_{2}$ is the relative
velocity, $\widehat{\boldsymbol{\sigma}}$ is a unit vector along the line of
centers of the two spheres at contact, $\Theta$ is the Heaviside step
function, and the double primes denote pre-collisional velocities. The
relationship between pre-collisional
$(\mathbf{v}_{1}^{\prime\prime},\mathbf{v}_{2}^{\prime\prime})$ and post-
collisional $(\mathbf{v}_{1},\mathbf{v}_{2})$ velocities is

$\mathbf{v}_{1,2}^{\prime\prime}=\mathbf{v}_{1,2}\mp(\widehat{{\boldsymbol{\sigma}}}\cdot{\mathbf{g}}_{12})\widehat{\boldsymbol{\sigma}}.$
(3)

Moreover, we have assumed in Eq. (1) that the particles of the molecular gas
can be subjected to the action of the (nonconservative) external force
$\mathbf{F}_{g}$.

The relevant hydrodynamic fields of the molecular gas are the number density
$n_{g}(\mathbf{r};t)$, the mean flow velocity $\mathbf{U}_{g}(\mathbf{r};t)$,
and the temperature $T_{g}(\mathbf{r};t)$. These fields are defined in terms
of the velocity distribution function $f_{g}$ as

$\left\\{n_{g},n_{g}\mathbf{U}_{g},dn_{g}T_{g}\right\\}=\int
d\mathbf{v}\left\\{1,\mathbf{v},m_{g}V_{g}^{2}\right\\}f_{g}(\mathbf{v}),$ (4)

where $\mathbf{V}_{g}=\mathbf{v}-\mathbf{U}_{g}$ is the peculiar velocity of
the molecular gas. Note that in Eq. (4) the Boltzmann constant
$k_{\text{B}}=1$. We will take this value throughout the paper for the sake of
simplicity. The balance equations for the hydrodynamic fields $n_{g}$,
$\mathbf{U}_{g}$, and $T_{g}$ are Garzó and Santos (2003)

$D_{t}^{g}n_{g}+n_{g}\nabla\cdot\mathbf{U}_{g}=0,$ (5)
$\rho_{g}D_{t}^{g}\mathbf{U}_{g}+\nabla\cdot\mathsf{P}_{g}=\boldsymbol{\mathcal{F}}_{g}^{U},$
(6)
$\frac{d}{2}n_{g}D_{t}^{g}T_{g}+\nabla\cdot\mathbf{q}_{g}+\mathsf{P}_{g}:\nabla\mathbf{U}_{g}={\mathcal{F}}_{g}^{T},$
(7)

where $D_{t}^{g}\equiv\partial_{t}+\mathbf{U}_{g}\cdot\nabla$ is the material
time derivative for the molecular gas, $\rho_{g}=m_{g}n_{g}$ is the mass
density of molecular gas,

$\mathsf{P}_{g}=m_{g}\int
d\mathbf{v}\;\mathbf{V}_{g}\mathbf{V}_{g}f_{g}(\mathbf{v})$ (8)

is the pressure tensor,

$\mathbf{q}_{g}=\frac{m_{g}}{2}\int
d\mathbf{v}\;V_{g}^{2}\mathbf{V}_{g}f_{g}(\mathbf{v})$ (9)

is the heat flux,

$\boldsymbol{\mathcal{F}}_{g}^{U}=\int
d\mathbf{v}\;\mathbf{F}_{g}f_{g}(\mathbf{v}),\quad{\mathcal{F}}_{g}^{T}=\int
d\mathbf{v}\;\mathbf{V}_{g}\cdot\mathbf{F}_{g}f_{g}(\mathbf{v})$ (10)

are the production terms of momentum and energy, respectively, due to the
external force $\mathbf{F}_{g}$.

###  II.2 Granular gas

Since the particles of the granular gas collide among themselves and with
particles of the molecular gas, its distribution function
$f(\mathbf{r},\mathbf{v};t)$ in the low-density regime verifies the kinetic
equation

$\frac{\partial f}{\partial t}+\mathbf{v}\cdot\nabla
f+\frac{\partial}{\partial\mathbf{v}}\cdot\left(\frac{\mathbf{F}}{m}f\right)=J[f,f]+J_{\text{BL}}[f,f_{g}],$
(11)

where $\mathbf{F}$ is the external force acting on solid particles. As said in
Sec. I, the Boltzmann collision operator $J[f,f]$ gives the rate of change of
the distribution $f$ due to binary inelastic collisions between granular
particles while the Boltzmann-Lorentz collision operator
$J_{\text{BL}}[f,f_{g}]$ accounts for the rate of change of the distribution
$f$ due to elastic collisions between granular and molecular gas particles.
The explicit form of the nonlinear Boltzmann collision operator $J[f,f]$ is
Garzó (2019)

$\displaystyle J[\mathbf{v}_{1}|f,f]$ $\displaystyle=$
$\displaystyle\sigma^{d-1}\int d\mathbf{v}_{2}\int
d\widehat{\boldsymbol{\sigma}}\Theta(\widehat{{\boldsymbol{\sigma}}}\cdot{\mathbf{g}}_{12})(\widehat{\boldsymbol{\sigma}}\cdot{\mathbf{g}}_{12})$
(12)
$\displaystyle\times\left[\alpha^{-2}f({\mathbf{v}}_{1}^{\prime\prime})f({\mathbf{v}}_{2}^{\prime\prime})-f({\mathbf{v}}_{1})f({\mathbf{v}}_{2})\right],$

where in Eq. (12) the relationship between pre-collisional
$(\mathbf{v}_{1}^{\prime\prime},\mathbf{v}_{2}^{\prime\prime})$ and post-
collisional $(\mathbf{v}_{1},\mathbf{v}_{2})$ velocities is

$\mathbf{v}_{1,2}^{\prime\prime}=\mathbf{v}_{1,2}\mp\frac{1+\alpha}{2\alpha}(\widehat{{\boldsymbol{\sigma}}}\cdot{\mathbf{g}}_{12})\widehat{\boldsymbol{\sigma}}.$
(13)

The form of the linear Boltzmann-Lorentz collision operator
$J_{\text{BL}}[f,f_{g}]$ is Garzó (2019); Résibois and de Leener (1977)

$\displaystyle J_{\text{BL}}[\mathbf{v}_{1}|f,f_{g}]$ $\displaystyle=$
$\displaystyle\overline{\sigma}^{d-1}\int d\mathbf{v}_{2}\int
d\widehat{\boldsymbol{\sigma}}\Theta(\widehat{{\boldsymbol{\sigma}}}\cdot{\mathbf{g}}_{12})(\widehat{\boldsymbol{\sigma}}\cdot{\mathbf{g}}_{12})$
(14)
$\displaystyle\times\left[f({\mathbf{v}}_{1}^{\prime\prime})f_{g}({\mathbf{v}}_{2}^{\prime\prime})-f({\mathbf{v}}_{1})f_{g}({\mathbf{v}}_{2})\right],$

where $\overline{\sigma}=(\sigma+\sigma_{g})/2$ and in Eq. (14) the
relationship between
$(\mathbf{v}_{1}^{\prime\prime},\mathbf{v}_{2}^{\prime\prime})$ and
$(\mathbf{v}_{1},\mathbf{v}_{2})$ is

$\mathbf{v}_{1}^{\prime\prime}=\mathbf{v}_{1}-2\mu_{g}(\widehat{{\boldsymbol{\sigma}}}\cdot{\mathbf{g}}_{12})\widehat{\boldsymbol{\sigma}},\quad\mathbf{v}_{2}^{\prime\prime}=\mathbf{v}_{2}+2\mu(\widehat{{\boldsymbol{\sigma}}}\cdot{\mathbf{g}}_{12})\widehat{\boldsymbol{\sigma}}.$
(15)

Here,

$\mu_{g}=\frac{m_{g}}{m+m_{g}},\quad\mu=\frac{m}{m+m_{g}}.$ (16)

As in the case of the molecular gas, the relevant hydrodynamic fields of the
granular gas are the number density

$n(\mathbf{r};t)=\int d\mathbf{v}\;f(\mathbf{v}),$ (17)

the mean flow velocity

$\mathbf{U}(\mathbf{r};t)=\frac{1}{n(\mathbf{r};t)}\int
d\mathbf{v}\;\mathbf{v}f(\mathbf{v}),$ (18)

and the _granular_ temperature

$T(\mathbf{r};t)=\frac{m}{dn(\mathbf{r};t)}\int
d\mathbf{v}\;V^{2}f(\mathbf{v}),$ (19)

where $\mathbf{V}=\mathbf{v}-\mathbf{U}$ is the peculiar velocity of the
granular gas. Note that in general the mean flow velocity $\mathbf{U}$ of
solid particles can be different from the mean flow velocity $\mathbf{U}_{g}$
of particles of the molecular gas. In fact, the difference
$\mathbf{U}-\mathbf{U}_{g}$ may induce a non-vanishing contribution to the
heat flux Gómez González and Garzó (2022).

The macroscopic balance equations for the granular gas are obtained by
multiplying Eq. (11) by $\left\\{1,\mathbf{v},mV^{2}\right\\}$ and integrating
over velocity. After some algebra, one gets the result

$D_{t}n+n\nabla\cdot\mathbf{U}=0,$ (20) $\rho
D_{t}\mathbf{U}+\nabla\cdot\mathsf{P}=\boldsymbol{\mathcal{F}}^{U}+\boldsymbol{\mathcal{F}}[f],$
(21)
$\frac{d}{2}nD_{t}T+\nabla\cdot\mathbf{q}+\mathsf{P}:\nabla\mathbf{U}={\mathcal{F}}^{T}-\frac{d}{2}nT\left(\zeta+\zeta_{g}\right).$
(22)

Here, $D_{t}=\partial_{t}+\mathbf{U}\cdot\nabla$ is the material derivative of
solid particles, $\rho=mn$ is the mass density of solid particles, and the
pressure tensor $\mathsf{P}$ and the heat flux vector $\mathbf{q}$ are given,
respectively, as

$\mathsf{P}=\int d\mathbf{v}\;m\mathbf{V}\mathbf{V}f(\mathbf{v}),$ (23)
$\mathbf{q}=\int d\mathbf{v}\;\frac{m}{2}V^{2}\mathbf{V}f(\mathbf{v}).$ (24)

The production terms of momentum and energy due to the external force
$\mathbf{F}$ are

$\boldsymbol{\mathcal{F}}^{U}=\int
d\mathbf{v}\;\mathbf{F}f(\mathbf{v}),\quad{\mathcal{F}}^{T}=\int
d\mathbf{v}\;\mathbf{V}\cdot\mathbf{F}f(\mathbf{v}).$ (25)

Since the Boltzmann–Lorentz collision term $J_{g}[f,f_{g}]$ does not conserve
momentum, then the production of momentum term $\boldsymbol{\mathcal{F}}[f]$
due to collisions between the granular and molecular particles is in general
different from zero. It is defined as

$\boldsymbol{\mathcal{F}}[f]=\int
d\mathbf{v}\;m\mathbf{V}J_{\text{BL}}[f,f_{g}].$ (26)

This production term vanishes when $\mathbf{U}=\mathbf{U}_{g}$ Gómez González
and Garzó (2022). In addition, the partial production rates $\zeta$ and
$\zeta_{g}$ are given, respectively, as

$\zeta=-\frac{m}{dnT}\int d\mathbf{v}\;V^{2}\;J[\mathbf{v}|f,f],$ (27)
$\zeta_{g}=-\frac{m}{dnT}\int
d\mathbf{v}\;V^{2}\;J_{\text{BL}}[\mathbf{v}|f,f_{g}].$ (28)

The cooling rate $\zeta$ gives the rate of kinetic energy loss due to
inelastic collisions between particles of the granular gas. It vanishes for
elastic collisions. On the other hand, the term $\zeta_{g}$ gives the transfer
of kinetic energy between the particles of the granular and molecular gas and
it cancels out when the granular and molecular gas are at the same temperature
($T_{g}=T$).

It is quite apparent that the balance equations (5)–(7) and (20)–(22) for the
molecular and granular gas, respectively, do not constitute a closed set of
equations for the hydrodynamic fields unless one expresses the fluxes, the
production momentum term, and the cooling rates in terms of those fields.
Here, we will consider an specific non-equilibrium state: the USF state.

##  III Granular particles immersed in a molecular gas under USF

We assume that the system (granular particles plus molecular gas) are
subjected to USF. At a macroscopic level, this state is characterized by a
linear profile of the $x$ component of the flow velocities along the $y$ axis,
constant densities $n_{g}$ and $n$, and uniform temperatures $T_{g}$ and $T$:

$n_{g}\equiv\text{const},\quad n\equiv\text{const},$ (29) $\nabla T_{g}=\nabla
T=0,$ (30) $U_{g,i}=U_{i}=a_{ij}r_{j},\quad a_{ij}=a\delta_{ix}\delta_{jy},$
(31)

$a$ being the _constant_ shear rate. Since the only spatial gradient present
in the system is the shear rate, the heat fluxes $\mathbf{q}_{g}$ and
$\mathbf{q}$ vanish and the pressure tensors $\mathsf{P}_{g}$ and $\mathsf{P}$
are the relevant fluxes in the problem. They measure the transport of momentum
across the system. In addition, the production of momentum term
$\boldsymbol{\mathcal{F}}[f]$ vanishes in the USF problem since
$\mathbf{U}=\mathbf{U}_{g}$.

On the other hand, according to Eq. (7), it is quite apparent that in the
absence of an external force the USF for the molecular gas is not a stationary
state since the temperature increases with time due to viscous heating. Here,
as mentioned in Sec. I, a nonconservative external force of the form Evans and
Morriss (1990)

$\mathbf{F}_{g}=-m_{g}\xi\mathbf{V}$ (32)

is introduced to achieve a steady state. The quantity $\xi$ is chosen to be a
function of the shear rate by the condition that the temperature $T_{g}$
reaches a constant value in the long time limit. Analogously, the granular gas
is also subjected to this sort of Gaussian thermostat so that,
$\mathbf{F}=-m\xi\mathbf{V}$. Note that the parameter $\xi$ is common for both
species. The use of this kind of thermostat is quite usual in sheared
molecular mixtures Garzó and Santos (2003). Since $\mathbf{F}_{g}$ and
$\mathbf{F}$ are both proportional to the peculiar velocity $\mathbf{V}$, then
$\boldsymbol{\mathcal{F}}_{g}^{U}=\boldsymbol{\mathcal{F}}^{U}=\mathbf{0}$,
and

$\frac{\mathcal{F}_{g}^{T}}{n_{g}T_{g}}=\frac{\mathcal{F}^{T}}{nT}=-d\xi.$
(33)

###  III.1 Rheology of the molecular gas

As the state of the molecular gas is not disturbed by the presence of granular
particles, it is more convenient to determine its rheological properties
before considering the USF state for the solid particles. At a microscopic
level, the USF state becomes spatially homogeneous when the velocities of the
particles are referred to the Lagrangian frame moving with the flow velocity
$\mathbf{U}$. In this new frame, the distribution function of the molecular
gas adopts the form $f_{g}(\mathbf{r},\mathbf{v})=f_{g}(\mathbf{V})$ and the
Boltzmann equation (1) in the presence of the external force (32) reads

$-\frac{\partial}{\partial V_{i}}\left(a_{ij}V_{j}+\xi V_{i}\right)f_{g}({\bf
V})=J_{g}[f_{g},f_{g}].$ (34)

The relevant transport coefficients of the USF problem are obtained from the
knowledge of the non-zero elements of the pressure tensor $\mathsf{P}_{g}$.
These elements can be determined by multiplying the Boltzmann equation (34) by
$m_{g}{\bf V}{\bf V}$ and integrating over ${\bf V}$. The result is

$a_{ik}P_{g,kj}+a_{jk}{P}_{g,ki}+2\xi P_{g,ij}={A}_{g,ij},$ (35)

where we have introduced the collisional moment $\mathsf{A}_{g}$ as

${A}_{g,ij}=m_{g}\int d{\bf V}\;V_{i}V_{j}J_{g}[{\bf V}|f_{g},f_{g}].$ (36)

From Eq. (35), in particular, one gets the balance equation for the
temperature $T_{g}$:

$aP_{g,xy}+d\xi p_{g}=0,$ (37)

where $p_{g}=n_{g}T_{g}$ is the hydrostatic pressure of the molecular gas.

The determination of the collisional moment $\mathsf{A}_{g}$ for arbitrary
shear rate requires the knowledge of the velocity distribution function
$f_{g}(\mathbf{V})$. This is a quite formidable task. However, one expects to
get a good estimate of this collisional moment by using Grad’s approximation
Grad (1949)

$f_{g}({\bf V})\to f_{g,\text{M}}({\bf
V})\left(1+\frac{m_{g}}{2T_{g}}\Pi_{g,ij}V_{i}V_{j}\right),$ (38)

where

$f_{g,\text{M}}({\bf V})=n_{g}\left(\frac{m_{g}}{2\pi
T_{g}}\right)^{d/2}\exp\left(-\frac{m_{g}V^{2}}{2T_{g}}\right)$ (39)

is the Maxwellian distribution of the molecular gas and

$\Pi_{g,ij}=\frac{P_{g,ij}}{p_{g}}-\delta_{ij}$ (40)

is the traceless part of the (dimensionless) pressure tensor. With the trial
distribution (38), the collisional moment $\mathsf{A}_{g}$ can be explicitly
evaluated. When one neglects nonlinear terms in the tensor
$\boldsymbol{\Pi}_{g}$, the expression of $\mathsf{A}_{g}$ can be written as
Garzó (2002); Santos  _et al._ (2004)

$A_{g,ij}=-\nu_{g}p_{g}{\Pi}_{g,ij},$ (41)

where the collision frequency $\nu_{g}$ is

$\nu_{g}=\frac{8\pi^{(d-1)/2}}{(d+2)\Gamma(d/2)}n_{g}\sigma_{g}^{d-1}\sqrt{\frac{T_{g}}{m_{g}}}.$
(42)

Figure 1: Plot of the nonlinear shear viscosity $\eta_{g}^{*}$ [panel (a)] and
first viscometric function $\Psi_{g}^{*}$ [panel (b)] for hard spheres ($d=3$)
as functions of the (reduced) shear rate $\widetilde{a}=a/\nu_{g}$. Symbols
refer to the DSMC results.

The non-zero elements of the pressure tensor
$\mathsf{P}_{g}^{*}=\mathsf{P}_{g}/(n_{g}T_{g})$ can be easily obtained from
Eq. (35) when one takes into account Eq. (41). They are given by

$P_{g,xx}^{*}=\frac{1}{1+2\widetilde{\xi}}\left[1+\frac{2\widetilde{a}^{2}}{(1+2\widetilde{\xi})^{2}}\right],$
(43) $P_{g,yy}^{*}=P_{g,zz}^{*}=\frac{1}{1+2\widetilde{\xi}},$ (44)
$P_{g,xy}^{*}=-\frac{\widetilde{a}}{(1+2\widetilde{\xi})^{2}},$ (45)

where we have introduced the dimensionless quantities

$\widetilde{a}=\frac{a}{\nu_{g}},\quad\widetilde{\xi}=\frac{\xi}{\nu_{g}}.$
(46)

The constraint $P_{xx}^{*}+(d-1)P_{yy}^{*}=d$ leads to a cubic equation
relating $\widetilde{\xi}$ and $\widetilde{a}$:

$\widetilde{a}^{2}=d\widetilde{\xi}(1+2\tilde{\xi})^{2}.$ (47)

The real root of Eq. (47) gives the shear-rate dependence of
$\widetilde{\xi}(\widetilde{a})$. It is given by

$\widetilde{\xi}(\tilde{a})=\frac{2}{3}\sinh^{2}\left[\frac{1}{6}\cosh^{-1}\left(1+\frac{27}{d}\widetilde{a}^{2}\right)\right].$
(48)

For small shear rates, $\widetilde{\xi}\sim\widetilde{a}^{2}$ while
$\widetilde{\xi}\sim\widetilde{a}^{2/3}$ for large shear rates Garzó and
Santos (2003).

Equations (43)–(45) along with Eq. (48) provide the shear-rate dependence of
the elements of the (reduced) pressure tensor $P_{g,ij}^{*}$. These
expressions (obtained from Grad’s moment method) coincide con the exact
results obtained for Maxwell molecules in the USF problem Ikenberry and
Truesdell (1956).

In order to characterize the non-Newtonian transport properties, it is usual
McLennan (1989) to introduce the (dimensionless) nonlinear shear viscosity
$\eta_{g}^{*}$ and the (dimensionless) first viscometric function
$\Psi_{g}^{*}$ defined as

$\eta_{g}^{*}(\widetilde{a})=-\frac{P_{g,xy}^{*}}{\widetilde{a}},\quad\Psi_{g}^{*}(\widetilde{a})=\frac{P_{g,yy}^{*}-P_{g,xx}^{*}}{\widetilde{a}^{2}}.$
(49)

We recall again that the collisional moment $A_{g,ij}$ has been computed by
neglecting nonlinear terms in the traceless pressure tensor $\Pi_{g,ij}$. This
approximation yields $P_{g,yy}^{*}=P_{g,zz}^{*}$. The evaluation of $A_{g,ij}$
retaining all the quadratic terms in $\Pi_{g,ij}$ have been reported in Refs.
Tsao and Koch (1995); Chamorro  _et al._ (2015). The inclusion of these
nonlinear contributions to $A_{g,ij}$ allows us to obtain the normal stress
differences $P_{g,yy}^{*}-P_{g,zz}^{*}$ in the plane orthogonal to the shear
flow. However, given that computer simulations Chamorro  _et al._ (2015) have
shown that the above difference is in general very small, one can conclude
that the expression (41) can be considered as accurate for obtaining the non-
transport properties of the molecular gas.

Figure 1 shows the shear-rate dependence of the rheological functions
$\eta_{g}^{*}(\widetilde{a})$ and $\Psi_{g}^{*}(\widetilde{a})$ for a three-
dimensional ($d=3$) molecular gas. Theoretical results are compared against
computer simulation results obtained from the direct simulation Monte Carlo
(DSMC) method Bird (1994). As expected Garzó and Santos (2003), $\eta_{g}^{*}$
and $|\Psi_{g}^{*}|$ are monotonically decreasing functions of the shear rate
$\widetilde{a}$. A shear thinning effect is observed for the shear viscosity
since the shearing inhibits the momentum transport: the actual value of
$|P_{g,xy}^{*}|$ is smaller than the one predicted by Newton’s law. With
respect to the viscometric function, we see that it is negative and hence,
$P_{g,xx}^{*}>P_{g,yy}^{*}$. This can interpreted as a breakdown of the energy
equipartition. It is quite also apparent that the (approximate) theoretical
results exhibit an excellent agreement with computer simulation results, even
for very large values of the shear rate.

##  IV Rheology of the granular gas

Once the rheological properties of the molecular gas have been determined, the
next step is to obtain the non-zero elements of the pressure tensor $P_{ij}$
defined by Eq. (23). In the steady USF state, the (homogeneous) distribution
function $f(\mathbf{V})$ of the granular gas obeys the kinetic equation

$-\frac{\partial}{\partial V_{i}}\left(a_{ij}V_{j}+\xi V_{i}\right)f({\bf
V})=J[f,f]+J_{\text{BL}}[f,f_{g}].$ (50)

Multiplying both sides of Eq. (50) by $mV_{i}V_{j}$ and integrating over
velocity, one achieves the equation

$a_{ik}P_{kj}+a_{jk}{P}_{ki}+2\xi P_{ij}={B}_{ij}+C_{ij},$ (51)

where

$B_{ij}=m\int d{\bf V}\;V_{i}V_{j}J[{\bf V}|f,f],$ (52) $C_{ij}=m\int d{\bf
V}\;V_{i}V_{j}J_{\text{BL}}[{\bf V}|f,f_{g}].$ (53)

As in the case of the molecular gas, the collisional moments $\mathsf{B}$ and
$\mathsf{C}$ cannot be exactly evaluated. Therefore, we employ the same
strategy as for $\mathsf{A}_{g}$; Grad’s moment method Grad (1949) is
considered to provide a good estimate of $\mathsf{B}$ and $\mathsf{C}$. In
this approach, $f_{g}(\mathbf{V})$ is replaced by the distribution (38) while
the granular gas distribution $f(\mathbf{V})$ is replaced by the trial
distribution

$f({\bf V})\to f_{\text{M}}({\bf
V})\left(1+\frac{m}{2T}\Pi_{ij}V_{i}V_{j}\right),$ (54)

where

$\Pi_{ij}=\frac{P_{ij}}{p}-\delta_{ij}$ (55)

is the traceless part of the (dimensionless) pressure tensor $P_{ij}/p$ and

$f_{\text{M}}({\bf V})=n\left(\frac{m}{2\pi
T}\right)^{d/2}\exp\left(-\frac{mV^{2}}{2T}\right)$ (56)

is the Maxwellian distribution of the granular gas (i.e., defined at the
temperature $T\neq T_{g}$).

The corresponding collision integrals defining the tensors $B_{ij}$ and
$C_{ij}$ can be now computed by using the trial distributions (38) for $f_{g}$
and (54) for $f$. In this case and retaining only linear terms in
$\boldsymbol{\Pi}_{g}$ and $\boldsymbol{\Pi}$, after some algebra the
collisional moments $B_{ij}$ and $C_{ij}$ can be written as Garzó (2002)

$B_{ij}=-\nu
nT_{g}\left[\nu_{\eta}^{*}P_{ij}^{*}+\chi\left(\zeta^{*}-\nu_{\eta}^{*}\right)\delta_{ij}\right],$
(57) $\displaystyle C_{ij}$ $\displaystyle=$
$\displaystyle-2nT_{g}\frac{m}{m_{g}}\mu\left(\frac{1+\theta}{\theta}\right)^{3/2}\gamma\Big{(}X\delta_{ij}+\chi^{-1}YP_{ij}^{*}$
(58) $\displaystyle+ZP_{g,ij}^{*}\Big{)},$

where $P_{ij}^{*}=P_{ij}/(nT_{g})$, $\chi=T/T_{g}$ is the temperature ratio,
and $\theta=mT_{g}/(m_{g}T)$ is the ratio of the mean square velocities of
granular and molecular gas particles. Moreover, in Eqs. (57) and (58) we have
introduced the quantities

$\nu=\frac{8\pi^{(d-1)/2}}{(d+2)\Gamma(d/2)}n\sigma^{d-1}\sqrt{\frac{T}{m}},$
(59)
$\nu_{\eta}^{*}=\frac{3}{4d}(1+\alpha)\left(1-\alpha+\frac{2}{3}d\right),$
(60) $\zeta^{*}=\frac{\zeta}{\nu}=\frac{d+2}{4d}(1-\alpha^{2}),$ (61)
$\gamma=\frac{4\pi^{(d-1)/2}}{d\Gamma\left(\frac{d}{2}\right)}\left(\frac{m_{g}}{m}\right)^{1/2}\left(\frac{2T_{g}}{m}\right)^{1/2}n_{g}\overline{\sigma}^{d-1},$
(62) $X=-\frac{\mu_{g}}{d+2}\left(d+\frac{\chi-1}{1+\theta^{-1}}\right),$ (63)
$Y=\frac{1}{(d+2)(1+\theta)}\Big{[}d\mu_{g}+\frac{(d+2)+(d+3)(\chi-1)\mu_{g}}{1+\theta^{-1}}\Big{]},$
(64)
$Z=\frac{\mu_{g}\theta}{(d+2)(1+\theta)}\Big{[}d+(d+3)\frac{\chi-1}{1+\theta^{-1}}\Big{]}-\frac{\theta}{(1+\theta)^{2}}.$
(65)

It is quite apparent that the expression of the drift or friction coefficient
$\gamma$ is the same as the one obtained in the Brownian limit ($m_{g}/m\to
0$) when the molecular gas is at equilibrium ($f_{g}=f_{g,\text{M}}$). In
these conditions, a Kramers–Moyal expansion in the velocity jumps
$\delta\mathbf{v}=(2/(1+m/m_{g}))(\widehat{{\boldsymbol{\sigma}}}\cdot{\mathbf{g}}_{12})\mathbf{g}_{12}$
allows to approximate the Boltzmann–Lorentz operator $J_{\text{BL}}[f,f_{g}]$
by a Fokker–Planck operator Résibois and de Leener (1977); McLennan (1989);
Brey  _et al._ (1999); Sarracino  _et al._ (2010). Note that the expression
(62) of $\gamma$ differs from the macroscopic Stokes law describing the
Brownian motion of a massive particle in an equilibrium host fluid Bocquet
_et al._ (1994a, b).

As usual in the USF problem for granular suspensions Hayakawa  _et al._
(2017); Hayakawa and Takada (2019); Takada  _et al._ (2020), we introduce the
dimensionless shear rate

$a^{*}=\frac{a}{\gamma},$ (66)

and Eq. (51) in dimensionless form can be rewritten as

$\displaystyle
a_{ik}^{*}P_{kj}^{*}+a_{jk}^{*}{P}_{ki}^{*}+2\xi^{*}P_{ij}^{*}=-\nu^{*}\left[\nu_{\eta}^{*}P_{ij}^{*}+\chi(\zeta^{*}-\nu_{\eta}^{*})\delta_{ij}\right]$
$\displaystyle-2\frac{m}{m_{g}}\mu\left(\frac{1+\theta}{\theta}\right)^{3/2}\left(X\delta_{ij}+\chi^{-1}YP_{ij}^{*}+ZP_{g,ij}^{*}\right),$
(67)

where $\xi^{*}=\xi/\gamma$ and

$\nu^{*}=\frac{\nu}{\gamma}=\epsilon\chi^{1/2},\quad\epsilon=\frac{2^{d+2}d}{(d+2)\sqrt{\pi}}\phi\sqrt{T_{g}^{*}}.$
(68)

Here,

$\phi=\frac{\pi^{d/2}}{2^{d-1}d\Gamma\left(\frac{d}{2}\right)}n\sigma^{d}$
(69)

is the solid volume fraction and

$T_{g}^{*}=\frac{T_{g}}{m\sigma^{2}\gamma^{2}}$ (70)

is the reduced background temperature. Note that the density dependence of
$\nu^{*}$ arises from the scaling of the shear rate $a$ with the drift
coefficient $\gamma$. If we had reduced the shear rate for instance with the
collision frequency $\nu(T)$, then the above density dependence had been
removed. On the other hand, since we want to study the non-Newtonian transport
properties of the granular gas in thermal contact with a bath of elastic
particles at temperature $T_{g}$ it seems natural to scale $a$ with respect to
$\sqrt{T_{g}}$ instead of $\sqrt{T}$, as usual in dry granular gases Santos
_et al._ (2004). Moreover, given that we want to make a close comparison with
computer simulations Hayakawa  _et al._ (2017); Hayakawa and Takada (2019);
Takada  _et al._ (2020), the theory must employ the same input parameters as
in the simulation results. In this context, in order to express the
rheological properties of the molecular gas in terms of the (reduced) shear
rate $a^{*}$, one has to take into account the relations

$\widetilde{a}=\frac{\gamma}{\nu_{g}}a^{*},\quad\widetilde{\xi}=\frac{\gamma}{\nu_{g}}\xi^{*},$
(71)

where

$\frac{\gamma}{\nu_{g}}=\frac{d+2}{\sqrt{2}d}\left(\frac{\overline{\sigma}}{\sigma_{g}}\right)^{d-1}\frac{m_{g}}{m}.$
(72)

The equation for the temperature ratio $\chi$ can be obtained from Eq. (IV) as

$2a^{*}P_{xy}^{*}+2d\chi\xi^{*}+d\chi\left(\zeta^{*}+\zeta_{g}^{*}\right)=0,$
(73)

where $\zeta^{*}$ is defined by Eq. (61) and $\zeta_{g}^{*}=\zeta_{g}/\nu$ is
given by

$\zeta_{g}^{*}=2x(1-x^{2})\mu^{1/2}\epsilon^{-1},$ (74)

with

$x=\left(\mu_{g}+\mu\frac{T_{g}}{T}\right)^{1/2}.$ (75)

For elastic collisions ($\alpha=1$) and vanishing shear rates ($a^{*}=0$),
$\zeta^{*}=\xi^{*}=0$, and Eq. (73) simply yields $\chi=1$ (energy
equipartition), as expected. For inelastic collisions ($\alpha<1$) and
arbitrary shear rates, Eq. (73) must be numerically solved.

The elements $P_{xx}^{*}$, $P_{yy}^{*}$, and $P_{xy}^{*}$ can be easily
determined from Eq. (IV). In terms of the temperature ratio, they can be
written as

$P_{yy}^{*}=\frac{\Omega_{yy}}{\nu_{yy}},\quad
P_{xy}^{*}=\frac{\Omega_{xy}}{\nu_{yy}}-\frac{a^{*}}{\nu_{yy}}P_{yy}^{*},$
(76) $P_{xx}^{*}=d\chi-(d-1)P_{yy}^{*},$ (77)

where we have introduced the quantities

$\nu_{yy}=\nu^{*}\nu_{\eta}^{*}+2\xi^{*}+2\mu\theta\left(\frac{1+\theta}{\theta}\right)^{3/2}Y,$
(78)
$\Omega_{yy}=-\nu^{*}\chi\left(\zeta^{*}-\nu_{\eta}^{*}\right)-2\frac{m}{m_{g}}\mu\left(\frac{1+\theta}{\theta}\right)^{3/2}\left(X+ZP_{g,yy}^{*}\right),$
(79)
$\Omega_{xy}=-2\frac{m}{m_{g}}\mu\left(\frac{1+\theta}{\theta}\right)^{3/2}ZP_{g,xy}^{*}.$
(80)

The temperature ratio can be determined by substituting Eqs. (74) and (76)
into Eq. (73). This yields the equation

$a^{*}=\sqrt{\frac{d}{2}\frac{2\mu^{2}\left(\frac{1+\theta}{\theta}\right)^{1/2}(1-\chi)-\left(\nu^{*}\zeta^{*}+2\xi^{*}\right)\chi}{\frac{\Omega_{xy}/a^{*}}{\nu_{yy}}-\frac{\Omega_{yy}}{\nu_{yy}^{2}}}}.$
(81)

As occurs in previous works on granular suspensions Hayakawa  _et al._ (2017);
Hayakawa and Takada (2019, 2019); Takada  _et al._ (2020), the temperature
ratio $\chi$ cannot be expressed in Eq. (81) as an explicit function of the
(reduced) shear rate $a^{*}$ and the remaining parameters of the system.
However, the dependence of $\chi$ on the latter parameters can implicitly be
obtained from the physical solution to Eq. (81) as
$a^{*2}(\chi,\alpha,\sigma/\sigma_{g},m/m_{g},\phi,T_{g}^{*})$.

As for the molecular gas, non-Newtonian transport properties of the granular
gas are characterized by the nonlinear shear viscosity

$\eta^{*}=-\frac{P_{xy}^{*}}{a^{*}},$ (82)

and the normal stress difference

$\Psi^{*}=P_{xx}^{*}-P_{yy}^{*}.$ (83)

As expected, both quantities are highly _nonlinear_ functions of the (reduced)
shear rate $a^{*}$ and the parameter space of the system.

###  IV.1 Navier–Stokes shear viscosity

Before considering the shear-rate dependence of the non-Newtonian transport
properties of the granular suspension, it is quite instructive to consider the
limit of small shear rates (Navier–Stokes regime) when the molecular gas is at
equilibrium. This was the situation analyzed in Ref. Gómez González and Garzó
(2022).

For vanishing shear rates ($a^{*}=0$), $\xi^{*}=0$ and Eq. (IV) becomes
$\zeta^{*}+\zeta_{g}^{*}=0$. This yields the following cubic equation for $x$:

$2x(x^{2}-1)=\vartheta,\quad\vartheta=\frac{d+2}{4d}\mu^{-1/2}\epsilon(1-\alpha^{2}).$
(84)

The physical root of Eq. (84) can be written as Santos (2003)

$x=\Bigg{\\{}\begin{array}[]{cc}\frac{\sqrt{3}}{3}\left\\{\sqrt{3}\cos\left[\frac{1}{3}\sin^{-1}\left(\frac{3\sqrt{3}}{4}\vartheta\right)\right]+\sin\left[\frac{1}{3}\sin^{-1}\left(\frac{3\sqrt{3}}{4}\vartheta\right)\right]\right\\},&\vartheta\leq\frac{4\sqrt{3}}{9}\\\
\frac{2\sqrt{3}}{3}\cosh\left[\frac{1}{3}\cosh^{-1}\left(\frac{3\sqrt{3}}{4}\vartheta\right)\right],&\vartheta\geq\frac{4\sqrt{3}}{9}.\end{array}$
(85)

Thus, according to Eq. (75), the temperature ratio $T/T_{g}$ is

$\frac{T}{T_{g}}=\frac{m/m_{g}}{\left(1+\frac{m}{m_{g}}\right)x^{2}-1}.$ (86)

Equations (85) and (86) agree with those obtained in Ref. Gómez González and
Garzó (2022).

In the Navier–Stokes domain, $P_{g,ij}^{*}=\delta_{ij}$ and since
$\xi^{*}\propto\widetilde{\xi}\sim a^{*2}\to 0$ when $a^{*}\to 0$, then Eqs.
(76)–(80) lead to $P_{xx}^{\text{NS}}=P_{yy}^{\text{NS}}=nT$, as expected. In
addition, the shear stress

$P_{xy}^{\text{NS}}=-\eta_{\text{NS}}a,$ (87)

where the Navier–Stokes shear viscosity $\eta_{\text{NS}}$ can be written as

$\eta_{\text{NS}}=\frac{\eta_{0}}{\nu_{\eta}^{*}+\widetilde{\nu}_{\eta}},$
(88)

where

$\eta_{0}=\frac{(d+2)\Gamma(d/2)}{8\pi^{(d-1)/2}}\sigma^{1-d}\sqrt{mT}$ (89)

is the Navier–Stokes shear viscosity of an ordinary dilute gas of hard spheres
($\alpha=1$), $\nu_{\eta}^{*}$ is given by Eq. (60) and

$\widetilde{\nu}_{\eta}=2\frac{m}{m_{g}}\mu\left(\frac{1+\theta}{\theta}\right)^{3/2}Y\epsilon^{-1}\chi^{-3/2}.$
(90)

The expression (88) of $\eta_{\text{NS}}$ agrees with the one derived in Ref.
Gómez González and Garzó (2022) in the low-density regime from the
Chapman–Enskog method. This shows the self consistency of the present results
with those reported in the Navier–Stokes limiting case.

##  V Brownian limiting case

It is quite apparent that the results obtained in Sec. IV apply for arbitrary
values of the diameter and mass ratios. This is in fact one of the new added
values of the present work. However, in the case of particle-laden suspensions
Subramaniam (2020) where collisions play an essential role in the dynamics of
grains, the granular particles are much heavier than the particles of the
surrounding molecular gas ($m_{g}/m\to 0$). This is referred here as to the
Brownian limit.

As discussed in the Introduction section, given the technical difficulties for
describing a system constituted by two or more phases (gas-solid flows), a
coarse-grained approach is usually adopted for analyzing gas-solid flows. In
this approach, the influence of gas-phase on grains is incorporated in the
starting kinetic equation via an effective fluid-solid interaction force Koch
(1990); Gidaspow (1994); Jackson (2000). Based on the results obtained in
direct numerical simulations, for high-velocity (but low-Reynolds numbers)
gas-solid flows, the effective force is constituted by two terms in the USF
problem: (i) a drag force term proportional to the particle velocity and (ii)
a stochastic Langevin-like term. While the first term models the energy lost
by grains due to its friction on the viscous gas, the second term accounts for
the energy transferred to the solid particles due to their collisions with the
particles of the interstitial molecular gas. Thus, in the low-density regime,
the starting kinetic equation for studying the USF is the Boltzmann equation
plus a Fokker–Planck term. In fact, this latter term can be formally obtained
from the Boltzmann–Lorentz collision operator in the limiting Brownian case
when the molecular gas is at equilibrium. However, some papers Rodríguez  _et
al._ (1983); Pelargonio and Zaccone (2023) have shown that the conventional
Langevin equation is modified by the shear flow so that, these new
contributions should be incorporated in the coarse-grained description when
one analyzes the USF problem in granular suspensions.

Figure 2: Thermostat parameter $\xi^{*}=\xi/\gamma$ versus the (reduced) shear
rate $a^{*}=a/\gamma$ for hard spheres ($d=3$) with $\sigma=\sigma_{g}$ and
three different values of the mass ratio $m/m_{g}$.

The objective of this section is to assess the consistency between the results
obtained here (which are based on a collisional model) in the Brownian limit
with those previously obtained Hayakawa  _et al._ (2017); Hayakawa and Takada
(2019); Takada  _et al._ (2020) by employing the Langevin equation. An
important conclusion of those papers is that the non-Newtonian viscosity
$\eta^{*}$ exhibits an $S$ shape in a plane of stress–strain rate
(discontinuous shear thickening effect).

At a given _finite_ value of the (reduced) shear rate $a^{*}$, according to
Eqs. (71) and (72), $\widetilde{a}\propto m_{g}/m$ and $\widetilde{\xi}\propto
m_{g}/m$. This means that in the Brownian limiting case ($m_{g}/m\to 0$) the
shear rate $a$ in units of the collision frequency $\nu_{g}$ of the molecular
gas is very small. Consequently, the pressure tensor of the granular gas
behaves as $P_{g,ij}^{*}=\delta_{ij}$ and hence, the granular gas does not
perceive the presence of a strong shear field. Although not explicitly stated,
this is in fact one of the assumptions in the suspensions models employed by
starting from the conventional Langevin equation in the USF state Hayakawa
_et al._ (2017); Hayakawa and Takada (2019); Takada  _et al._ (2020); Kawasaki
_et al._ (2014); Tsao and Koch (1995); Sangani  _et al._ (1996); Heussinger
(2013); Saha and Alam (2017, 2020).

In the Brownian limit, $\mu_{g}\to 0$, $\mu\to 1$, $\chi\equiv\text{finite}$,
and so $\theta\sim m/m_{g}\chi^{-1}$, $X+Z\sim-m_{g}/m$, and
$Y\sim\theta^{-1}\sim(m_{g}/m)\chi$. In addition, the thermostat parameter
$\xi^{*}=(\nu_{g}/\gamma)\widetilde{\xi}$ is negligibly small when $m\gg
m_{g}$. This is illustrated in Fig. 2 where $\xi^{*}$ is plotted versus
$a^{*}$ for $d=3$, $\sigma=\sigma_{g}$, and different values of the mass
ratio. Since we see in Fig. 2 that $\xi^{*}\to 0$ as $m_{g}/m\to 0$, we can
take $\xi^{*}=0$ in Eqs. (76)–(81). Thus, after taking carefully the Brownian
limit ($m_{g}/m\to 0$) in the above equations, one achieves the expressions

$P_{yy}^{*}=\frac{2-\nu^{*}\chi\left(\zeta^{*}-\nu_{\eta}^{*}\right)}{2+\nu^{*}\nu_{\eta}^{*}},$
(91) $P_{xy}^{*}=-\frac{a^{*}}{2+\nu^{*}\nu_{\eta}^{*}}P_{yy}^{*},$ (92)
$P_{xx}^{*}=\frac{2\left[d(\chi-1)+1\right]+\nu^{*}\chi\left[(d-1)\zeta^{*}+\nu_{\eta}^{*}\right]}{2+\nu^{*}\nu_{\eta}^{*}},$
(93)
$a^{*}=\sqrt{\frac{d}{2}\frac{\nu^{*}\zeta^{*}+2\left(1-\chi^{-1}\right)}{\nu^{*}\left(\nu_{\eta}^{*}-\zeta^{*}\right)+2\chi^{-1}}}\left(2+\nu^{*}\nu_{\eta}^{*}\right).$
(94)

Equations (91)–(94) are consistent with the results obtained in Refs. Hayakawa
_et al._ (2017); Hayakawa and Takada (2019); Gómez González and Garzó (2019b)
from a coarse-grain approach.

In summary, we can conclude that the expressions of the rheological properties
derived here from a suspension model that accounts for the grain-gas
collisions reduce in the Brownian limit ($m\gg m_{g}$) to those obtained in
previous works Hayakawa  _et al._ (2017); Hayakawa and Takada (2019); Gómez
González and Garzó (2019b) when the influence of the interstitial gas on
grains is accounted for by means of an effective external force.

##  VI Some illustrative systems

The dependence of the (reduced) temperature $\chi$, the non-Newtonian shear
viscosity $\eta^{*}$, and the viscometric function $\Psi^{*}$ on the (reduced)
shear rate $a^{*}$ is shown in Figs. 3 and 4 for inelastic ($\alpha=0.9$) and
elastic ($\alpha=1$) collisions, respectively. Apart from studying the shear-
rate dependence of the rheological properties, our main goal here is to assess
the impact of the mass ratio $m/m_{g}$ on them. Additionally, Eq. (72)
introduces the diameter ratio $\sigma/\sigma_{g}$ as a “new” parameter for
consideration. To ensure that the effect of inelastic collisions is on par
with that of the (elastic) Boltzmann-Lorentz collision operator, the (reduced)
background temperature $T_{g}^{*}\propto\nu/\gamma$ is chosen to be constant
and approximately close to unity. In this specific instance, $T_{g}^{*}=1$.
The selection of $T_{g}^{*}$ as a free parameter imposes the following
constraint between the diameter ratio ($\sigma/\sigma_{g}$) and mass ratio
($m/m_{g}$):

$\frac{\sigma_{g}}{\sigma}=\left[\left(\frac{\sqrt{\pi}}{4\sqrt{2}}\frac{n}{n_{g}}\sqrt{\frac{m}{m_{g}}}\frac{1}{\phi\sqrt{T_{g}^{*}}}\right)^{1/(d-1)}-1\right].$
(95)

As mentioned in Sec. IV, an explicit density dependence emerges due to the
scaling of the background temperature with the known quantity $\gamma$. In
this paper, we consider a very dilute gas with $\phi=0.0052$. Furthermore,
since the granular gas is present in tracer concentration, the number density
of granular particles $n_{g}$ is chosen to be an order of magnitude greater
than the number density of the molecular gas $n$. More specifically, we take
$n/n_{g}=10^{-3}$. It is important to highlight that Eq. (95) ensures
$\xi^{*}\to 0$ as $m_{g}/m\to 0$, thereby maintaining the consistency of Eqs.
(76)–(81) with those expressions derived from the Fokker–Planck operator Gómez
González and Garzó (2022) in the Brownian limit ($m_{g}/m\to 0$).

Figure 3: Plots of the (reduced) granular temperature $\chi$ [panel (a)], the
non-Newtonian shear viscosity $\eta^{*}$ [panel (b)], and the viscometric
function $\Psi^{*}$ [panel (c)] as a function of the (reduced) shear rate
$a^{*}$ for $\alpha=0.9$ and different values of the mass ratio $m/m_{g}$:
$10^{3}$ (blue lines), $10^{4}$ (green lines), $10^{5}$ (red lines), and the
Brownian limiting case (dotted lines). Here, $T^{*}_{g}=1$ , $d=3$, and
$\phi=0.0052$. The filled symbols refer to DSMC simulations, while the open
ones refer to event-driven Langevin simulations Hayakawa  _et al._ (2017),
both performed in the Brownian limit.  Figure 4: Plots of the (reduced)
granular temperature $\chi$ [panel (a)], the non-Newtonian shear viscosity
$\eta^{*}$ [panel (b)], and the viscometric function $\Psi^{*}$ [panel (c)] as
a function of the (reduced) shear rate $a^{*}$ for $\alpha=1$ and different
values of the mass ratio $m/m_{g}$: $10^{3}$ (blue lines), $10^{4}$ (green
lines), $10^{5}$ (red lines), and the Brownian limiting case (dotted lines).
Here, $T^{*}_{g}=1$ , $d=3$, and $\phi=0.0052$. The filled symbols refer to
DSMC simulations, while the open ones refer to event-driven Langevin
simulations Hayakawa  _et al._ (2017), both performed in the Brownian limit.

The principal conclusion drawn from Fig. 3 and 4 is that variations in the
mass ratio $m/m_{g}$ do not significantly alter the observed trends in prior
studies Hayakawa  _et al._ (2017); Gómez González and Garzó (2019b, 2020)
within a coarse-grained approach. Notably, there is a discontinuous transition
in all rheological properties with an increase in shear rate. Specifically,
Panel (b) of Figs. 3 and 4 underscores the presence of DST in shear viscosity
$\eta^{*}$ under sufficiently large mass ratios. Namely, the non-Newtonian
shear viscosity $\eta^{*}$ discontinuously increases/decreases (at a certain
value of $a^{*}$) as the (scaled) shear rate gradually increases/decreases.
Quantitatively, it becomes apparent that, for a constant value of $a^{*}$,
larger mass ratios marginally increase the magnitude of the $\eta^{*}$
discontinuity. The application of a shear flow impels particles to move
together at the same velocity $U_{x}$. Over time, collisions between particles
in each layer of the flow induce momentum transmission to the $y$ axis.
Heavier particles are provided with more kinetic energy to maintain a constant
velocity profile, leading to increased viscosity and temperature, as evident
in Panel (a) of Fig. 3 and 4.

Analytically, Eq. (72) indicates that $\widetilde{a}\to 0$ when $m_{g}/m\to 0$
for a finite value of $a^{*}$. Thus, Fig. 1(a) asserts that the viscosity of
the molecular gas increases. Additionally, the balance between collisional
cooling due to the inelasticity of grain collisions and bath thermalization
depends solely on temperature, as $T_{g}^{*}$ is held constant. However, an
increase in the mass ratio $m/m_{g}$ elevates temperature due to the higher
kinetic energy required to move the entire system at the same velocity.
Consequently, the collision frequency $\nu$ rises, leading to more collisions
and facilitating momentum transmission. This effect combined with the
increased viscosity of the molecular gas causes the viscosity of the
suspension to increase. Although similar trends are noted for $\Psi^{*}$, it
is noteworthy that the combined influence of $m/m_{g}$ and $a^{*}$ on the
viscometric function $\Psi^{*}$ is substantial. While this quantity remains
negligible for low shear rates, it abruptly increases for moderately small
shear rates (e.g., $a^{*}\approx 0.1$).

Theoretical predictions are compared against event-driven simulations Hayakawa
_et al._ (2017) and DSMC results carried out in this paper for the Brownian
limiting case ($m/m_{g}\to 0$). Some technical details of the application of
the DSMC method to the system studied in this paper are provided in the
supplementary material of Ref. Gómez González and Garzó (2022) available at
https://doi.org/10.1017/jfm.2022.410.

It is quite apparent the the agreement between theory and computer simulations
is generally favorable, with the exception of a localized region proximate to
the transition point. In this region, MD simulation data indicate a more
pronounced transition compared to the Boltzmann results. Given that the
selected reference frequency is $\gamma$, it is conjectured that this slight
disparity is primarily attributable to the marginal density dependence of the
(reduced) collision frequency $\nu^{*}=\nu/\gamma$. This effect is not
effectively accounted for in Boltzmann kinetic theory.

In the absence of shear or with minimal shear, no such amount of energy is
required to move the fluid at a constant velocity $U_{x}$. Pure collisional
effects from the isolated action of the background fluid and inelasticity
emerge. As granular particles become heavier, the inertial effect of mass
causes the grains to necessitates more collisions with the background fluid
for thermalization. This produces a decrease in granular temperature and
viscosity, as observed in Fig. 5 for small values of the (reduced) shear rate
$a^{*}$. Remarkably, shear-thinning effects are observed for sufficiently
small values of both $m/m_{g}$ and $a^{*}$.

Figure 5: Plot of the non-Newtonian shear viscosity $\eta^{*}$ as a function
of the (reduced) shear rate $a^{*}$ for $\alpha=0.9$ and different values of
the mass ratio $m/m_{g}$: $10^{3}$ (blue line), $10^{4}$ (green line),
$10^{5}$ (red line), and the Brownian limiting case (dotted line). Here,
$T^{*}_{g}=1$ , $d=3$, and $\phi=0.0052$.

Another important effect to consider is the impact of inelasticity on
rheology. To explore this, our focus is on the Brownian limit where the
influence of $\alpha$ is isolated from the mass ratio. Figure 6 illustrates
the rheological properties for various values of $\alpha$. Firstly, it is
observed that inelasticity attenuates the DST transition in the non-Newtonian
viscosity $\eta^{*}$. This is primarily attributed to the collisional cooling
resulting from interactions between grains, leading to a decrease in the
(reduced) temperature $\chi$. As the most noticeable effect, this temperature
decrease reduces collision frequency. Consequently, momentum transfer becomes
more challenging, causing a viscosity decrease for a given $a^{*}$. We also
see that the effect of $\alpha$ on the viscometric function $\Psi^{*}$ is
smaller than the one found for the temperature $\chi$ and the non-Newtonian
viscosity $\eta^{*}$. Moreover, a good level of agreement between theory and
DSMC simulations is shown.

Figure 6: Plots of the (reduced) granular temperature $\chi$ [panel (a)], the
non-Newtonian shear viscosity $\eta^{*}$ [panel (b)], and the viscometric
function $\Psi^{*}$ [panel (c)] in the Brownian limit as a function of the
(reduced) shear rate $a^{*}$ for different values of $\alpha$: $0.5$ (blue
lines), $0.6$ (green lines), $0.7$ (red lines), $0.8$ (orange lines), $0.9$
(purple lines), and $1$ (black lines). Here, $T^{*}_{g}=1$ , $d=3$, and
$\phi=0.0052$. The symbols refer to DSMC simulations.

##  VII Summary and concluding remarks

In recent years, there has been a growing interest in employing kinetic theory
tools to determine the non-Newtonian transport properties of inertial
suspensions under simple shear flow. Numerous studies, drawing on the
Boltzmann kinetic equation (applicable to very dilute systems) and/or the
Enskog kinetic equation (suitable for moderately dense systems), have made
substantial contributions to this field. In particular, several works Tsao and
Koch (1995); Sangani  _et al._ (1996); Garzó  _et al._ (2012); Chamorro  _et
al._ (2015); Hayakawa  _et al._ (2017); Saha and Alam (2020); Hayakawa and
Takada (2019); Gómez González and Garzó (2019b, 2020) have derived explicit
expressions for the shear-rate dependence of the kinetic temperature, non-
Newtonian viscosity, and viscometric functions.

Among various gas-solid flows, the particle-laden suspensions Gidaspow (1994);
Jackson (2000); Koch and Hill (2001); Tenneti and Subramaniam (2014); Fullmer
and Hrenya (2017); Wang (2020); Liu  _et al._ (2020) pose an intriguing
problem. In such suspensions, dilute and immiscible particles are immersed in
a denser fluid, causing hydrodynamic interactions to become less relevant
Subramaniam (2020); Batchelor (1972). Consequently, the dynamics of solid
particles are influenced by thermal fluctuations in the fluid, with
interparticle, external, and Brownian forces prevailing Brady and Bossis
(1988). Assuming the number density of the solid phase is significantly
smaller than that of the surrounding fluid, the latter remains practically
unaffected by the presence of the solid particles. In this scenario, the
external fluid may be treated as a thermostat (or bath) at a fixed temperature
$T_{g}$. Thus, gas-solid interactions are effectively modeled via a fluid-
solid force in the starting kinetic equation Koch (1990); Gidaspow (1994);
Jackson (2000).

Some models for granular suspensions Louge  _et al._ (1991); Tsao and Koch
(1995); Sangani  _et al._ (1996); Wylie  _et al._ (2009b); Parmentier and
Simonin (2012); Heussinger (2013); Wang  _et al._ (2014); Saha and Alam
(2017); Alam  _et al._ (2019); Saha and Alam (2020) consider only isolated
body resistance through a linear drag law, while others Puglisi  _et al._
(1998); Garzó  _et al._ (2012) include an additional Langevin-type stochastic
term. Analytical results from these models, validated against computer
simulations, generally exhibit good agreement under conditions of practical
interest. However, questions arise about whether these models accurately
reproduce real situations and if they can be recovered using purely kinetic
arguments.

In this work, we have started from a suspension model that consider real
collisions between grains and particles of the background gas in the starting
kinetic equation. The influence of the gas phase on solid particles is
introduced via the (elastic) Boltzmann-Lorentz collision operator
$J_{g}[f,f_{g}]$. As in previous works Hayakawa  _et al._ (2017); Hayakawa and
Takada (2019); Takada  _et al._ (2020), our initial objective has been to
determine approximate expressions for the rheological properties of the
molecular gas by using Grad’s moment method Grad (1949). Specifically, we have
computed the collisional moment $\mathsf{A}_{g}$ defined in Eq. (36) through
Grad’s distribution (38). The knowledge of this collisional moment enables the
derivation of the explicit forms for the non-Newtonian shear viscosity
$\eta_{g}$ and the viscometric function $\Psi^{*}_{g}$ of the gas phase Garzó
(2002); Santos  _et al._ (2004). The anticipated shear-thinning effect in the
shear viscosity is observed Garzó and Santos (2003), a phenomenon corroborated
by the excellent agreement between the results obtained from Grad’s method and
DSMC simulations.

Following the characterization of the molecular gas, we have also computed by
means of Grad’s moment method the collisional moments $\mathsf{B}$ and
$\mathsf{C}$ of the granular gas defined in Eqs. (57) and (58), respectively.
After performing algebraic manipulations, we have expressed the (reduced)
granular temperature $\chi$ and the (reduced) relevant components of the
pressure tensor $P^{*}_{ij}$ in terms of the parameter space governing the
problem. This parameter space includes the coefficient of normal restitution
$\alpha$, the (reduced) shear rate $a^{*}$, the (reduced) background
temperature $T_{g}^{*}$, the volume fraction $\phi$, and the diameter
$\sigma_{g}/\sigma$ and mass $m_{g}/m$ ratios. Consequently, a noteworthy
contribution of our findings lies in providing an explicit dependence of the
rheological properties on the above ratios, especially in the case of the mass
ratio $m_{g}/m$.

The results derived here for arbitrary values of the mass ratio show no new
surprises relative to the earlier works for gas-solid flows Hayakawa  _et al._
(2017); Gómez González and Garzó (2019b): the flow curve for the non-Newtonian
viscosity $\eta^{*}$ exhibits an $S$-shape, and hence, DST is present.
Nonetheless, from a more quantitative level, the results obtained in this work
illustrate the amplifying influence of the DST effect with increasing the mass
ratio $m/m_{g}$. Notably, when the ratio $m/m_{g}$ reaches a sufficiently
large value, the conclusions derived in this study agree with those previously
established in the context of the Brownian limit from the Fokker–Planck
operator Hayakawa  _et al._ (2017); Gómez González and Garzó (2019b). This
agreement serves to validate the application of the conventional Langevin
equation in the analysis of non-Newtonian transport within dilute granular
suspensions, even when subjected to high shear rates Pelargonio and Zaccone
(2023). Another remarkable result lies in the mass ratio values for which
agreement is reached with the coarse-grained results. In contrast to the
situation observed in the context of the Navier–Stokes transport coefficients,
where convergence to the Brownian limit was attained for relatively small mass
ratios ($m/m_{g}\approx 50$) Gómez González and Garzó (2022), the rheological
scenario requires mass ratios exceeding $10^{6}$ to observe this collapse.
This emphasizes the intricacy of the collisional model, not only in its
capacity to elucidate the Fokker–Planck model as a limiting case derived from
purely kinetic principles but also for its ability for representing a diverse
spectrum of systems whose rheology may be not fully captured by the existing
effective models.

Another noteworthy observation related to the variability of the mass ratio
$m/m_{g}$ lies in the ability to completely modify the rheological behavior of
the suspension. This is manifested in a transition from shear thinning to
thickening for a critical mass ratio when we consider sufficiently low values
of $a^{*}$ as showed in Fig. 5.

Furthermore, the results depicted in Figs. 3 and 4 highlight the commendable
efficacy of both Grad’s method and DSMC simulations in capturing the shear-
rate dependence of the rheological properties. Remarkably, a perfect agreement
emerges among the results obtained from Grad’s method, DSMC, and MD
simulations in the Brownian limit. Despite this agreement, slight disparities
manifest in a limited region near the transition point, where MD simulation
data indicate a more pronounced transition than the Boltzmann results suggest.
This minor disparity is attributed primarily to the constraints of the
Boltzmann equation in incorporating small density corrections to $\eta^{*}$ in
the vicinity of this transition point. In addition, although the origin of the
DST has been attributed to structural properties and mutual friction between
grains Brown and Jeager (2014); Bourrianne  _et al._ (2022), our ability to
replicate the DST effect using the Boltzmann kinetic equation (which is only
applicable to extremely dilute systems) is noteworthy.

Taking advantage of the excellent agreement observed between MD and DSMC
computer simulations, we can exploit the latter to explore new situations in
which to validate Grad’s results. This is done in Fig. 6, where the
rheological properties are plotted vs the shear rate in the Brownian limit for
different coefficients of restitution. Firstly, we observe a commendable
agreement between theoretical predictions and simulation data, underscoring
the robustness of Grad’s results even in scenarios characterized by moderate
inelasticities. On the other hand, as expected, the influence of inelasticity
manifests in a dampening of the DST transition. Although this dissipative
effect is evident in the viscometric function $\Psi^{*}$ as well, its impact
is comparatively less pronounced.

It is apparent that the theoretical results presented here are relevant to
make a comparison with computer simulations. The observed agreement in the
Brownian limit prompts the expectation that such alignment extends to cases
involving finite mass ratios $m/m_{g}$. We intend to pursue these simulations
in the near future. Another prospective project involves extending the current
findings to finite densities by considering the Enskog kinetic equation.
Within this context, an intriguing question arises: does the variability of
the mass ratio influence the transition from DST to CST as density increases,
as observed in the Fokker–Planck model? Exploration along this avenue is
planned for future work.

###### Acknowledgements.

We acknowledge financial support from Grant PID2020-112936GB-I00 funded by
MCIN/AEI/10.13039/501100011033, and from Grant IB20079 funded by Junta de
Extremadura (Spain) and by ERDF A way of making Europe.

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