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Journal of Non-Newtonian Fluid Mechanics 239 (2017) 62–72
Contents lists available at ScienceDirect
Journal of Non-Newtonian Fluid Mechanics
journal homepage: www.elsevier.com/locate/jnnfm
Steady-shear rheological properties for suspensions of axisymmetric
particles in second-order fluids
J. Férec
a , ∗, E. Bertevas b , B.C. Khoo
b , G. Ausias a , N. Phan-Thien
b
a Institut de Recherche Dupuy de Lôme (IRDL), Univ. Bretagne Sud, FRE CNRS 3744, IRDL, F-56100 Lorient, France b Department of Mechanical Engineering, National University of Singapore, 119260, Singapore
a r t i c l e i n f o
Article history:
Received 27 June 2016
Revised 10 December 2016
Accepted 17 December 2016
Available online 21 December 2016
Keywords:
Normal stress difference coefficients
Constitutive equation
Viscoelastic media
Fokker–Planck equation
Shear flow
Rod suspensions
a b s t r a c t
Following Leal who gave the motion of a slender axisymmetric rod in a second-order fluid, we derived
a complete rheological constitutive equation for dilute and semidilute slender rod suspensions in a vis-
coelastic solvent based on a cell model. Numerical solutions for the Fokker–Planck equation are obtained
for simple shear flows at low and large Peclet numbers using a finite volume method, hence avoiding
the need for closure approximations. The second normal stress difference coefficient of the solvent plays
a non-negligible role in the particle contribution to the stress as well as on the rod orientation dynam-
ics: a spread of the particle orientation in the flow-vorticity plane and an enhancement of the alignment
along the vorticity direction are predicted when increasing the second normal stress difference coef-
ficient. Brunn extended the Leal analysis to dumbbells and tri-dumbbells, for which both normal stress
difference coefficients have to be considered. The original Pipkin diagram is finally modified to help guide
the choice of relevant constitutive equations for particles in viscoelastic fluids.
© 2016 Elsevier B.V. All rights reserved.
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1. Introduction
The rheological characterization of rod-filled media is of ma-
jor concern to many industries, such as printing and papermaking,
petroleum, polymer processing, aerospace, bioengineering, phar-
maceutical industry, construction, ceramics, food, etc. Indeed, the
behavior of the suspension is usually significantly different from
that of the suspending fluid. The orientation distribution of rods
induced by the flow field strongly influences certain macroscopic
physical properties such as the rheological behavior of the suspen-
sion, which itself governs the flow pattern. A large body of work
in the literature has focused on the study of rod-filled Newtonian
liquids, in which the rheological effects and the orientation evolu-
tion of the rods are described [1,2] . Despite the fact that almost all
solvents used in the industry are viscoelastic by nature, the under-
standing and especially the modeling of the rheological behavior of
rod-filled viscoelastic media remain a formidable challenge. Due to
their complexity, only a limited amount of studies has attempted
to embark on such an endeavor.
Constitutive equations for rod filled viscoelastic systems may
generally be considered as a two-component fluid, in which the
∗ Corresponding author.
E-mail address: [email protected] (J. Férec).
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http://dx.doi.org/10.1016/j.jnnfm.2016.12.006
0377-0257/© 2016 Elsevier B.V. All rights reserved.
otal stress of the composite can be assumed as [3]
= −P δ + τm + τ p , (1)
here P is the isotropic pressure, δ is the identity tensor, τm is the
atrix contribution and τp is the particle contribution to the extra
tress tensor.
.1. Newtonian suspending fluids
When dealing with a Newtonian solvent of viscosity η0 , the
article contribution to the extra stress tensor ( τp ) at low rod vol-
me fraction φ takes the following general form [4]
p = η0 φ[μ1 a 4 : ˙ γ + μ2
(˙ γ · a 2 + a 2 · ˙ γ
)+ μ3 ̇ γ + 2 μ4 a 2 D r
], (2)
here ˙ γ is the deformation rate tensor. a 2 and a 4 are respec-
ively the second- and fourth-order orientation tensors [5] which
re commonly used to describe the average rod orientation state
n an efficient and concise way, without any significant loss of in-
ormation. The coefficients { μi , i = 1 , 2 , 3 , 4 } in Eq. (2) are geomet-
ic shape factors (see Table 1 in [2] ), and D r is the rotary diffu-
ivity due to Brownian motion. For slender rods, particle thickness
an be ignored and this is achieved by setting μ2 and μ3 equal
o zero. If the particles are large enough so that Brownian motion
an be ignored, the last term containing D r can be omitted. For in-
tance, Sepehr et al. [6] have theoretically checked this assumption
or short glass fiber suspensions, where the particle aspect ratio
s close to 20. Once μ , μ and μ (or equivalently D r ) are set
2 3 4
J. Férec et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 62–72 63
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o zero and μ1 is suitable chosen, Eq. (2) reduces to the expres-
ion of Dinh and Armstrong [7] , where the particle thickness has
een neglected in the derivation. Three regimes of rod concentra-
ions related to characteristic particle dimensions are proposed in
he literature [8] : dilute, for which φ < D
2 / L 2 ; semidilute D
2 / L 2 <
< D / L and concentrated φ > D / L , where L and D are respectively
he length and the diameter of the particle.
Particle motion in a Newtonian fluid was investigated theoret-
cally by Jeffery [9] , who solved the creeping flow equations for a
igid ellipsoid freely suspended in an infinite Newtonian fluid. In
simple shear flow, Jeffery’s solution shows that the particle cen-
er translates with the local fluid velocity and rotates in a time-
ependent periodic orbit about the vorticity axis of the flow [see
q. (6)]. Bretherton [10] indicates that the period of rotation for
ny axisymmetric particle is given by T r = 2 π( a r + a −1 r ) / ̇ γ , where
r = L / D is the particle aspect ratio and ˙ γ is the applied bulk shear
ate. Note that Jeffery’s theory is supported by extensive experi-
ental results [11–13] .
The orientation dynamics of a population of rods is commonly
odeled through a time evolution equation of the second-order
rientation tensor. This requires the use of closure approximations
o express higher order orientation tensors [14–17] . For non-dilute
od suspensions in Newtonian fluids, most theories make use of
he following expression
D a 2 Dt
= −1
2
( ω · a 2 − a 2 · ω ) +
λ
2
(˙ γ · a 2 + a 2 · ˙ γ − 2 a 4 : ˙ γ
)+ 2 D r
(δ − 3 a 2
), (3)
here ω is the vorticity tensor, λ = (a 2 r − 1) / (a 2 r + 1) is a shape
actor and D / Dt denotes the material derivative. The first two terms
n the left-hand side of Eq. (3) represent the hydrodynamic con-
ribution derived from the Jeffery’s equation and are valid for di-
ute suspensions of ellipsoids in a Newtonian fluid at low Reynolds
umbers. In order to describe concentrated non-Brownian particle
uspensions, Folgar and Tucker [18] suggested modeling particle-
article interactions by means of D r = C I | ̇ γ | , where C I is an inter-
ction coefficient [19,20] and | ̇ γ | is the effective deformation rate.
.2. Viscoelastic suspending fluids
With the prospect of modeling phenomena in composite pro-
essing, the general form of the constitutive equation cited above
as been extended in various manners to include the effect of
he viscoelastic polymer matrix on the suspension behavior. Ex-
ept for a few studies in which rods are omitted and therefore the
hole suspension is treated as a homogeneous viscoelastic fluid
21] , the particle contribution to the extra stress tensor, τp , is sim-
ly obtained by replacing the Newtonian viscosity in Eq. (2) by that
f the matrix η0 ≡ ηm ( ̇ γ , t ) , which can be shear rate-dependent,
ime-dependent or both. As for the evolution equation of a 2 , the
xpression in Eq. (3) is used without modification.
Fan [22] derived a constitutive equation in the general frame-
ork of phase-space kinetic theory. In this study, the suspending
uid was assumed to behave as an Oldroyd-B fluid. Assuming that
olymer chain motion was more strongly hindered in a direction
rosswise to the rod axis compared to the lengthwise direction,
nteractions between fluid and rods were modeled by means of
n anisotropic resistance coefficient [23,24] . Azaiez [3] used the ki-
etic theory of elastic dumbbells and a rod orientation-dependent
riction factor to develop constitutive equations for fiber suspen-
ions in polymer solutions based on the FENE-P (Finitely Extensi-
le Non-linear Elastic - Peterlin), FENE-CR (Finitely Extensible Non-
inear Elastic - Chilcott and Rallison), and Giesekus models. Ait-
adi and Grmela [25] assumed that the viscoelastic matrix be-
avior is governed by a second-order conformation tensor and
btained its time-evolution equation from the generalized Pois-
on bracket formalism. Their choice for the Helmholtz free en-
rgy function yields a FENE-P type viscoelastic matrix. This work
as then extended by Ramazani and co-authors [26,27] , who in-
roduced fiber-matrix interactions through anisotropic expressions
or the mobility tensor. A similar approach was adopted by [28] to
stablish a rheological model for semi-flexible fiber suspensions in
olymeric fluids described by a FENE-P model. Beaulne and Mit-
oulis [29] used the K-BKZ integral constitutive equation with mul-
iple relaxation times as proposed by [30] for the polymer ma-
rix. Some authors [31,32] applied the multi-mode Giesekus model
33] to predict the strain rate-dependent viscoelastic behavior of
he polymer matrix.
Nevertheless, none of the theories cited above considered the
ffects of the normal stress differences exhibited by the viscoelas-
ic matrix. In view of the state of current approaches, the questions
emaining open as to what should models include are: do rod sus-
ensions behave differently in a viscoelastic matrix as compared
o a Newtonian matrix, and does the suspending fluid elasticity
ontributes additional components to the particle stress tensor? In
ost of the previous studies, the rod orientation dynamic is based
n the Jeffery’s equation, which was derived for Newtonian fluids.
hat would be the effect of elasticity on fiber orientations?
Recently, D’Avino and Maffettone [34] compiled an exhaus-
ive literature review on particle dynamics in viscoelastic liq-
ids. Numerical simulations of the motion of spherical and el-
ipsoidal particles in viscoelastic liquids are addressed as well as
ome experimental results. Pioneer experimental work was car-
ied out by Saffman [35] , who observed that rods immersed in
non-Newtonian fluid undergoing Couette flow align along the
orticity axis. The same conclusions were reached over several
ecades by Mason and coworkers [36–39] , by Fuller and cowork-
rs [40,41] using linear dichroism measurements, by Iso and co-
uthors [42,43] for weakly and highly elastic fluids, and by Gunes
t al. [44] , who coupled rheo-optical methods and flow microscopy
o analyze the dynamics of spheroidal particles. However, no com-
lete model was clearly identified as only few theoretical studies
eal with the behavior of rods in a viscoelastic fluid.
A leading modeling study was conducted by Leal [45] , who de-
ived the motion of a slender rod in a second-order fluid (SOF) un-
ergoing simple shear flow. The velocity field produced by the rod
as expressed as a perturbation from the Newtonian flow solution.
s compared to the Jeffery’s solution, the particle still translates
ith the local undisturbed velocity of the suspending fluid but its
rientation time evolution involves the second normal stress differ-
nce coefficient of the fluid [Eq. (7)]. Later, Brunn [46,47] derived
nalogous equations for rigid tri-dumbbells [Eq. (8)] and 1st-order
umbbells [Eq. (9)], for which the condition of a non-zero second
ormal stress difference can be relaxed. Harlen and Koch [48] con-
idered an Oldroyd-B fluid and showed that the first normal stress
ifference is responsible for the particle alignment along the vor-
icity axis. Hence, all these theories predict a particle drift towards
he vorticity axis, instead of following a closed orbit as suggested
y Jeffery’s analysis in Newtonian fluids. This drift appears to arise
rom the normal stress differences of the fluid.
Despite good quantitative agreements between Leal’s theoreti-
al predictions and experimental results, few models have emerged
o describe the rheological behavior of such systems. The orienta-
ion distribution of rods in a SOF in simple shear flow has been an-
lyzed at the asymptotic limit of weak [49] and strong [50] Brown-
an diffusion. In the latter, the Fokker–Planck equation is solved for
he near-equilibrium conditions by using spherical harmonics and
runcating the resulting infinite series. The non-Newtonian proper-
ies of the fluid result in a narrower distribution of particles near
he shear plane, but cause a spread in the orientation in the flow
irection due to the drift towards the vorticity axis. Using a finite
64 J. Férec et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 62–72
Table 1
Expressions for λ, β and χ from models reported in the literature.
Authors λ β χ Comments Eq.
[9] a 2 r − 1
a 2 r + 1 0 0 Ellipsoids (Newtonian fluid) (6)
[45] 1 − 2 , 0 | ̇ γ | 8 η0
0 Slender particles (2 nd -order fluid) (7)
[47] a 2 r − 1
a 2 r + 1
λ2 | ̇ γ | 4 η0
( 1 , 0 − 2 2 , 0 ) β
(1
a 2 r − 1
)Rigid tri-dumbbells (2 nd -order fluid) (8)
[46] a 2 r e
− 1
a 2 r e + 1
| ̇ γ | 4 η0
[ 1 , 0
(1 +
3
4 a r
)− 2 2 , 0 ] 0 1 st -order dumbbells (2nd-order fluid)
a r e a r
=
√
3
2
(1 +
3
8 a −1
r + O ( a −2 r ) ...
)(9)
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difference scheme, this result was confirmed by Kamal and Mutel
[51] , who also showed that the effect of elasticity is negligible at
small Peclet number. However, none of this work led to any rigor-
ous expression for the particle stress contribution in a SOF with
the exception of Chung and Cohen [50] who, unfortunately, did
not provide any solution for the rheological resistance coefficients.
Hence, the matrix viscoelasticity only enters these models through
its effect on the orientation distribution.
In the present article, we derive a rheological model for slender
rod suspensions in a second-order fluid based on the Leal theory.
Following this result, the effect of the non-Newtonian nature of the
fluid on rod orientation in filled systems is investigated: How do
the matrix viscoelastic properties alter the rod dynamics as com-
pared to Newtonian fluids and what are the evolutions of the ma-
terial functions in shear flows? These are some of the questions
we wish to address. Hence, analyses of the model predictions for
steady-state shear flows with weak and strong rod interaction are
presented. To avoid the questionable accuracy of closure approxi-
mations, the Fokker–Planck equation associated with the Leal the-
ory is numerically solved using a finite volume method. We then
apply a cell model to extend our result to semidilute slender rod
suspensions in a SOF, in a manner similar to the work of Dinh and
Armstrong [7] for slender rod suspensions in a Newtonian liquid.
Finally, a modification of the Pipkin diagram is suggested to pro-
vide guidance on a proper choice of a relevant constitutive equa-
tion for a viscoelastic suspension.
2. Model equations for slender bodies
2.1. Orientation evolution of a single particle in a SOF
Different models describing the particle evolution in second-
order fluids are found in the literature. Such fluids exhibit a con-
stant shear viscosity η0 as Newtonian fluids do, but have non-zero
first and second normal stress coefficients, namely 1, 0 and 2, 0
[33] . This model is may be a good choice for investigating the ef-
fects of finite normal stress differences. Here, it can be expressed
by the following equation
τm = η0 ̇ γ − 1 , 0
2
D ˙ γ
Dt + 2 , 0 ̇ γ · ˙ γ , (4)
where D / D t is the upper convective derivative. It is now shown
that Jeffery’s, Leal’s and Brunn’s theories describing the orientation
evolution of a particle can be recast in the general form
˙ p = −1
2
ω · p +
λ
2
(˙ γ · p − ˙ γ : ppp
)− β
2 | ̇ γ | ˙ γ : pp
(˙ γ · p − ˙ γ : ppp
)− χ
2 | ̇ γ | (
˙ γ2 · p − ˙ γ2 : ppp
), (5)
where p denotes a unit vector directed along the main particle axis
and ˙ p represents its material derivative. Different expressions for
the parameters λ, β and χ are summarized in Table 1 .
In what follows, we focus on investigating Leal’s theory which
assumes that the particle is a slender body, but our general ap-
proach is also applicable for Brunn’s results. Leal’s solution is valid
or creeping flows where Re � De � 1. Re is the Reynolds number
ratio of hydrodynamic forces to viscous forces) and De is the Deb-
rah number (ratio of the intrinsic relaxation time of the fluid to
he rotational relaxation time of the rod). Please note that the 2nd-
rder model has been used because of its tractability (not much
rogress can be made with more complex model), and we hope
hat it can offer some qualitative information into the material be-
avior, whilst keeping in mind its instabilities in unsteady flows.
The Leal model predictions have been calculated for a single
oint in simple shear flow (subscripts 1, 2 and 3 stand for flow,
elocity gradient and vorticity directions, respectively). The rod ini-
ial orientation is √
3 / 3 for the three components of unit vector p ,
he applied shear rate is set at ˙ γ = 1 s −1 and Eq. (7) is solved us-
ng explicit time integration for β = 0, 0.5 and 1, respectively. Fig. 1
epicts the evolution the components of p as a function of defor-
ation ( γ = ˙ γ t) and shows that the rod will tend to orient in the
orticity direction with increasing the dimensionless second nor-
al stress coefficient ( β), as observed experimentally (see above).
.2. Model formulation for a rod population
To deal with a rod population instead of a single rod, Advani
nd Tucker [5] introduced the orientation distribution function,
( ϕ, θ ), for which ψ( ϕ , θ )d ϕ d θ represents the probability of find-
ng a rod between in the configuration between ϕ and ( ϕ + d ϕ),
nd between θ and ( θ + d θ ). The two spherical coordinates ϕ and
are related to the Cartesian components ( p 1 ; p 2 ; p 3 ) of the unit
ector p through p 1 = sin θ cos ϕ, p 2 = sin θ sin ϕ, and p 3 = cos θ .
he probability distribution function (PDF) must respect the nor-
alization condition and must be periodic as there is no distinc-
ion between the head and the tail of a rod. Moreover, the PDF
volution may be modeled by convection-diffusion scalar transport
quation which describes the fact that when a fiber leaves a cer-
ain orientation, it must adopt another one. From [52] , the conti-
uity relation for the PDF can be expressed in homogeneous flows
s
Dψ
Dt = −∇ p ·
(˙ p ψ
)+ C I | ̇ γ | ∇
2 p ψ, (10)
here ∇ p is the differential operator ∂ / ∂ p and corresponds to a
-operator on the surface of a unit sphere. The first term is the
onvective contribution resulting from the hydrodynamic forces,
nd the second one is the diffusion due to the rod-rod interaction.
t should be noted that the diffusivity, as expressed in Eq. (10) , can
nly be orthogonal to p .
In order to derive the evolution equation for the second-order
ensor, a 2 , Eq. (10) is multiplied by the dyadic product of p and
hen integrated over all possible directions of p ; with the concur-
ent use of Leal’s expression for ˙ p [Eq. (7)] (an alternative method
s proposed in Appendix A , where Eq. (5) is employed) and after
ome straightforward calculations, we find that
D a 2 Dt
= −a 4 : ˙ γ − 1
2
β
| ̇ γ | (
˙ γ · a 4 : ˙ γ + ˙ γ : a 4 · ˙ γ − 2 ̇ γ : a 6 : ˙ γ)
+ 2 D r
(δ − 3 a 2
), (11)
J. Férec et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 62–72 65
Fig. 1. Evolution of the Cartesian components of the unit vector p as a function of deformation ( β = 0 on the left; β = 0.5 in the center; and β = 1 on the right).
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here a 4 and a 6 are the fourth- and sixth-order orientation ten-
ors [5] . Eq. (11) may be made dimensionless using the Peclet
umber defined as P e = 1 / C I .
.3. Numerical method
The presence of even orientation tensors up to the sixth-rank in
q. (11) requires knowledge of their time evolution to fully com-
lete the model. Eq. (11) describes the evolution of a 2 and, un-
ortunately, deriving the rate equation for a 4 and a 6 will intro-
uce orientation tensors up to the 10th order which, again, poses
closure problem. This problem commonly arises when equations
or moments are derived from the distribution function. Since a 2 ould contain sufficient information to describe the rod orienta-
ion states, its time evolution is deemed to be sufficient to pre-
ict the microstructure change. Using solely the rate equation for
2 [Eq. (11)] implies the need for a closure approximation for
6 ( a 6 contains information about lower order orientation tensors
hrough the application of the normalization condition of the PDF
5] ), which are available in the literature [2] . However, testing the
ifferent approximation closures and their consequences on the
heological properties is not in the scope of this work.
Instead of employing disputable closure approximations, com-
lete solutions of the Fokker–Planck equation [Eq. (10)] , combined
ith Leal’s expression [Eq. (7)], are obtained numerically using a
nite volume method. Based on the work of [53] , the convective
uxes are upgraded to consider the elastic contribution in Eq. (7).
he half surface of the unit sphere, which is sufficient to repre-
ent the domain of possible orientations (again, there is no dis-
inction between the head and the tail of a rod), is discretized with
qually distributed nodes in the ϕ- and θ-directions ( ϕ and θ are
he two spherical coordinates). To deal with strong flows, a fine
rid composed of 150 × 150 nodes (extended to 250 × 250 nodes
or Pe > 10 3 ) and a power-law differencing scheme were adopted.
or a given Pe and starting with an initial guess of 1/4 π (isotropic
tate), the time integration is performed by means of the Crank–
icolson method and a steady-state solution is considered to be
eached when the absolute error between two consecutive time it-
rations is less than 10 −5 . Once the PDF is numerically computed,
he orientation tensor components are straightforwardly obtained.
For large values of Pe , the strong convective character of the
okker–Planck equation [Eq. (10)] may represent a numerical prob-
em. When appropriate and in order to gain insight into the model
redictions, especially at high Pe number flows, we shall also pro-
eed with a particle-based simulation (the acronym PBS, which
tands for particle-based simulation, is used in the following text)
pproach, which consists in following the orientation evolution of
finite number of rods and properly computes averaged prop-
rties [more details can be found in [54,55] . Starting with a set
f random initial orientation for N rods ( N = 10,0 0 0 was found to
e a sufficient number), the ordinary differential Eqs. (5) and (7),
or each particle, are solved and the components of a are out-
2ut as functions of time. Note that the steady-state regimes, when
eached at a deformation of Pe = 100, are assumed to correspond
o the Pe = ∞ limit.
. Stress determination
Prior to proceeding with the derivation of the stress tensor, note
hat in what follows, the forms proposed are only valid for small
eborah number ( De ), where De is the ratio of the intrinsic relax-
tion time of the fluid to the rotational relaxation time of the rod
see Section 5 below).
.1. Giesekus form
The particle-contributed stress for a dilute suspension of slen-
er rods in a SOF may be derived conveniently using the Giesekus
orm [52] . It can be shown to be given by
p =
nζ
4
[a 4 : ˙ γ − 1
2
β
| ̇ γ | (2 ̇ γ : a 6 : ˙ γ − ˙ γ · a 4 : ˙ γ − ˙ γ : a 4 · ˙ γ
)
+ 6 C I | ̇ γ | a 2 ], (12)
here n is the number of rods per unit volume and ζ is the
esistance coefficient (see Section 3.2 below for its expression).
ote that the term proportional to the identity tensor has been
emoved from Eq. (12) to be included in the hydrostatic pres-
ure contribution. Furthermore, the last term in Eq. (12) is ne-
lected as 10 −2 ≤ C I ≤ 10 −4 [20] . In addition, the calculations given
n Appendix B shows that the formula for the extra stress tensor
nd the internal-structure equation related to the elasticity effects
ncountered by a SOF, guarantee their compatibility with thermo-
ynamics [56] .
Based on the results above, the total stress tensor for slender
ods suspended in a SOF can now be expressed as
= −P δ + η0 ˙ γ +
1 , 0
2
[ ( ∇ v ) † · ˙ γ +
˙ γ · ∇ v ]
+ 2 , 0 ˙ γ · ˙ γ
+
nζ
4
[a 4 : ˙ γ+
2 , 0
16 η0 ( 2 ̇
γ : a 6 : ˙ γ− ˙ γ ·a 4 : ˙ γ− ˙ γ : a 4 · ˙ γ )
], (13)
here ∇v † is the velocity gradient tensor [ 33 ], with † denoting
he transpose operation. Note that the expression of the stress ten-
or for slender bodies suspended in a Newtonian fluid is recovered
hen 1, 0 =2, 0 = 0.
For the case of steady shear flow, the particle contribution to
he dimensionless shear stress and dimensionless normal stress
ifferences are directly obtained by expanding Eq. (12) and can be
ritten as
p, ∗12
=
4 τ p 12
nζ | ̇ γ | = 2 sgn ( ˙ γ ) a 1122 − β( 4 a 111222 − a 1112 − a 1222 ) , (14)
66 J. Férec et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 62–72
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i
e
t
i
f
o
c
t
a
a
m
π
−
a
f
τ
t
t
a
t
t
N
p
a
N
p, ∗1
=
4 τ p 11
nζ | ̇ γ | −4 τ p
22
nζ | ̇ γ | = 2 sgn ( ˙ γ ) ( a 1112 − a 1222 )
− 4 β( a 111122 − a 112222 ) , (15)
N
p, ∗2
=
4 τ p 22
nζ | ̇ γ | −4 τ p
33
nζ | ̇ γ | = 2 sgn ( ˙ γ ) ( a 1222 − a 1233 )
− 4 β( a 112222 − a 112233 ) + 2 βa 1122 , (16)
where sgn denotes the sign function. For the sake of clarity, the
asterisk denoting dimensionless variables is omitted. Note that Eqs.
(14) –( 16 ) show that key components of the fourth- and sixth-order
orientation tensors are also needed to evaluate the stress material
functions.
3.2. Resistance coefficient for semiconcentrated suspensions
So far, only dimensionless stress components have been given.
The resistance coefficient ζ , which remained unspecified is now
derived. The slender-body theory [57,58] is used to express the pa-
rameter ζ in the particle-contributed stress contribution for non-
dilute systems [Eq. (12)] . In order to achieve this, the effect of
neighboring particles can be approximated by an equivalent cylin-
drical boundary around the test particle. In this case, the cell
model approach simplifies the problem to a single-particle theory.
A rod is represented by a straight and rigid cylinder of length L and
diameter D , coaxially embedded in a SOF (whose three viscomet-
ric material functions are the viscosity η0 , the first normal stress
coefficient 1, 0 and the second normal stress coefficient 2, 0 ),
and is bounded by a cylindrical cell of radius h and of the same
length L . A local coordinate system { e r , e θ , e z } whose origin is lo-
cated at the rod center is introduced and in which e z is directed
along the rod axis and the radial unit vector e r points into the
fluid domain. In the annular region, the relative velocity of the
fluid with respect to the rod along the cell boundary ( r = h ), w ( s ),
is expected to be undisturbed by the particle and is only a function
of s , where s represents an arc length measured along the rod axis
with s = 0 at the center of the rod. A no-slip condition is applied
at the fiber surface ( r = D /2) and the velocity gradient is assumed
constant along the rod. The flow is presumed to be incompress-
ible, laminar and steady-state and the pressure gradient and body
forces are neglected. For the problem under consideration, we pos-
tulate a velocity field of the form v z ( r ), where the only nonzero
component is in the z -direction. This leading term represents the
zeroth-order approximation of the velocity field for a r 1 [59–62] .
This self-consistent model is usually presumed to capture the es-
sential physics up to moderate concentrations, for which we expect
to find one rod in a volume h 2 L , thus leading to φa 2 r = O ( L 2 / h 2 ) .
Under these assumptions, the resulting flow field is a longitudinal
shearing flow given by the following expression
v z = w ( s ) ln ( 2 r /D )
ln ( 2 h /D ) . (17)
Then, from the Cauchy stress tensor, the total force per unit
length locally exerted by a SOF on the particle surface is given by
d f ( s ) = 2 π
(η0
w ( s )
ln ( 2 h /D ) e z − 2 , 0
w
2 ( s )
R ln
2 ( 2 h /D )
e r
)ds. (18)
This result shows two orthogonal contributions, one from the
shear stress and one from the normal stress. It is now conve-
nient to consider a set of rectangular Cartesian at the rod center
on the coaxial axis. Thus, the first unit base vector p is collinear
to e z and the second one lies in the plane formed by the vectors
p and ˙ γ · p . This vector is related to e r by e r = αu / ̇ γ : pp , where
u = ( δ − pp ) · ( ̇ γ · p ) is normal to p ( u · p = 0 ) . The quantity ˙ γ : pp
s introduced to render the vector u dimensionless and α is used
o normalize e r . At a position s along the rod, the local velocity rel-
tive to the bulk motion is found to be w (s ) = s ̇ γ : pp / 2 . In order
o express the deviatoric stress tensor on the rod surface averaged
ver the particle length, a force balance on a particle segment is
rst performed to derive the tension force. Then, in a given volume
ontaining a large number of rods, the sum of the contribution of
ach particle is obtained by an ensemble average with respect to
he distribution function of p . Thus, the total particle contribution
o the stress tensor obtained is written as
p =
η0 φa 2 r
3 ln ( 2 h /D ) ˙ γ : a 4
+
2 , 0 φa 3 r α
12 ln
2 ( 2 h /D )
(2 ̇ γ : a 6 : ˙ γ − ˙ γ : a 4 · ˙ γ − ˙ γ · a 4 : ˙ γ
), (19)
here the compact notations have been used for the orientation
ensors.
For dilute suspensions, the particles are hydrodynamically in-
ependent and therefore the average lateral spacing between par-
icles, h , is greater than, or of the order L . Thus, the coefficient
n (2 h / D ) in Eq. (19) may be replaced by ln (2 a r ) [59] . We are now
ble to compare the stress expression derived from Leal’s theory
nd our self-consistent model, as both are valid for dilute systems.
omparison of Eqs. (12) and ( 19 ) leads to the desired expressions
or ζ and α, namely ζ = η0 πL 3 / 3 ln ( 2 a r ) and α = ln ( 2 a r ) / 4 a r .
For semiconcentrated suspensions, a first approximation con-
ists in assuming the form ln (2 h / D )/4 a r for α since ln (2 h / D ) is
ore restrictive than ln (2 a r ) but still respects the zeroth-order ap-
roximation for a r 1. The error in the expression for the stress
s expected to be O ( ln ( 2 h /D ) −1 ) . Eq. (19) can be viewed as an
xtension of the Dinh and Armstrong model [7] for semiconcen-
rated rods in a second-order suspending fluid. In their theory, h
s considered to depend on the rod orientation states: h = R √
π/φor perfectly aligned orientation and h = Rπ/ 2 φa r for 3D random
rientation. Continuum theory requires that the coupling coeffi-
ients are allowed to depend on scalar invariants of the orienta-
ion tensors [63] . Therefore, in our development, we follow the
pproach of [64] , which assumes that the average distance from
given rod to its nearest neighbor is linear in terms of the scalar
easure of orientation, that is h = f a 2 h aligned + ( 1 − f a 2 ) h random
for
/ 4 a 2 r < φ < π/ 4 a r and h = h aligned for π /4 a r ≤ φ < π /4, where
f a 2 = 1 − 27 det a 2 [14] .
In summary, the total stress of the composite is given by σ =P δ + τm + τ p , where τm is the second-order fluid contribution
nd the related rod contribution to the extra stress tensor, τp , is
ound to be
p =
η0 φa 2 r
3 ln ( 2 h /D ) ˙ γ : a 4
+
2 , 0 φa 2 r
48 ln ( 2 h /D )
(2 ̇ γ : a 6 : ˙ γ − ˙ γ : a 4 · ˙ γ − ˙ γ · a 4 : ˙ γ
). (20)
For dilute suspensions, the average lateral spacing between par-
icles, h , is replaced by L . The equation describing the rod orienta-
ion in a SOF is given by Eq. (11) , in which the rod-rod interactions
re considered [18] . In SOFs, the convective part of the rod orien-
ation equation was found to be equal for dilute and semiconcen-
rated regimes. Note that this observation was also made by [7] for
ewtonian suspending fluids. Finally, the proposed model is com-
lete provided appropriate closure approximations for a 4 and a 6 re employed.
J. Férec et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 62–72 67
∝
Fig. 2. Effect of the dimensionless second normal stress coefficient ( β) on the nor-
malized steady-state shear stress for the rod contribution, τ p 12
, as functions of Pe .
4
4
s
i
n
c
i
T
i
1
s
t
e
t
r
A
m
r
t
b
c
s
s
I
t
d
(
e
n
s
m
t
m
n
f
n
t
a
a
∝
Fig. 3. Effect of the dimensionless second normal stress coefficient ( β) on the nor-
malized steady-state first normal stress difference for the rod contribution, N p 1
, as
functions of Pe .
β
βββ
Fig. 4. Effect of the dimensionless second normal stress coefficient ( β) on the nor-
malized steady-state second normal stress difference for the rod contribution, N p 2
,
as functions of Pe .
o
N
4
r
(
F
o
fl
θ
T
i
s
θ
p
o
a
r
c
r
. Model prediction for steady-state simple shear flow
.1. Shear stress and normal stress differences
The rod contribution to the dimensionless steady-state shear
tress, τ p 12
, in Newtonian (NF) and second order suspending flu-
ds are compared in Fig. 2 at different Pe . The effect of elasticity is
egligible at small Pe ( < 10 −1 ) and for any values of β , the results
onverge toward the solution for an isotropic state due the strong
nteractions which is given by the analytical solution τ p 12
= 2 / 15 .
he most pronounced effects of the fluid elasticity are observed at
ntermediate and large Pe . In the region of Pe between 10 −1 and
0, τ p 12
increases slightly with elasticity toward a maximum: the
uspension resists the flow and the isotropic orientation distribu-
ion of rods is destroyed. The maxima correspond to peaks in en-
rgy dissipation caused by peculiar configurations of rod orienta-
ion. For NF, this maximum is reached when the largest fraction of
ods is found in close-alignment with the principal axis of strain.
t Pe greater than 10, shear-thinning behavior is observed and is
ore pronounced as β becomes large: the rods orient toward di-
ections where the energy dissipation is the lowest. Notice that for
he same Pe , the stronger the fluid elasticity is, the lower τ p 12
will
e. When Pe tends to infinity, the model predicts no shear stress
ontribution from the rods: particles are aligned in the plane of
hear. In slender body theories, the particle thickness is neglected
o that it is invisible to the flow when it lies in a plane of shear.
n the range of Pe investigated, τ p 12
in a NF does not reach zero but
his plateau is confirmed by [7] for the same constitutive law.
The particle contributions to the dimensionless normal stress
ifferences, N
p 1
and N
p 2
, have also been evaluated according to Eqs.
15) and ( 16 ), and are depicted in Figs. 3 and 4 , respectively. A gen-
ral observation is that with increasing the elasticity ( β), the mag-
itude of N
p 1
is lower than the one found in the NF, and an oppo-
ite behavior is noticed for N
p 2
in the range of Pe analyzed. Further-
ore, N
p 1
and N
p 2
exhibit strong overshoots at moderate Pe . For N
p 1
,
he maxima shift toward lower Pe when increasing β , whereas the
axima for N
p 2
are detected at the same Pe of about 5.5. At high Pe ,
o particle contribution to both dimensionless normal stress dif-
erences is observed. In the case of β = 0, large values of Pe have
ot been reached for N
p 1
to vanish although this has been reported
heoretically by [7] . At low Pe , N
p 1
is also null and a small neg-
tive contribution is predicted for β = 0.5 when Pe is between 1
nd 4.5. As for N
p 2
, its low Pe magnitude is found to be dependent
n β: if an isotropic orientation state is considered, Eq. (16) gives
p 2
= 2 β/ 35 .
.2. Rod orientation state
As mentioned previously, the model can provide information
egarding the microstructure orientation state.
The effect of the dimensionless second normal stress coefficient
β) on the equilibrium rod orientation distribution is shown in
ig. 5 at Pe = 100 and as ψ is symmetric, only half the surface
f the unit sphere is depicted. It appears that the non-Newtonian
uid properties cause a spread in the orientation distribution in
-direction, and a narrower distribution around the shear planes.
he spread in the θ-direction is a result of the elastic effects forc-
ng slender rods to drift towards the vorticity axis. The effect is
imilar to a stretching force on the orientation distribution in the
-direction.
The second-order moment of the orientation distribution, a 2 ,
rovides some efficient information about the microstructure
rientation state. The non-zero steady-state components of a 2 re plotted as functions of Pe , for β = 0, 0.2 and 0.5 in Figs. 6 –8 ,
espectively. An analysis of the second-order orientation tensor
omponent a 11 (where 1 is the flow direction), shows that the
ods suspended in a NF ( β = 0) align in the flow direction with
68 J. Férec et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 62–72
Fig. 5. Steady-state orientation distribution, ψ , for slender rod suspensions in a NF and SOFs at Pe = 100 ( β = 0 on the left; β = 0.2 in the center; and β = 0.5 on the right).
The subscripts 1, 2 and 3 stand for flow, velocity gradient and vorticity directions, respectively.
Fig. 6. Non-zero steady-state components of a 2 as a function of Pe for suspension
in a Newtonian fluid ( β = 0).
Fig. 7. Non-zero steady-state components of a 2 as a function of Pe for suspension
in a SOF ( β = 0.2).
s
d
(
c
i
t
a
r
m
o
(
r
d
a
fi
t
(
t
r
h
l
r
a
a
increasing Pe ( a 11 → 1), and no rods are found in the velocity
gradient and vorticity directions as a 22 and a 33 tends to 0. When
β � = 0, a 11 seems to reach a plateau at large Pe and its value
decreases with increasing β ( a 11 ≈ 0.73 for β = 0.2 as compared
to a 11 ≈ 0.69 for β = 0.5). For both SOF systems, a 22 reaches zero
for the highest Pe values investigated here: this confirms that the
second normal stress pushes rods that lie in the gradient direction
towards the vorticity direction (the plateau values for a 33 increase
with β). The Pe at which we observe the overshoot in the a 12
components are relatively closed to the Pe for the overshoot in
the shear stress ( Fig. 2 ). Hence, this also confirms the previous
statement that the shear stress overshoots are found to take place
for a particular distribution of orientation which sees the largest
fraction of particles aligned close to the axis of maximum rate of
deformation (45 ° from the flow axis).
In order to verify the steady-state solutions in the case of high
Pe flows, we aim to analytically solve Eq. (11) in simple shear flow
assuming quadratic closure approximations for both a 4 and a 6 , and
setting C I = 0 . Note that quadratic closures are known to be ex-
act for fully-aligned rods only. Amongst the seven solutions found,
three of them respect the properties of the second-order orienta-
tion tensor and are given in the Table 2 .
Solution #1 corresponds to uniaxial orientation, where all rods
are aligned in a single direction, namely the flow direction (1-
direction). Solution #2 suggests particular orientation states where
ome rods can be oriented in the velocity gradient direction (2-
irection) under the influence of a second normal stress coefficient
β � = 0) for purely convective flows. Obviously, solution #1 is re-
overed for a Newtonian fluid when β = 0. The last solution (#3)
ndicates that rods lie in the plane formed by the velocity and vor-
icity directions and suggests that particles may eventually orient
long the vorticity axis (3-direction). m 1 must be a positive and
eal number ranging between 0 and 1 in order to respect the nor-
ality condition and m 2 must a real number representing the tilt
f the principal axes of a 2 from the 1-direction in the 1–3 plane
the maximum value that can be assigned to m 2 is 0.5, which cor-
esponds to a rotation of 45 °). However, the results of solution #3
o not allow us to express numerical values for the components of
2 as there exists an infinite combinations m 1 and m 2 .
The analytic solutions derived above do not allow us to con-
rm the results observed for a 2 at large values of Pe , for which
he strong convective character of the Fokker–Planck equation [Eq.
10)] may not be resolved adequately. Therefore, PBS are performed
o obtain the steady-state values for a 11 , a 22 and a 33 which are
eported in Figs. 6 –8 . The limiting values obtained from PBS ex-
ibit a shift as compared to solutions from the Fokker–Planck at
arge Pe , except for β = 0. The Fokker–Planck results appear to
each a plateau, which is probably an artifact as the PDF tends to
Dirac-delta function. Stochastic simulations were also conducted
nd these seem to be able to span the gap between the limiting
J. Férec et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 62–72 69
Table 2
Real analytical solutions of Eq. (11) assuming quadratic closure approximations and C I = 0.
Solution a 11 a 22 a 33 a 12 a 13 a 23
#1 1 0 0 0 0 0
#2 9 r − 4
9 s 1 / 3 − s 1 / 3 +
1
3 1 − a 11 0 −β + 3 βa 22 − 2 βa 2 22 0 0
#3 m 1 0 1- m 1 0 m 2 0
where r =
5 β2 +1
12 β2 ; s =
√
( r − 4 9 )
3 + ( r − 91 216
) 2 − r +
91 216
; 0 ≤ m 1 ≤ 1 and | m 2 | ≤√
m 1 ( 1 − m 1 )
depending how the principal axis of a 2 are oriented.
Fig. 8. Non-zero steady-state components of a 2 as a function of Pe for suspension
in a SOF ( β = 0.5).
Fig. 9. Non-zero steady-state components of a 2 as a function of β for Pe = ∞ ob-
tained with stochastic simulations.
v
g
a
s
a
t
p
a
b
T
a
5
c
t
i
d
s
e
f
t
t
h
b
o
b
a
o
i
fl
c
o
c
i
t
d
t
c
c
d
m
fi
W
b
l
t
W
b
e
l
c
c
s
v
t
I
a
n
t
c
d
alues. However, the stochastic results do not appear to be conver-
ent and therefore we omit them from Figs. 6 –8 .
Fig. 9 reports the non-zero steady-state components of a 2 (i.e.
11 and a 33 ) obtained from PBS as functions of the dimensionless
econd normal stress coefficient, β . By increasing β , more rods
lign in the vorticity direction at the expense of the flow direc-
ion. Moreover, all particles remain lying in the flow-vorticity ( 1–3 )
lane as a 22 = 0, which follows the form of solution #3. Amongst
ll solutions for m 1 , one possibility is to use the expression given
y solution #2 and this hypothesis is only valid for high Pe flows.
he results shown in Fig. 9 are found to be in good qualitative
greement with the data obtained from the PBS.
. Pipkin diagram for rods suspended in viscoelastic media
The Pipkin diagram is a general way to depict the unfilled vis-
oelastic material behavior at various frequency and strain ampli-
ude regimes [65] . In this framework, the Deborah number ( De )
s plotted versus the Weissenberg number ( Wi ): De represents the
egree to which elasticity plays a role in the transient fluid re-
ponse, whereas Wi measures the extent to which nonlinearity is
xhibited in the response. Therefore, Pipkin diagram delineates dif-
erent flow regimes and helps in the choice of relevant constitu-
ive equations [66] . However, a similar framework does not exist
o characterize elongated rod filled materials which can also ex-
ibit both viscous and elastic nonlinearities simultaneously.
Dealy [67] suggested defining the Deborah number as the ratio
etween a characteristic time of the fluid, λm , and the duration of
bservation. Any axisymmetric particle rotates with a period given
y T SB r = 2 πa r / | ̇ γ | for slender bodies [68] , which is used to char-
cterize the duration of observation. This leads to define the Deb-
rah number as De = λm /T SB r . Therefore, three different character-
stic behaviors are to be mentioned: (a) for De � 1, the suspending
uid whose characteristic time (i.e., relaxation time) is very short,
an be assumed Newtonian; (b) for De ≈ 1, the viscoelastic nature
f the material (e.g., through its viscometric properties) has to be
onsidered in the particle stress contribution to the total stress and
n the rods dynamics; and (c) for large Deborah number ( De 1),
he rod microstructure does not have time to evolve with the fluid
eformation: particles orientation dynamics can be omitted and
herefore the whole suspension is treated as a homogeneous vis-
oelastic fluid.
When dealing with the non-linearity of the viscoelastic matri-
es for rod suspensions, the Weissenberg number is also intro-
uced and involves the product of a characteristic rate of defor-
ation, namely | ̇ γ | , and a characteristic time of the fluid, λm , de-
ned by 1, 0 / η0 (or equivalently by 2, 0 / η0 ). Therefore, when
i = λm | ̇ γ | >> 1 , the effects of normal stress differences in the
ehavior of rod dynamics must be considered.
Following the discussion above, we proposed a diagram simi-
ar to Pipkin diagram, but applied to rods suspended in viscoelas-
ic fluids, as shown in Fig. 10 . The vertical axis represents the
eissenberg number and the horizontal axis the Deborah num-
er. Constitutive equations dealing with Newtonian fluids are rel-
vant when De = W i << 1 . For larger De but still in the region of
ow Wi , the viscosity involved in the determination of the parti-
le stress contribution must be time dependent only, as the mi-
rostructure is not affected by the flow. Increasing De further, the
uspension behavior should be treated as that of a homogeneous
iscoelastic fluid with its material function enhanced with the par-
icle volume fraction (the rod microstructure remains unchanged).
n regions where Wi > 1, the effects of normal stress differences
re to be considered in both rod stress contribution and rod dy-
amics and have to be time dependent when De ∼ 1. Noting that
he diagram presented is not drawn to scale, Fig. 10 sums up these
oncepts.
As observed for the rotation of spheres, there are essential
ifferences between a second-order fluid and other models for
70 J. Férec et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 62–72
De1
Wi 1
Newtonian suspensions
Ani
sotro
pic
non-
linea
r ela
stic
solid
s
Vis
com
etric
flow
s(S
tress
tens
or a
nd o
rient
atio
n dy
nam
ics
may
dep
end
on η
0, Ψ
1,0
and Ψ
2,0)
Linear viscoelasticity(η0 is time dependent)
Non-linear viscoelasticity(η0, Ψ1,0 and Ψ2,0 are time dependent)
Fig. 10. Pipkin diagram for rod suspensions delineates relevant forms for constitu-
tive equation.
w
f
A
s
(
p
[
τ
w
s
s
w
b
b
a
b
−
τ
w
n
A
e
τ
r
f
a
G
t
t
t
s
m
viscoelastic fluids [69,70] , and this should be probably true for
the slender rods. What is more, we report different expressions
for slender particles suspended in a second-order fluid although
particle shapes are not similar.
6. Concluding remarks
Constitutive equations have been proposed in order to describe
the rheological behavior of axisymmetric particles suspensions in
viscoelastic fluids as well as their microstructure evolution. The de-
velopment is based on Leal’s theory [45] who studied the motion
of a slender rod in a second-order fluid. Next, the orientation dis-
tribution of particles in such systems is obtained from numerical
solutions of the generalized Fokker–Planck equation, in which par-
ticle angular velocities are provided by the Leal model. From these
results and with the introduction of orientation tensors, an equa-
tion of change for the rod orientation as well as the particle contri-
bution to the total stress tensor is derived. The coefficients describ-
ing the resistance on the rod and present in the latter are deter-
mined from a cell model, which are valid for dilute and semidilute
rod suspensions.
At low Pe , the strong interaction effects lead to a near isotropic
state and the weak viscoelastic effects are dominated by the strong
diffusion in orientation space. The most pronounced effects aris-
ing from the fluid viscoelasticity are observed at intermediate and
large Pe . As for a single rod, the second normal stress difference
tends to push a rod population which lies in the gradient direction
toward the vorticity axis. This drift is accompanied by an induced
spread of the rods in the flow-vorticity plane. Moreover, the degree
of rods alignment in the vorticity direction is found to be directly
related to the magnitude of the second normal stress difference.
The condition that only a non-zero second normal stress differ-
ence induces the drift of a slender rod is relaxed by Brunn [46,47] ,
who extended the Leal analysis to dumbbells and tri-dumbbells for
which the particle thickness is considered. Finally, in order to fa-
cilitate the choice of the relevant terms required in a constitutive
model, we propose an adaptation of the original Pipkin diagram to
the case of rod suspensions in viscoelastic media.
Acknowledgments
This work has been performed while J. Férec was on sabbat-
ical leave at the National University of Singapore (NUS), as visit-
ing Professor at the Department of Mechanical Engineering. J. Férec
ishes to thank his hosts Prof. N. Phan-Thien and Prof. B.C. Khoo
or their kind hospitality and a very stimulating environment.
ppendix A. Derivation of the stress tensor
In this section, we report for completeness the derivation of the
tress tensor using the Brunn’s model for rigid tri-dumbbells [Eq.
8)]. The contribution to the stress tensor due to the presence of
articles in a SOF is defined according to the Giesekus expression
52] as follows
= −nζ
4
D a 2 Dt
, (A1)
here D / D t is the upper convective derivative and a 2 is the
econd-order orientation tensor [5] . The time evolution of a 2 is
imply calculated as follows
D a 2 Dt
= 〈 ̇ p p 〉 + 〈 p ̇ p 〉 , (A2)
here D / Dt represents the material derivative and the angular
rackets denote the ensemble average with respect to the distri-
ution function. Substituting Eq. (8) in the previous equation and
fter some calculations, we obtain
D a 2 Dt
= −1
2
( ω · a 2 − a 2 · ω ) +
λ
2
(˙ γ · a 2 + a 2 · ˙ γ − 2 ̇ γ : a 4
)− β
2 | ̇ γ | (
˙ γ : a 4 · ˙ γ + ˙ γ · a 4 : ˙ γ − 2 ̇ γ : a 6 : ˙ γ)
− χ
2 | ̇ γ | (
˙ γ2 · a 2 + a 2 · ˙ γ2 − 2 ̇ γ2 : a 4 ). (A3)
Hence, the contribution to the stress tensor due to the particles
ecomes
4 τ
nζ=
λ − 1
2
(˙ γ · a 2 + a 2 · ˙ γ
)− λ ˙ γ : a 4
− β
2 | ̇ γ | (
˙ γ : a 4 · ˙ γ + ˙ γ · a 4 : ˙ γ − 2 ̇ γ : a 6 : ˙ γ)
− χ
2 | ̇ γ | (
˙ γ2 · a 2 + a 2 · ˙ γ2 − 2 ̇ γ2 : a 4 ). (A4)
For slender bodies ( a r → ∞ ), Eq. (A4) reduces to
=
nζ
4
[˙ γ : a 4 − β
2 | ̇ γ | (2 ̇ γ : a 6 : ˙ γ − ˙ γ : a 4 · ˙ γ − ˙ γ · a 4 : ˙ γ
)],
(A5)
here β = ( 1 , 0 − 2 2 , 0 ) | ̇ γ | / 4 η0 . This expression shows that both
ormal stress coefficients play a role even for slender bodies.
ppendix B. Compatibility verification with thermodynamics
The general expression for the particle contribution, τ p , to the
xtra stress tensor in a SOF can be split in two parts, τ p = τ p NF
+p SOF
, where τ p NF
and τ p SOF
are the Newtonian and SOF contributions,
espectively. We will focus our argumentation on the derivation
or the SOF contribution in term of thermodynamic considerations,
s the one for the Newtonian contribution is deeply discussed in
rmela et al. [56] .
First, the internal microstructure is chosen to be described by
he dyadic of product of p , where p is a unit vector directed along
he main axis of the rod. Following [71–73] , the volume conserva-
ive form of the time evolution equation of the conformation ten-
or, pp , and the stress tensor, τ p SOF
, according to the GENERIC for-
ulation can be presented in the following forms
D pp
Dt
∣∣∣SOF
= −�pp · ∂A
∂ pp
, (B1)
J. Férec et al. / Journal of Non-Newtonian Fluid Mechanics 239 (2017) 62–72 71
τ
w
a
a
e
A
w
d
e
w
n
s
i
t
w
γ
e
w
τ
t
i
t⟨w
s
t
t
c
o
i
ρ
t
e
w
w
�
a
o
v
i
i
E
R
[
[
[
[
[
[
[
[
p SOF
= −2 pp · ∂A
∂ pp
, (B2)
here A is the Helmholtz free energy function and � is known
s the mobility. The model is completed when the expression of A
nd � are given.
Now, we proceed to the specification of the Helmholtz free en-
rgy function. We consider the following expression
=
ρ
2
(˙ γ : pp
)2 , (B3)
here ρ is a model parameter. It remains to calculate the Volterra
erivative. After a long but straightforward calculation, the differ-
ntial operator is found to be
∂ ( pp )
∂ pp
= spsp + pssp + spps + psps + pttp + ptpt + tptp + tppt ,
(B4)
here p, s and t are the unit vectors in the spherical coordi-
ate system [52] . When the following property is used, δ − pp =s + uu , this can be rearranged to
∂ ( pp )
∂ pp
= p δp +
t (p δp
)+
(p δp
)t +
t (p δp
)t − 4 pppp , (B5)
n which the exponent t preceding/following a tensor indicates
ransposition of the first/last two indices. The double dot product
ith the strain rate tensor, ˙ γ , leads to
˙ : ∂ ( pp )
∂ pp
= 2 ̇ γ · pp + 2 pp · ˙ γ − 4 ̇ γ : pppp . (B6)
We are now able to express the Volterra derivative of the free
nergy function. With the help of the chain rule of differentiation,
e obtain
∂A
∂ pp
= ρ(
˙ γ : pp
)˙ γ :
∂ ( pp )
∂ pp
= 2 ρ(
˙ γ · pppp : ˙ γ+ ̇ γ : pppp · ˙ γ − 2 ̇ γ : pppppp : ˙ γ). (B7)
Hence, the stress tensor becomes
p SOF
= −2 pp · ∂A
∂ pp
= 4 ρ(− ˙ γ · pppp : ˙ γ − ˙ γ : pppp · ˙ γ + 2 ̇ γ : pppppp : ˙ γ
). (B8)
We may consider Eq. (B8) to be the rod contribution to
he stress in a fluid which consists of a large number of non-
nteracting particles in a SOF at each material point. Such an in-
erpretation permits us to rewrite Eq. (B8) as follows
τ p SOF
⟩= 4 ρn
(− ˙ γ · a 4 : ˙ γ − ˙ γ : a 4 · ˙ γ + 2 ̇ γ : a 6 : ˙ γ
), (B9)
here the angular brackets denote the ensemble average with re-
pect to the distribution function and n is the rod number concen-
ration. Eq. (B9) represents the particle stress contribution due to
he elasticity properties encountered in the SOF. Consequently, by
omparing the previous expression [Eq. (B9)] with the second term
f Eq. (12) , it is found to be the same when the model parameter
s
=
ζ
32
β
| ̇ γ | . (B10)
Definitions for ζ , β and | ̇ γ | are given in the main text. As for
he internal-structure equation, Eq. (B1) results in, when an the
nsemble average is performed
D a 2 Dt
∣∣∣SOF
= −�2 ρ(
˙ γ · a 4 : ˙ γ + ˙ γ : a 4 · ˙ γ − 2 ̇ γ : a 6 : ˙ γ), (B11)
hich is also exactly the 2nd term of Eq. (11) in the manuscript
hen
=
1
4 ρ
β
| ̇ γ | =
8
ζ. (B12)
This shows that the free energy function given in Eq. (B3) is
n appropriate potential when considering the elasticity effects
n slender rods suspended in a SOF. These investigations in-
olve that the same formulas have been derived from mechan-
cs and thermodynamic considerations, therefore their compatibil-
ty is guaranteed. As for the Newtonian contributions (1st terms in
qs. (11) and ( 12 )), we refer the reader to Grmela et al. [56] .
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