Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 4.7.1 - Analysis of rotational diffusion of non-Brownian rigid fibers in shear flow
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00115
 Material Information
Title: 4.7.1 - Analysis of rotational diffusion of non-Brownian rigid fibers in shear flow Colloidal and Suspension Dynamics
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Salahuddin, A.
Wu, J.
Aidun, C.K.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: lattice-Boltzmann method
rotational diffusion
rigid and flexible fiber suspension
 Notes
Abstract: The recent developed particle level simulation method (Wu & Aidun (2010a,b)) is implemented in this study to investigate the orientation diffusion of rigid and the orientation and rheology of flexible rod-like fibers in semidilute suspension of creeping shear flow, with a Newtonian fluid medium. For rigid fibers, an anisotropic, weak rotary diffusion model proposed by Rahnama et al. (1995) is tested to evaluate the rotational diffusivity ratio, R = D D , from a one parameter fit of the model to the simulated orbit-constant Cb-distribution in semidilute regime. The diffusivity ratios from simulation data compare very well with Stover et al. (1992)’s experiments and Rahnama et al. (1995)’s ‘hydrodynamic interaction theory’. But, the simple anisotropic weak diffusion model can not describe the violation of Stokes flow symmetry which is triggered by the presence of a small but detectable amount of non-hydrodynamic (mechanical) fiber-fiber interaction in creeping flow in semidilute regime and thus emphasizes the need for a more sophisticated model to predict orientation diffusion. For flexible fibers, the decrease of fiber stiffness (bending ratio BR) is shown to cause increasing asymmetry in -distribution (here, is the meridian angle in the flow-gradient plane). The impact of this asymmetry on first normal stress difference, NB 1 is discussed in the light of Batchelor (1971)’s theory.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Resource Identifier: 471-Salahuddin-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Analyses of rotational diffusion of rigid fibers and orientation effect on rheology of
flexible fibers in shear flow with direct numerical simulation.


Asif Salahuddin and Jingshu Wu and C.K. Aidun

Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30318, USA
gth823e@mail.gatech.edu and wujingshu@gatech.edu and cyrus.aidun@me.gatech.edu
Keywords: lattice-Boltzmann method, rotational diffusion, rigid and flexible fiber suspension




Abstract

The recent developed particle level simulation method (Wu & Aidun (2010a,b)) is implemented in this study to
investigate the orientation diffusion of rigid and the orientation and rheology of flexible rod-like fibers in semidilute
suspension of creeping shear flow, with a Newtonian fluid medium. For rigid fibers, an anisotropic, weak rotary
diffusion model proposed by Rahnama et al. (1995) is tested to evaluate the rotational diffusivity ratio, R (-) ,
from a one parameter fit of the model to the simulated orbit-constant Cb-distribution in semidilute regime. The
diffusivity ratios from simulation data compare very well with Stover et al. (1992)'s experiments and Rahnama et
al. (1995)'s hydrodynamicc interaction theory'. But, the simple anisotropic weak diffusion model can not describe
the violation of Stokes flow symmetry which is triggered by the presence of a small but detectable amount of
non-hydrodynamic (mechanical) fiber-fiber interaction in creeping flow in semidilute regime and thus emphasizes the
need for a more sophisticated model to predict orientation diffusion. For flexible fibers, the decrease of fiber stiffness
(bending ratio BR) is shown to cause increasing asymmetry in p-distribution (here, p is the meridian angle in the
flow-gradient plane). The impact of this asymmetry on first normal stress difference, NB is discussed in the light of
Batchelor (1971)'s theory.


Introduction


Fiber Orientation in Suspension


The understanding of orientation distribution and mi-
crostructure of rigid and flexible fiber suspensions are
very important in paper-making and composite produc-
tion. The well-known Jeffery (1922)'s solution of the
motion of a single ellipsoid in suspension completely ne-
glected effects such as fluid and particle inertia, Brown-
ian motion and particle-particle interactions, all of which
may be present in industrially important semidilute to
concentrated suspension systems. By virtue of their abil-
ity to simulate these effects in suspensions, particle-level
simulations are useful and serve as complements to the-
oretical and experimental approaches in research.

In the past, some notable investigations on diffusiv-
ity of suspensions have mostly been performed (An-
czurowski & Mason (1967a,b)) at concentrations lower
than aimed to be studied in this research. Also, the orien-
tation diffusivity was considered to be isotropic (Folgar
& Tucker (1984)).


Jeffery (1922) showed that in simple shear flow, a single
fiber rotates indefinitely about the vorticity axis along
one of an infinite number of periodic, closed orbits.
For almost any rigid body of revolution including cir-
cular cylinders, the orbit period (T) will be: T
27 (r + 1/re) /j, where j is the shear rate and a semi-
empirical relation (Cox (1971)) replaces fiber aspect-
ratio rp L/d with an effective aspect-ratio, re
1.24r (lnrp)1/2.
The spherical co-ordinate system used to describe
fiber orientation is defined in Fig. 1. The integration of
time evolution for 0 and p yields:


tan 6 2 1/
(r6 cos2 + sin2 9)1/

tan = re tan (27 + K)
(T


where K is the phase angle. The orbits of the ends of the
fiber are a symmetrical pair of spherical ellipses defined







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Voricity


Flow


Figure 1: The co-ordinate system with a fiber centered
at the origin. The unit vector, p is parallel to the fiber's
axis. Typical orbits for a slightly prolate spheroid (fiber)
are shown with solid blue lines.


C = tan00 C os2 0o + r sin2 yo
v r2


(3)


where, C is the orbit constant and subscript '0' denotes
the initial orientation. The value C 0 corresponds
to perfect alignment of the fiber with the vorticity di-
rection of the flow, whereas C = o corresponds to
rotation in the flow gradient (x-y) plane. It is advan-
tageous to characterize the orbits with an orbit constant,
SC/(1 + C) [0, 1].
Fiber suspensions are broadly divided into three
regimes: in the dilute regime, nL3 << 1, the semidi-
lute regime is defined by nL3 >> 1 but nL2d << 1.
And finally, in concentrated regime c, > d/L, (c
n7dd2L/4 for cylindrical fibers). Here, n is the fiber
number density, d is the fiber diameter and L is the fiber
length. The fiber bending ratio (BR) is a measure of
nondimensional stiffness of the flexible fiber defined by
Forgacs & Mason (1959) as:

Ey (In 2r, 1.5)
BR = (4)
2(/,)r

Here, Ey is the fiber's Young's modulus. It is to be
noted that, the fibers used in this study are non-colloidal
with P6clet number, Pe oc, so the effect of Brown-
ian motion is negligible. The particle Reynolds number,
Re = L2 /v, is very small (the fibers are on the order
of microns) and therefore, the fiber inertia is also negli-
gible.

Rotary Diffusivity

As a simplification to the detailed accounting of fiber-
fiber interaction for rod-like fibers, the small changes in
orientation of one fiber because of the presence of it's


neighbors can be accounted for with a rotary diffusion.
Folgar & Tucker (1984) introduced a phenomenological
model which describes the effects of fiber interactions
in terms of an isotropic, rotary diffusivity, D,. Leal
& Hinch (1971) tried to develop a solution for steady
state C-distribution by accounting for a weak scalar
rotary diffusivity but failed to reproduce Anczurowski
& Mason (1967b)'s experimental C-distribution. Rah-
nama et al. (1995) generalized Leal & Hinch (1971)'s
solution of the Burgers evolution equation by including
the orientation dependent anisotropic diffusivity tensor,
D,. So, the differential probability distribution function
2 (0, y, t) can be stated as:

Q +V, (pJ ) -Vp.(D, VQ) -V (Q2ph) (5)

where, Vp is the gradient operator in orientation space,
pij is the rotation rate of the fiber as described by Jef-
fery (1922), and ph is the drift velocity. The original
eqn. (5) was re-examined to consider the consequences
of the action of a small amount of weak anisotropic dif-
fusion in the limit as time, t oc and drift velocity
was neglected. In the (C, T) orbit co-ordinates (see Leal
& Hinch (1971) for this co-ordinate definition), where
T is the phase angle, the orientation distribution can be
separated into two parts:
2 (C, ) f (C) (C, T) (6)
In this expression f (C) is the unknown distribution
function describing the population amongst the Jeffery
orbits; g (C, T) is the distribution around an individual
orbit, C. In the large aspect-ratio limit, the integral
expression for f (C) was analytically solved to yield:
C) 4RC where R=( D) Do is the
7r(4RC2+1)3/2 D "
proportionality between gradients of 2 (0, y) in the 0-
direction and the flux of probability in the 0-direction.
A large value of R implies that the fibers are aligned
near the vorticity axis, as the effect of Doo is to push
fibers away from the flow direction and toward vortic-
ity or toward decreasing values of orbit constant. The
f (C) can be transformed to give the differential proba-
bility distribution function, p (Cb) for steady state orbit
constant distribution as:


p(Cb)


4RCb
(4R l Cb)]2 1 3/2
,4R [Cb/( C)+ (1


Ob)


(7)
where fo p (Cb) dCb = 1. Eqn. (7) would be referred to
as the 'Anisotropic Diffusivity Model'.

LBM-EBF Method

In this study, we use a novel particle-level numerical
simulation method (Wu & Aidun (2010a,b)) to analyze











suspension of both rigid and flexible fibers. The im-
portant aspects of the implemented LBM-EBF method
are: simulations include fiber-fiber contact and lubrica-
tion forces, the fluid and fibers are two-way coupled with
direct numerical simulation and the physical properties
of flexible fibers are directly related to simulation pa-
rameters. In this method, the no-slip boundary condition
at the fluid-solid interface is based on external boundary
force (EBF) method. The details of this method are pre-
sented in previous papers (Wu & Aidun (2010a,b)).

Results and Discussion

Validation of the lattice-Boltzmann approach with the
EBF method for fiber suspension is presented in previ-
ous publications (Wu & Aidun (2010a,b)). The focus
of this paper is to investigate the anisotropic diffusiv-
ity model Eqn. (7) for large aspect-ratio rigid fibers in
semidilute regime. The present report would also dis-
cuss the effect of fiber stiffness on rheology and orienta-
tion of flexible fiber suspension to some extent.
An unbounded shear domain is implemented based
on the Lees-Edwards boundary condition (LEbc) (Lees
& Edwards (1972)) to improve computational efficiency
by removing wall effects. The computational domain is
5L x 5L x 4L and the suspending fibers have diameter
of d = 0.4 LBM unit lattice size.

Orientation Diffusivity of Rigid Fibers in
Suspension

The Cb at each timestep is divided into n-no. of bins,
n
where ZdCb(i) 1.0. A MATLAB function dis-
i= 1
tributes the fibers into i no. of bins based upon their
corresponding Cb(i) values. Then, the probability of
finding a specific fiber in i-th bin is calculated as:
no. of fibers in Cb() ri / \ i
P Cb) ttal no of fibers The p ( b) is normal-
ized with the integral area under the histogram. Sim-
ilarly, the probability distribution, p (y), is calculated,
n
where, Jdo(i) 7r.
i= 1
Stover et al. (1992) experimentally measured rigid
fiber orientation in a cylindrical couette device using
an isorefractive suspension of matched densities, with
few observable opaque test fibers. They found that the
semidilute Cb-distributions were more uniformly dis-
tributed than the dilute Cb-distributions of the exper-
iments conducted by Anczurowski & Mason (1967b).
Anczurowski and Mason's experiments (in cylindri-
cal counter-rotating couette device) in infinitely dilute
regime heavily favored lower orbit constants. Accord-
ing to Stover et al. (1992), the distribution of Cb in the


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


nL3 Stover's Hydrodyn. LBM-EBF
experiment Interaction simulation
Theory
rp 31.9 rp 31.9 r, 32.0
45 3.16 2.55 2.58
18 1.38 3.12 3.31
10 2.85 3.50 2.86
5 2.18 3.50 2.45

Table 1: Comparison of Diffusivity-Ratio, R (")


steady state has a peak in the range of 0 < Cb < 0.5
for both dilute and concentrated systems. Yamane et al.
(1994)'s numerical simulations (with only short-range
hydrodynamic interaction) and Fan et al. (1998)'s nu-
merical simulations (with both the short and long-range
hydrodynamic interactions) showed ili.ii sc.idv state Cb-
distribution shifted towards higher values (0.42 ~ C ~
1.12 or 0.296 ~ Cb ~ 0.528) than observed in Stover
et al. (1992)'s experiments. The reason of this discrep-
ancy is probably due to statistical errors in Fan et al.
(1998)'s numerical simulation induced by the choice of
small-sized domain (0.5 < 1 < 1.0, where, 12 is the do-
main height in gradient direction) to save computational
cost which rules out the possibility of fiber orientation
perpendicular to the (x-y) plane.
In this study, a set of LBM-EBF simulations are
performed at non-dimensional volume concentrations,
nL3= 5, 10, 18 and 45 with rigid fibers of rp=32.0. The
Cb-distributions with the LBM-EBF simulations for this
nL3 range in semidilute regime demonstrate good pre-
dictions of the Stover et al. (1992) experiments as can
be noted in Fig. 2. The Cb-distributions in Fig. 2 for
LBM-EBF are smoothed over the last two orbit periods
simulated. The peaks of the Cb-distributions remain in
the range 0.15 < Cb < 0.4 for LBM-EBF which is con-
sistent with the report of Stover et al. (1992) experiments
in semidilute regime.
To determine the diffusivity ratio R=- ) in
semidilute regime, Stover et al. (1992) fitted the
'anisotropic diffusivity model' (eqn. (7)) to their ex-
perimental Cb-distribution with R as an adjustable pa-
rameter. They also fitted the 'anisotropic diffusivity
model' to Anczurowski & Mason (1967b)'s dilute ex-
perimental Cb-distribution. The fibers with rp=18.4
for Anczurowski & Mason (1967b)'s experiment show
a plateau at about ( ) =17 for the apparent dilute
regime (nL3 =0.016, 0.066). But, a steep drop to another
plateau, at about ( 2,,) =1.5 was observed for Stover et
al. (1992)'s semidilute regime experiments.
Rahnama et al. (1995) theoretically studied the ori-








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


S 01 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Cb

SLBM-EBF simulation
Best fitted 'anisoopic diffusivity
2 del'to BM-EBPF imlim dta
x with R=2.86
x Stover et. al. (1992) experiment

x r =32.0


'-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cb
25
x vert. al. (1992) experiment
SLBM-EBF imulation
Best fitned 'aniso*tre cdifffaivity







0C 01 0.2 03 04 05 06 0.7 0.8 0.9
2 mdel'tIBM-EBF im d&ta






1 .5.0


S 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x SW .a. (1992) experen t
*LBM-EBF simulation
2 -Best fitted *amtsopop diffu-vity
mdel'tomM-F imlaim dita


IBrM=32.0
X



0 0.1 02 0.3 0.4 Cb0.5 0.6 0.7 0.8 0.9


Figure 2: p(Cb) with rp=32.0 for nL3=5, 10, 18 and 45
from top to bottom sub-figure respectively. The exper-
imental Cb-distributions are within 80% confidence in-
terval. The solid curves demonstrate the one-parameter
fit (with parameter R) of 'anisotropic diffusivity model'
to LBM-EBF simulation data. The best-fitted R values
are indicated in corresponding legends.


x Stovetal. (1992) expiment
* LBM-EBF simulation
Best fitted 'anisotropic diffusivity
mdel'tolBM-EBF aimdim ta
tth tR2. 4

=32.O


entation diffusivity in dilute to semidilute regime with
hydrodynamicc interaction theory'. In this theory hy-
drodynamic, orientation diffusivity was obtained from
an ensemble average of the fiber-fiber interactions. The
steady-state fiber orientation distribution is controlled
by the anisotropy and orientation dependence of the
diffusivity. The steady state and transient fiber ori-
entation distributions are derived using a perturbation
analysis for weak hydrodynamic orientation diffusion.
For computational convenience, Rahnama et al. fitted
the 'anisotropic diffusivity model' to the Cb-distribution
calculated with hydrodynamicc interaction theory' and
obtained the best-fit value of R. Rahnama et al. used
an iterative solution with an initial guess of R (averaged
R=2.4 from Stover et al.'s experiments for rp=31.9 was
used as an initial guess) to calculate the Cb-distribution
with hydrodynamicc interaction theory'. We also fitted
the 'anisotropic diffusivity model' to the Cb-distribution
from LBM-EBF simulation and predicted the best-fit R
values in semidilute regime. A nonlinear curve fitting
method is used for this purpose.

Table 1 summarizes R=() D values by Stover et
al. (1992) experiments, hydrodynamicc interaction the-
ory' and LBM-EBF simulations. The R values in the
semidilute regime from nL3=1-45 fall in the same range
for theory and experiment. The quantitative compari-
son between LBM-EBF simulation and experiments are
good, considering statistical uncertainties in the experi-
ment. However, Stover et al. (1992)'s experimental val-
ues of R do not show any systematic dependence on
the volume concentration, nL3 and aspect ratio, rp.
Whereas, the hydrodynamicc interaction theory' reveals
a dependence of diffusivity ratio, R, on nL3 and as-
pect ratio, rp: for a fixed aspect ratio (rp=31.9), the
value of R decreases with increasing nL3. Physically,
this means that the fibers shift closer toward the flow di-
rection with increasing nL3. According to Rahnama et
al. (1995), this shift resulted from the anisotropic hy-
drodynamic screening incorporated in the renormalized
Green's function derived by Shaqfeh and Fredrickson
(1990). The LBM-EBF predicted R values first increase
slightly from nL3=5-18 and then it decreases with in-
creasing concentration following the theoretical predic-
tion. The physical reason behind this can be explained
by studying change of (Cb) values with concentration.

From Fig. 3 it is seen that, the (Cb) for Stover et
al. (1992) experiments and LBM-EBF simulations de-
crease with increase of nL2d (the parameter nL2d is
used since, it is a more relevant measure of concen-
tration in semidilute regime). Physically this means
the fibers are shifting towards the vorticity axis and
as a result R value also increases (diffusion occurring
primarily in flow-vorticity plane) with concentration.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


C Stovr t al epetnment (1992), r31 9
LBM EBF smulation, r =32 0
Hydrodynac minteraction theory


Figure 3: Average value of orbit constant, (Cb) as a
function of nL2 d. The prediction of hydrodynamicc in-
teraction theory' is indicated with a solid line.



But, as the concentration increases more (nL3>18 or
nL2d > 0.56253)), the fiber-fiber hydrodynamic inter-
actions dominate and also there can be presence of some
mechanical contacts among fibers. Both of these effects
cause extra fiber flipping and as a result the fibers move
closer to the flow direction ((Cb) increases) and the dif-
fusivity ratio R decreases (i.e. diffusion occurring pri-
marily in the flow-gradient plane). So, the (Cb) values
of Stover et al. (1992)'s experiment, hydrodynamicc in-
teraction theory' and LBM-EBF simulation fall close to
each other for higher values of concentration, nL3 45
(nL2d z 1.41). Now, the apparent discrepancy be-
tween theory and experiment at low concentrations can
be due to the lower accuracy of renormalized Green's
function at low concentrations (which is used to account
for hydrodynamic screening in hydrodynamicc interac-
tion theory'). Actually the function shows better accu-
racy at higher values of nL3 (Shaqfeh and Fredrickson
(1990)). Perhaps at lower concentration, a two-fiber the-
ory would work better. On the flip-side, through a pri-
vate communication with Dr. Koch, the authors came
to know that during Stover et al. (1992)'s experiments,
the group of researchers did not to pay too much atten-
tion to these trends of diffusivity ratio, R not be-
ing too sure if those were statistically ,igiilik.ii and
rather paid more attention just to the overall order of
magnitude of diffusivity ratio R. So, we can strongly
claim that, LBM-EBF simulations produced excellent
results to verify the range of R in semidilute regime
and also shows encouraging results to physically ex-
plain the trend of R in semidilute suspension of non-
colloidal rigid fibers. The LBM-EBF simulation has the
algorithm to count the number of mechanical contacts
present in the suspension during a flow situation. We
observed a small amount of mechanical contacts in the
semidilute regime which breaks the Stokes flow symme-
try (results not shown here for conciseness). This phe-
nomenon was also observed in Stover et al. (1992) and
Petrich et al. (2000)'s semidilute regime experiments


r,=16
c,=.o053
nL=17.3


-BR=2940
- -BR=1.47
-*BR=0.15


0.2 0.4 0.6 0.8
~In


0 0.2 0.4 0.6 0.8
tIu


Figure 4: The p-distribution for flexible fiber suspen-
sions with aspect ratio, r = 16 (top) and r = 32 (bot-
tom) with different bending ratios, BR, and for a fixed
volume fraction, c, = 0.053 with LBM-EBF simula-
tion.


where the p-distribution showed asymmetry across the
flow-vorticity plane due to non-hydrodynamic mechan-
ical contacts among fibers and thus violated the Stokes
flow symmetry. This phenomenon can not be explained
by the weak diffusion considered in 'anisotropic diffu-
sivity model' (Eqn. (7)) used to predict diffusivity ra-
tio, R, since weak diffusion should preserve the symme-
try of Stokes flow. This indicates that a complete de-
scription of hydrodynamic interactions is more complex
and sophisticated than the simple 'anisotropic diffusiv-
ity model' used to fit the orbit-constant distribution. Al-
though the model was very good fit to the p (Cb) it can
not describe the asymmetry in p ( ).


Rheology and Orientation of Flexible Fibers in
Suspension

In this section, the change in fiber theological properties
with change of fiber flexibility is explained by study-
ing the p-distribution, p (4) of the fiber suspension for
different aspect-ratios, rp and bending ratio BR and







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


those information are theoretically related to Batchelor
(1971)'s work.
Batchelor (1971)'s relation for the contribution of sus-
pended fibers in dilute suspension without any Brownian
motion, yields the first normal stress difference, N1 as:


N B = jfiberf (KPI. )


fiberr (sin 4 sin 4 ))
*--(<(si Ism14p))


Where, the superscript 'B' refers to Batchelor's the-
ory and fiber is a function of concentration, orienta-
tion distribution and fiber geometry. Eqn. (8) shows
that if the suspension has no direct physical contacts
between fibers and in the absence of fiber-deformation
(rigid fiber), the fiber orientation dispersion, p (p) would
necessarily be symmetric about (x-z) plane. And in these
cases, first normal stress difference, Nf given by Batch-
elor's theory would be zero, since it is an odd function of
/. So, if direct contact between fibers exists or if fibers
are deformable, Nf will not vanish.
The p-distribution in Fig. 4 with LBM-EBF simula-
tion clearly gives evidence that for decreasing bending
ratio, BR (i.e. by increasing fiber flexibility), the mean
orientation angle (p) will become slightly less than 7r/2;
and this small asymmetry of the fiber orientation distri-
bution makes Nf non-zero (+ve). So, by reviewing the
Batchelor's theory above and by giving computational
evidence of asymmetry in p-distribution due to change
of BR, here it is proved that the first normal stress differ-
ence is strongly dependent on the fiber flexibility. On the
other hand, from Eqn. (8), it is clear that the change of
0 is also very important for the first normal stress differ-
ence. NA increases with 0, where 0 is directly related to
the orbit constant Cb (Eqn. (1)). But, the concept of orbit
constant is not well-defined for flexible fibers (Skjetne
et al. (1997)), since the geometry of the fiber vary with
time. For flexible fibers Jeffery's orbit become unstable
and C will tend to drift (for the most part) either to 0
or oo as evidenced in experiments (Arlov et al. (1958))
and numerical simulations (Skjetne et al. (1997)), de-
pending on the initial C value, while intermediate values
also being observed. To the authors' knowledge, no ro-
tational diffusion model exists which accounts for these
disturbed Jeffery's orbits for flexible fibers.
It is worth to point out here that, the effect of vol-
ume fraction on relative shear viscosity of suspension
p,,e of flexible fibers has been discussed in previous Wu
& Aidun (2010a) paper with LBM-EBF simulations. It
was shown that with increase of volume fraction the rel-
ative shear viscosity, pre increases which can be at-
tributed to the increased fiber-fiber interaction due to
fiber crowding and fiber deformation with shear strain.


Conclusion


(. I /*' ))


The steady-state distribution of orbit-constant, p (Cb), of
rigid cylindrical fibers suspended in a Newtonian fluid
subjected to simple shear flow has been predicted for
several concentrations (nL3=5-45) in semidilute regime
with LBM-EBF simulation. The 'anisotropic diffusivity
model' is fitted to the simulated p (Cb) which gives best-
fit diffusivity ratios (R values) in the range of 2.45 ~
R ~ 3.31 and thus verifies the range of R observed in
experiment and theory. But, the small amount of asym-
metry in p (4) as was observed in simulation and experi-
ments questions the simplicity of 'anisotropic diffusivity
model' and emphasizes the need for the development of
a more sophisticated model. Also, in future, it would
be interesting to numerically investigate the time corre-
lation function of Cb(t), defined as (Cb(t)Cb(t T)),
where, T is the delay time, t is the time measured dur-
ing the simulation, and the angle brackets denote aver-
ages over all available values. By analyzing the decay
of (Cb(t)Cb(t T)) as a function of T, the temporal
stochastic fluctuations of Cb(t) can be explored and it
would be a good test of the magnitude of hydrodynamic
rotary diffusivity.
The LBM-EBF simulation with flexible fibers shows
fiber stiffness has strong impact on suspension rheology.
Also, the effect of fiber stiffness on the first normal stress
difference, NB is demonstrated based on analyzing the
p-distribution and Batchelor (1971)'s theory. It is proved
that the influence of fiber orientation due to change in
fiber stiffness is a major contributor to the variation in
theological properties.


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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Wu, J. and Aidun, C.K., A method for direct simulation
of flexible fiber suspensions using lattice-Boltzmann
equation with external boundary force, International
Journal of Multiphase Flow, Vol. 36, pp. 202-209, 2010a

Wu, J. and Aidun, C.K., Simulating 3D deformable par-
ticle suspensions using lattice Boltzmann method with
discrete external boundary force, International Journal
for Numerical Methods in Fluids, Vol. 62, pp. 765-783,
2010b

Yamane, Y. and Kaneda, Y. and Dio, M., Numerical sim-
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