7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Terminal Velocity of Particles Falling in NonNewtonian Yield Pseudoplastic Fluids using
a Macroscopic Particle Model
Divyamaan Wadnerkar1, Madhusuden Agrawal2 and Vishnu Pareek1
1Department of Chemical Engineering, Curtin University of Technology, WA
2ANSYS Inc, Houston, TX
divyamaan.wadnerkar@student.curtin.edu.au
Keywords: Macroscopic Particle Model, NonNewtonian Fluids, Terminal Velocity, CFD, Drag Force, Direct Numerical
Simulation
Abstract
Terminal velocities of particles in nonNewtonian fluids are often based on the extrapolations of known Newtonian behaviour
or some empirical correlations for each new fluid. Such estimations are full of uncertainties. The available models for
hydrodynamic simulation in CFD packages are either not sufficient or have a high computational requirement. then this paper
we have conducted detailed simulation of falling spherical particles in a stagnant nonNewtonian fluid using a novel
Macroscopic Particle Model (MPM) approach. The terminal velocities predicted using this model were in close agreement
with the experimental data. The particle fall behaviour for Newtonian and nonNewtonian fluids were also compared. Due to
the high viscosity in low shear regions initially, the velocity in nonNewtonian case was always lower than the Newtonian
cases.
Introduction
law velocity (Vs) and generated a three parameter model.
Particle motion in nonnewtonian fluids finds direct
application in various industries. In nonNewtonian fluids,
shear stress versus shear rate curve is not linear or if it is
linear, it does not pass through the origin. The viscosity of
such fluids is a function of the applied shear, that is, with a
change in the shear, there is a new value of apparent
viscosity. In this paper, we have numerically studied fall of
a ball in a viscoplastic fluid which has a time dependent
viscosity and possess a yield stress. The rate of shear of
these fluids is purely a function of the current value of shear
stress. The stressstrain curve of these fluids may be linear
or nonlinear, once the yield stress has been achieved. The
examples of viscoplastic systems are ore slurries, paints,
mud, food stuffs, cosmetic creams, hair gels, crystallizing
lava, polymer solutions, etc.
There are several studies on particles settling in fluids
[Cox & Brenner (1967), Bungay & Brenner (1973), Brunn
(1982), Turton & Clark (1987), Madhav & Chhabra (1994),
Atapattu (1995), Di Felice (1996), Beaulne (1997), Nguyen
(1997), Chhabra (2003)]. A review of the particle falling in
both Newtonian and nonNewtonian fluids has been
conducted by Chhabra (2006). Experimental determinations
show that for the settling velocities of spherical particles,
with particle Reynolds numbers of less than 10, can be
adequately described theoretically (Reynolds & Jones,
1989). Subsequently, Haider and Levenspiel (1989) worked
out correlations for the prediction of terminal velocities and
drag coefficient for spherical particles with RMS deviation
2.4% and nonspherical particles with RMS deviation of
35%.
Nguyen (1997) carried out an asymptotic analysis for
calculation of terminal velocity of rigid spheres from stokes
= 1+ A (1 + 0.079Ar0.749)0.755
V 96
According to the author, this new model was capable of
describing the terminal velocity for a wide range of
Archimedes numbers.
Di Felice (1996) reported a relationship for the prediction of
bound terminal velocity (ut) in terms of unbound terminal
velocities (ut,) and the ratio of particle to cylinder diameter
(Q) and they proposed it to be applicable to viscous, inertial
and intermediate flow regimes:
ut (1A3 )a
Uto i 10.33A
where a in terms of Reynolds number (Ret,) for unbound
fluid is given by,
3.3a
a = 0.1Ret
a 0.85
Chhabra (2003) considered a sphere settling on the axis of a
long cylindrical tube and identified Reynolds numbers
that mark the transitions from the viscous to intermediate
and from the transitional to fully inertial flow regimes as
functions of X. The functional dependence of the wall factor
on the spheretotube diameter ratio (Q) and the sphere
Reynolds number were examined. They derived a wall
factor depending on X and Re in the intermediate zone and
depending on X at very low and very high values of Re.
Kelissidis (2'"'4) determined correlations in order to
predict the terminal velocities of nonnewtonian
pseudoplastic fluids and compared the results with
previously published data (Kelessidis & Mpandelis, 2004;
Turton & Clark, 1987), and derived:
.18 /[ 0.824/n 0.3210.412 .214
1= + (4)
Wilson et al ,' 11 1) devised 'equivalent newtonian viscosity'
for nonnewtonian fluids dependent on the fluid rheology,
which was then used for the direct prediction of terminal
velocity. According to them, the accuracy of predicted
velocities was adequate for engineering purposes. The
equivalent Newtonian viscosity was given by:
[eq = T/y
where y' is the shear rate at
determined by the rheogram. t is
given by:
T = 
and, Superficial shear stress,
T (Pp Pf)dp
6
the particular t an
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
will be computationally expensive. Therefore, in this paper,
we have used an MPM model to simulate the motion of a
particle falling in a stagnant nonNewtonian fluid. The
model predictions have then been compared with the
experimental data as well as the terminal velocity prediction
of Newtonian fluid with an equivalent viscosity as
suggested by Wilson et al. (2003).
Nomenclature
Ar
CD
(5) C1
C2
dp
id is dp
d.
g
K,
n
(6) Re
Ret,
Rv,
Rp,(
(7) Rm,.
T
u1
Wilson et al(2003) used a value of 0.3 for 4 to calculate the
equivalent viscosity.
In this study we have used a macroscopic particle model
(MPM) for predicting terminal velocities in a yield
pseudoplastic fluid. MPM is a quasi direct numerical
simulation over the static computational cells and captures
the precise hydrodynamics in particulate flows. MPM is
based on lagrangian frame of reference in which equations
of motion are solved for each individual particle. Each
particle is assumed to span several computational cells; a
solid body velocity that describes the particle motion
translationall and rotational) is fixed in these cells to
effectively transfer the momentum to liquid phase due to
particle motion. Drag and lift forces as well as torque on
the particle are explicitly calculated based on velocity,
pressure and shear stress distribution in fluid cells around
the particle. Pressure, viscous as well as virtual mass
components are included in particle forces/torque
calculations. New velocities and positions of the particles
are calculated at each time step taking drag force and lift
force into consideration.
In comparison with voluminous literature available for drag
formulations and the movement of sphere in quiescent
Newtonian fluids, only a few papers relate with the
hydrodynamics of the sphere falling in nonNewtonian
fluids and its terminal velocity. Current methods
forpredicting terminal velocity are based on extrapolations
from known Newtonian behaviour or some empirical
correlations for each new fluid or case. The discrete particle
model (DPM) in FLUENT considers particle as a point
mass which makes it unsuitable to take into account the
actual drag and lift forces. A direct numerical simulation
(DNS) with dynamic meshing around a moving particle
along with six degree of freedom (6DOF) algorithm, is
expected to be most accurate method for this purpose but it
t
ut,
U.
U*
V
Vs
u, v,
w
Archimedes Number
Drag Coefficient
Intermediate parameter
Intermediate parameter
Particle Diameter
Dimensionless Diameter
Accelaration due to gravity
Consistency Index
Power law Index
Reynolds Number
Reynolds Number for Unbound Fluid
Viscous Component of Drag Force
Pressure Component of Drag Force
Virtual Mass Component of Drag Force
Time
Bound Terminal Velocity
Unbound Terminal Velocity
Dimensionless Velocity
Terminal Velocity of Rigid Spheres
Stokes Law Velocity
Velocities in x, y and z directions respectively
Greek letters
a An Intermediate Parameter
y' Shear Rate
X Sphere to Cylindrical Tank Diameter ratio
Peq 'Equivalent Newtonian Viscosity' for falling
spheres
rlc Consistency Parameter for Casson Model
p, Particle Density
pf Fluid Density
T Shear Stress
Mean Superficial Shear Stress
To Yield Stress for Casson Fluid
Th Yield Stress for HB Fluid
Hydrodynamics Model
For the flow of nonNewtonian fluid, the laws
conservation apply in the same manner as for t
Newtonian fluids. The conservation of mass can
described by the continuity equation (Versteeg
Malalasekera, 1995):
ap+ div(pu) = 0
at
and the equation for momentum conservation in three
dimensions is as follows (Versteeg & Malalasekera, 1995):
xcomponent of momentum equation:
Du a a(p +rxx ) adzx S
P + y + +SMy
Dt ax dy 0z
(8.a.)
ycomponent of momentum equation:
Dv acy a(P+Tyy) a9zy
P = + + SMy (8.b.)
Dt ax 9y 0z y
zcomponent of momentum equation:
Dw axz ia(P+Tzz) + yz
Dt ax d ay +S (8..)
For the calculation of drag between the particle and
surrounding fluid, a force balance on the particle was
carried out, which consisted of viscous, pressure and
virtual mass forces (Agrawal, Ookawara, & Ogawa, 2009).
Viscous component is calculated on the basis of shear
stress distribution around the particle and is given by the
equation:
 (i
Ry,i = Zsurface i (Tj 2 ( (9)
cells  I I (
Pressure component is due to the pressure distribution of the
fluid surrounding the particle. It is calculated using the
nodal values of pressure in the cells surrounding the particle
as given in equation (10):
Rp,i = surface (Pd2 _7 x) (10)
cells
Applying the momentum conservation on the system, the
virtual mass component of fluid force is calculated. The
integral of rate of change of momentum is obtained by
equation (11) considering the fluid cells within the particle
volume.
Rmi = Volume mf Vf,i Zvolume mrVfy (11)
cells cells At
The flow is governed by the usual conservation equations of
mass and momentum for an incompressible fluid under
isothermal conditions. The constitutive equations that
relates the stress and deformation are either the Casson
equation or the HerschelBulkley equation. Casson model is
a two parameter model describing the behaviour of the fluid.
It is written as:
(Tyx)1/2 = (T)1/2 + (cYyxyx/2 > TT
Yyx = 0 Tyx < Tc
The yield stress in this equation is given by rc and pc is
called the consistency parameter.
On the other hand, HerschelBulkley is a three parameter
model. It is given as:
Tyx = To + Kc(&yx) Tyx > T
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
yx = 0 Tyx < T
Where, Tr4 is the yield stress, Kc is the consistency index
and n is the powerlaw index.
Note that when the shear stress r falls below ty, a solid
structure is formed (unyielded). Also when the powerlaw
index is unity and the consistency index is equivalent to the
plastic viscosity, the HerschelBulkley model reduces to the
Bingham model.
Methodology
The physical domain comprises of a free falling spherical
particle in quiescent Newtonian and nonNewtonian fluids
without any wall effects. A structured meshed cylindrical
geometry with dimensions 2cm diameter and 6 cm height
was constructed in Gambit. These cases were then simulated
in FLUENT 6.1 using MPM.
Drag factor specification for momentumdeficit rate in x, y
and z coordinates were given as unity. The particleparticle
and particlewall collision factors were not applicable
because of the presence of only one particle throughout the
simulation and the absence of any wall. The particle
diameters between 0.006m to 0.01m and having density in
the range of 7638 kg m3 to 8876 kg m3 were investigated.
The particles were initially positioned right at the centre of
the domain to minimize any impact due to walls. With all
these parameters defined, the MPM model was initiated by
injecting the particles in a similar way as in DPM.
Both Newtonian and nonNewtonian cases, with constant
fluid density of 999 kg m3 and same "equivalent viscosity"
were studied. In nonNewtonian fluids, it was defined
using the HerschelBulkley model. For the Newtonian case,
the equivalent viscosity of the fluid was calculated using the
method of Wilson et al. (2003).
Before running simulations with the nonNewtonian method,
initial simulations were run for different grid sizes, time
steps and domain size for a Newtonian fluid.
Rheological Considerations
Wilson et. al. (2003) conducted experiments for particles
falling in stagnant nonNewtonian fluids. The rheology of
these nonNewtonian fluids was determined and was
represented using the Casson model.
The theological property of a fluid can be incorporated in
FLUENT in three different ways. Firstly, if viscosity is
constant, then its value can be provided. Secondly, viscosity
can be provided in the form of parameters of a model
already available in FLUENT. Finally, it can be introduced
as a user defined function, which provides the value of
viscosity at each node of the geometry at each time step.
Currently, Casson's theological model, which is most
appropriate for pseudoplastic fluids, is not available in
FLUENT. Therefore, theological properties of the fluid
were incorporated in FLUENT as a three parameter
HerschelBulkley model. The stress versus strain data were
collected from the Casson parameters and a best fit
parameters for HerschelBulkley model were calculated
using MATLAB. The apparent viscosities using the
Casson and the HerschelBulkley model are compared in
Figure 1.. For the range of data studied, RMS error between
the predictions using two models was 0.0276, which is in
tolerable limits.
100 Casson Model
Herschel Bulkley Model
10
0 20 40 60 80 100 120
Shear Rate (s1)
Figure 1: The comparison of apparent viscosity to strain
curve for the nonNewtonian fluid with given Casson
model and modelled Herschel Bulkley model.
Results and Discussions
Figure 2(a) shows the initial position of a 0.01m diameter
spherical particle, which is falling in the domain under the
effect of gravity. Since there is no other force acting on the
particle, it moves down along the axis without any lateral
deviation. The particle also imparts momentum to the fluid
in its vicinity. A longitudinal crosssection of the velocity
contours during the fall is given in Figure 2(b).
0
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
and on the other hand lower time steps rendering accuracy,
need very high computational power and time. As can be
seen from Figure 4, the time step of 103 s shows a greater
deviation in the terminal velocities from the values
calculated using Stokes law as compared to those with time
steps 104 s and 10 s. However, time steps of 104 s and
105 s gave very similar results, therefore timestep size of
104 s was considered as an optimum solution for setting up
the base case.
3a1%
Figure 2(b): Velocity Profile in longitudinal crosssection
of the domain.
0.20
0.18
0.16
r 0.14
1 Particle Velocity
E 0.12
0.10
10.06
0.04
0.02
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Time Step (in s)
Figure 3: Change in Particle velocity in Newtonian fluid
with time ( pp=8876 kg /m, pf=999 kg/m3, p=0.905 Pa.s,
Dp= 0.00635 m).
010
Figure 2(a): Particle Falling in the domain ( Dp= 0.01 m,
Length = 0.06m, Diameter = 0.02m).
The velocity at each timestep was noted and the simulation
was run until the terminal velocity reached a plateau as
shown in Figure 3. The rate of change of particle velocity
was initially high and as the particle approaches near the
terminal velocity, it asymptotically tends to a zero value.
A timestep size study is crucial to get a timestep
independent solution for simulations. A large time step
consumes less computational power but its accuracy is low
008
E 006
c
0 004
002
Time Step 103
Time Step 104
000 0
000 002 004 006 008 010
Time (in s)
Figure 4: Effect of time step on the particle velocity
prediction (pp=1550 kg /m3, pf=1000 kg/m3, g=1 Pa.s, Dp=
0.01 m).
The MPM has recently been improved (Agrawal et al.,
2009) by incorporating a new drag formulation. Figure 4
provides a comparison between predictions of the old and
new models as a function of grid size in a Newtonian. The
results were compared with the theoretical velocity obtained
from Stokes law. It is clear that the old model significantly
overpredicted the terminal velocities when compared to the
predictions using the Stokes law. With the newer drag
formulation, although terminal velocities were somewhat
underpredicted but they were significantly closer to those
predicted using the Stokes law.
A grid dependence study is necessary in order to minimize
numerical errors in the predictions. In the MPM, high grid
points per particle can provide more accurate results at the
expense of increased computational requirements. So, it
becomes necessary to find an optimum grid resolution
without compromising the numerical accuracy. Figure 5
show that for the newer drag formulation, all three grid
resolutions gave nearly similar results. Therefore, it may be
concluded that for this study, a 10 gridpoints per particle
mesh was an optimal choice. Higher grid points per particle
only resulted in higher computational requirements without
any considerable advantage in the accuracy. Therefore, for
the remaining simulations in this study a mesh with 10
gridpoints per particle was used.
0.06
U)
E 0.04
0
>
0.02
0.00 o 
0.00
0.02 0.04 0.06 0.08 0.10
Time (in s)
Figure 5: Effect of number of gridpoints per unit particle on
the particle velocity prediction( pp=1550 kg /m3, pf=1000
kg/m3, =l 1Pa.s, Dp= 0.01 m).
Domain size is another important parameter of this study. As
the particle diameter was increased to approach tothe
diameter of domain, it is expected to give errors in
predictions due to the nonavailability of enough node
points in the vicinity of particle across the diameter.
Therefore, simulations were conducted to study the effect of
domain size. Figure 6(a) compares velocity profile of a
particle in a Newtonian fluid inside a 2 cm diameter
cylindrical domain (D2cm, H 6cm) with that inside a
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
relatively larger cuboidal domain (L5cm, W5cm, H15cm).
It is clear that increasing the domain size did not have any
noticeable effect on the velocity profiles as the RMS error
between the two predictions was only 0.0061, which is
within tolerable limits. Therefore, for the simulation of
particles in the Newtonian fluid the smaller domain was
adequate, as it required significantly less computational
power as compared to the larger domain which had over
half a million grid points. As shown in Figure 6(b), similar
simulations were performed for the motion of particles in
nonNewtonian fluids which showed a very high sensitivity
towards the change in the domain size. Domains having a
fixed height of 15 cm with width and depth of of 2cm, 3cm,
4 cm, 5cm and 6cm were investigated. It is clear that for this
study (having particle diameter of 1 cm), domains with
width and depth of 2 cm somewhat overpredicted the
velocity when compared with the larger domains.
Therefore, for simulations inside nonNewtonian fluids,
larger domains must be used. However, all the domains with
widths and depths greater than 4cm gave nearly identical
results. Therefore, for the remaining simulations in this
study, a cuboidal domain with 4 cm width and depth (L4
cm, W4 cm, H15 cm) was used.
0.30
0.25
S0.20
E
0.15
5 0.10
0.05
0.00 
0.00
0.05 0.10 0.15 0.20 0.25
Time (in s)
Figure 6(a): Effect of domain extent on the particle
velocity prediction for Newtonian Fluids ( pp=7638 kg /m3,
pf=999 kg/m3, p=0.736 Pa.s, Dp= 0.00794 m).
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.00
0.02 0.04 0.06 0.08 0.10
Time (in s)
Figure 6(b): Effect of domain extent on the particle velocity
prediction for NonNewtonian Fluids ( pp=7638 kg /m,
pf=999 kg/m3, p=0.736 Pa.s, Dp= 0.00794 m).
Effect of Particle Diameter on the Terminal Velocity
in Newtonian Fluids
As can be seen in Figure 7, that for particle diameters of
0.007 m and less, there was a reasonable agreement between
the MPM predictions and those calculated using the
Stokes law. Further increase in the diameter resulted in
deviation from the calculated value. The MPM model under
predicts the fall velocity of particle with diameter greater
than one mentioned above.
0000 0002 0004 0006 0008 0010 0012 0014 0016 0018
Particle Diameter (in m)
Figure 7: The Predictions of terminal velocity with particle
diameter ( pp=1500 kg /m3, pf=1000 kg/m3, g=lPa.s).
Comparison of Fall Velocities for Newtonian and
NonNewtonian Fluids
The simulations for nonNewtonian fluids were conducted
using the operating conditions such as particle diameter,
particle density, fluid density, from Wilson et al. (2003).
They conducted experiments using nonNewtonian fluids
and evaluated an "equivalent viscosity" for predicting the
particle terminal velocities. Using the same constant
viscosity value and fluid density, simulations were
conducted for the Newtonian fluid as well. The plots for the
velocities were compared and shown in Figure 8. The
terminal velocity in case of the nonNewtonian fluids was
significantly lower when compared to the corresponding
Newtonian fluid. It was possibly because of the presence of
high viscosity due to the the yield stress of viscoplastic fluid
in the undisturbed fluid. The increase of the velocity of
particle results in shear thinning of the fluid. Due to this
shear thinning, the apparent viscosity surrounding the
sphere decreases and a bounded region of finite viscosity
forms. As a result, the rate of displacement increases and
gradually, the velocity of the sphere reaches the terminal
velocity. The rate of this process is slower when compared
with the constant viscosity case of Newtonian fluids.
It is clear from Figure 9 that the predictions for the terminal
velocities in case of Newtonian fluids were in close
agreement with those calculated using the correlation of
Turton and Levenspiel (1986):
240.413
CD = (1 + 0.173Re0.657) + 0.413(12)
The deviation in the predictions of the 1+1630nonNewtonian fluids
The deviation in the predictions of the nonNewtonian fluids
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
from Newtonian fluids can be attributed to the differences in
fluid characteristics.
030
025
' 020
E
0 15
" 0 10
 Newtonian
NonNewtonian
00 01 02 03 04
Time (in s)
Figure 8: The comparison of sphere free fall velocity
progression with time for a case of Newtonian and
nonNewtonian fluid simulation (Dp: 7.94mm pp=7638 kg
/m3, pf=999 kg/m3, p=0.736 Pa.s).
10
001 01
10 100
Figure 9: The comparison of Drag Coefficient for Newtonian
and nonNewtonian fluid simulation with respect to the
Reynolds Number along with the corresponding theoretical
values from Turton and Levenspiel (1986).
Comparison with the Experimental Data
Two fluids with different rheology were taken into
consideration, incorporated as Herschel Bulkley model and
simulated using CFD. The HB parameters for Fluid A were
Th= 16.0635, Kh= 1.4219 and n=0.5434 and that of Fluid B
were Th= 20.0589, Kh= 1.5198 and n=0.5374.The main
difference in these two fluids was the yield stress a particle
has to overtake in order to move. With these parameters,
Fluid B would always be more viscous for any similar
instance but would also have a higher rate of decrease of
viscosity than Fluid A once the yield stress is overcome.
Simulated terminal velocities for the particle in these two
fluids is plotted against the experimental results of Wilson
et al. (2003). Since most of the simulated data points lie in
the vicinity of the diagonal line, it may be concluded that
the MPM approach was adequate for predicting motion of
large particles in nonNewtonian fluids. As can be seen
 CFD Predictions
Stokes Law
Turton & Levenspiel (1986)
S o Newtonian CFD Predictions
S +* NonNewtonian CFD Predictions
from the graph, that for smaller terminal velocities, the
model slightly under predicts the terminal velocity and
reverse trend is observed for the larger particles.
E
. 025
A)
1 020
_0
S015
E 005
X 000
0
0
o Ox o Fluid A
x x Fluid B
00 005 010 015 020 025 030
CFD Particle velocity Predictions (in m s1)
Figure 10: The Validation of terminal velocity predictions
for NonNewtonian Fluids by MPM model with the
experimental values given by Wilson et. al.(2003).
In order to find out the effect of diameter on terminal
velocities in nonNewtonian fluids, one more fluid was
taken into account. This fluid C had Th= 18.1359, Kh=
1.7133 and n=0.566. Fluid A and fluid B had similar
properties except the difference in their yield stress values.
Fluid C had an intermediate yield stress, but is different in
terms of consistency index and power law index. With the
increase in shear stress, its shear rate increases rapidly and
therefore, it had more dominant shear thinning. Particles
of different diameters were dropped in these fluids and their
terminal velocities were plotted in Figure 11. As expected,
the terminal velocity in all three cases increased with the
particle diameter which is consistent with the
experimental results of Reynolds and Jones (1989).
035
0 30 o Fluid A, Pp7638 kg/m3
0 Fluid A, pp7792 kg/m3
0 25 Fluid B, Pp7638 kg/m3
U Fluid B, pp7792 kg/m3
S0 20 Fluid C, pp8000 kg/m3
a)
> 015
0 10
I 0
005
000
00065 00070 00075 00080 00085 00090 00095 00100 00105
Particle Diameter (in m)
Figure 11: The Comparison of velocity predictions for the
nonNewtonian fluids and their variation with particle
diameter ( pf=999 kg/m3 ).
The particle when suspended in fluid A showed a higher
terminal velocity when compared to that in fluid B under
the same conditions. If the curves of these two fluids are
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
compared, a definite trend in the variation can be seen. It
means, although the fluid properties are changed, but
influence of force applied by the particle on the velocity is
independent of the fluid property.
The velocities were also been compared for different
densities of particle having same diameter and using the
same fluid. It was observed that the increase in particle
density had a similar influence on the velocities to that of
particle diameter. It is because both the particle diameter
and density directly contribute to the shear applied on the
fluid.
025
020 
S 15 
c
o 010 //
3  p =7638kg/m3
Pp=8050 kg/m3
005 3
Sp=8463kg/m3
p=8876kg/m3
000
000 005 0 10 0 15 020 025 030
Time (in s)
Figure 12: The Comparison of velocity predictions for the
nonnewtonian fluids and their variation with fluid
properties (Dp: 7.94mm, pf=999 kg/m3, p=0.736 Pa.s).
Conclusions
In this study, the motion of particles in nonNewtonian
fluids has been investigated using a novel macroscopic
particle model (MPM) approach. Drag and lift forces as
well as torque on the particle were explicitly calculated
based on velocity, pressure and shear stress distribution in
fluid cells around the particle. New velocities and positions
of the particles are calculated at each time step taking drag
force and lift force into consideration. The theological
parameters given in Wilson et al. (2003) as Casson model
were remodelled as HerschelBulkley model. The
simulation results were assessed for the effect of time step,
domain size and grid points per particle. An optimum value
of 104 s was obtained for the time step size and 10 grid
points per particle as the optimal grid size. A domain with a
diameter of twice the particle diameter in Newtonian case
and four times in the case of nonNewtonian fluid was
required in order to facilitate the bounded region formation.
The MPM model under predicted the fall velocity of
particle with diameter greater than 0.007 m. The increase of
the velocity of particle resulted in shear thinning of the fluid.
As a result, the rate of displacement increased and gradually,
the velocity of the sphere reached to its terminal velocity. It
was observed that the particle velocities for Newtonian
fluids were greater than those in the case of nonNewtonian
fluids for the same equivalent viscosity. The deviation in the
predictions of the nonNewtonian fluids from experimental
values can be attributed to the mismatch in fluid properties
of Newtonian fluids. The results of simulations were
validated using the experimental data given in paper by
Wilson et al (2003) as well as Turton and Levenspiel (1986).
References
Agrawal, M., Ookawara, S., & Ogawa, K. (2009). Drag
Force Formulation in Macroscopic Particle Model and Its
Validation. Paper presented at the AIChE, Nashville, TN.
Atapattu, D. D. (1995). Creeping sphere motion in
HerschelBulkley fluids: flow field and drag. Journal of
nonNewtonian fluid mechanics, 59(23), 245.
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