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© 2016 The Korean Society of Rheology and Springer 207
Korea-Australia Rheology Journal, 28(3), 207-216 (August 2016)DOI: 10.1007/s13367-016-0021-8
www.springer.com/13367
pISSN 1226-119X eISSN 2093-7660
The effect of thixotropy on a rising gas bubble: A numerical study
Kayvan Sadeghy1,* and Mohammad Vahabi
2
1Center of Excellence in Design and Optimization of Energy Systems (CEDOES), School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran 11155-4563, Iran
2Department of Mechanical Engineering, College of Engineering, Central Tehran Branch, Islamic Azad University, Tehran 14676-86831, Iran
(Received February 10, 2016; final revision received June 19, 2016; accepted June 20, 2016)
The deformation of a single, two-dimensional, circular gas bubble rising in an otherwise stationary thixo-tropic liquid in a confined rectangular vessel is numerically studied using the smoothed particle hydrody-namics method (SPH). The thixotropic liquid surrounding the bubble is assumed to obey the Moore model.The main objective of the work is to investigate the effect of the destruction-to-rebuild ratio (referred to bythe thixotropy number in dimensionless form) in this model on the bubble's shape, velocity, and center-of-mass during its rise in the liquid. Based on the numerical results obtained in this work, it is found that thebubble moves faster in the Moore fluid as compared with its Newtonian counterpart. An increase in thethixotropy number is also shown to increase the bubble's speed at any given instant of time. The effect ofthixotropy number is found to be noticeable only when it is large. For Moore fluid, a large thixotropy num-ber means that the fluid is basically a shear-thinning fluid. Therefore, it is concluded that the shear-thinningbehavior of the Moore model easily masks its thixotropic behavior in the bubble rise problem. The effectof thixotropy number is weakened when the Reynolds number is increased.
Keywords: bubble rise, thixotropic fluid, WC-SPH method, Moore model, thixotropy number
1. Introduction
The motion of gas bubbles rising in a stationary liquid
is frequently encountered in many branches of engineer-
ing. One can mention, for example, biochemical engineer-
ing where contacting equipments such as bubble columns
are widely used for fermentation, wastewater treatment,
absorption, etc. Bubble rise is also encountered during oil
extraction, food processing operations, and bio-processes
(e.g., in pharmaceutical and environmental industries). A
good knowledge about the fundamentals of the bubble
hydrodynamics is essential for designing such processes.
Of primary importance is the bubble's residence time
which is needed when an equipment (e.g., bubble reactor)
is required to meet a desired specification. To achieve this
goal, it is imperative to evaluate the bubble's rise velocity.
For Newtonian fluids, this subject matter has been the
focus of extensive studies in the past, in both theoretical
and experimental domains, alike (Clift et al., 1978; Mag-
naudet and Eames, 2000; Krishna and van Baten, 1999).
Based on such studies, it is well-established that the
dynamics of bubbles rising in Newtonian fluids depends
primarily on the surface tension and the Reynolds number.
It is also known that in Newtonian fluids bubbles deform
from a spherical shape to an oblate ellipsoidal shape and
then to a spherical cap with increasing volume (Kulkarni
and Joshi, 2005). Despite its importance, the rise of gas
bubbles in non-Newtonian fluids have not been as thor-
oughly studied. In fact, currently, neither their dynamics
nor their shape are completely understood and this is
mainly because of the diversity of the non-Newtonian
behavior (Chhabra, 1993).
Due to the variety of non-Newtonian behavior, studies
carried out in the past have tried to focus on one effect at
a time. For example, Zhang et al. (2010) investigated the
bubble rise in a shear-thinning fluid obeying the Carreau
model and reached to the conclusion that the local change
in viscosity around the bubble strongly depends on the
bubble's shape and the zero-shear viscosity of the sur-
rounding liquid. It was also shown that the rise velocity is
higher in shear-thinning fluids as compared with Newto-
nian fluids of the same zero-shear viscosity. As to the
yield-stress effects, Sikorski et al. (2009) experimentally
showed that in Bingham fluids, only bubbles larger than a
critical radius may rise. The importance of the Bingham
number in decelerating and eventually immobilizing bub-
bles rising in a variety of viscoplastic materials (namely,
Bingham, Herschel-Bulkley, and Papanastasiou) has been
numerically demonstrated by Tsamopoulos et al. (2008),
and Dimakopoulos et al. (2013). As to viscoelastic effects,
it is long established that single bubbles rising in such liq-
uids may exhibit a velocity jump discontinuity, once a crit-
ical volume is exceeded (Astarita and Apuzzo, 1965). In
a recent work, it has been shown that this phenomenon
can be witnessed whenever a critical Weissenberg number
is exceeded (Pilz and Brenn, 2007). The jump in velocity*Corresponding author; E-mail: [email protected]
Kayvan Sadeghy and Mohammad Vahabi
208 Korea-Australia Rheology J., 28(3), 2016
was believed to be primarily caused by a negative wake
behind the bubble (Hassager, 1979). Pillapakkam et al.
(2007) showed that for viscoelastic fluids obeying Old-
royd-B model, the presence of an additional vortex ring
and the change in the velocity field in response to a
change in the bubble shape (which becomes asymmetric
and cusped-shaped) all contribute to the jump in the bub-
ble's velocity at a critical bubble volume. Their work,
however, cannot explain why, at times, air bubbles may
experience velocity jump despite the fact that there is no
negative wake present. In a more recent work, Fraggeda-
kis et al. (2016) have shown that use should preferably be
made of more realistic viscoelastic models in bubble rise
studies. They relied on the robust (exponential) Phan-
Thien/Tanner model and showed that the velocity jump
may occur even under creeping conditions, i.e., where no
wake is present (either positive or negative). As to the
effects of a fluid's thixotropy on a rising bubble, there
appears to be no published work in the open literature.
Having said this, it should be conceded that in an exper-
imental study Gueslin et al. (2006) have shown that thixo-
tropy may indeed affect the settling of spherical solid
particles in an aging fluid. But, because their fluid was
viscoplastic in addition to being thixotropic, it is hard to
say to what extent each behavior was responsible for the
observed effect. A fluid’s thixotropy is already known to
affect the dynamics of tiny spherical gas bubbles (Ahmad-
pour et al., 2011; Ahmadpour et al., 2013). To the best of
our knowledge, however, no such an analysis has previ-
ously been carried out for bubbles rising in thixotropic liq-
uids. In the present work, we intend to numerically
investigate the behavior of a buoyancy-driven large gas
bubble rising in a thixotropic fluid obeying the Moore
model (Moore, 1959). To achieve this goal, we rely on the
smoothed-particle-hydrodynamics method (SPH) for solv-
ing the equations of motion and capturing the shape of the
bubble.
To achieve its goals, the work is organized as follows:
we start with presenting the governing equations in its
most general form. We will then proceed with briefly
introducing the numerical method of solution used for
simulating the bubble motion, i.e., the weakly-compress-
ible SPH method. Numerical results are then presented
addressing the effects of thixotropic parameters appearing
in the Moore model on the shape and rise velocity of a gas
bubble. The work is concluded by highlighting its major
findings.
2. Governing Equations
For a single gas bubble rising in an arbitrary liquid under
isothermal conditions, the governing equations comprise
the continuity equation and the Cauchy equations of motion.
In Lagrangian framework, these equations take the form
(Sussman et al., 1994):
, (1a)
(1b)
where ρ is the liquid's density, is the velocity vector, D/
Dt is the material derivative, σ is the surface tension coef-
ficient, κ is the surface curvature, and δ is the Dirac’s
delta. Also, the subscript “d” represents the normal dis-
tance from the interface, with being the unit normal
vector of the interface. To focus on thixotropy effects, in
the present work, a purely-viscous thixotropic fluid model
is used to relate the stress tensor, , to the deformation-
rate tensor, . For such thixotropic fluids, the stress tensor
can be written as in which (unlike the Newtonian
fluids) the viscosity coefficient is time-dependent in addi-
tion to being shear-dependent. In most structural models,
the time-dependency is interlinked with the shear-depen-
dency introduced through invoking a scalar called the
structural parameter, λ. This parameter lies between zero
(denoting completely broken-down structures) and one
(denoting completely re-built structures) depending on the
level of structures left intact in the fluid. Fluid’s viscosity
at each instant of time is then assumed to be a function of
λ (Mujumdar et al., 2002). The simplest idea is the one
proposed by Moore (1959) who assumed that the fluid's
viscosity linearly depends on λ that is:
(2)
where μ0 is the viscosity corresponding to complete struc-
ture build-up (i.e., the zero-shear viscosity), and is the
viscosity corresponding to complete structure breakdown
(i.e., the infinite-shear viscosity). The main difference
between various structural models lies in the kinetic equa-
tion adopted for representing the time evolution of the
structural parameter, λ. In the Moore model, the kinetic
equation is of the following form (Moore, 1959),
(3)
where is the second invariant of the deforma-
tion-rate tensor, dij. In this equation, “a” and “b” are mate-
rial properties denoting (Brownian) structure build-up and
shear-induced structure break-down, respectively. Since
“a” has the dimension of reciprocal time and “b” is dimen-
sionless, the breakdown-to-rebuild ratio, b/a, can conve-
niently be interpreted as the characteristic time of the
Moore fluid. It is one of the main objectives of the present
work to investigate the effect of this ratio on the shape and
the rise velocity of a gas bubble. Before proceeding any
further, we would like to stress that for thixotropic fluids
obeying Moore model, the b/a ratio is important in two
DρDt------- + ρ ∇ ∇⋅⎝ ⎠
⎛ ⎞ = 0
ρDV
Dt-------- = ∇– p + ∇ τ
˜⋅ − σκδd( )n̂ + ρg
V
n̂
τ˜d
˜ τ˜ = 2μd
˜
μ t( ) = μ∞
μ0 μ∞
–( )+ λ
μ∞
dλdt------ = a 1 λ–( ) b– γ·λ
γ· = 2dijdji
The effect of thixotropy on a rising gas bubble: A numerical study
Korea-Australia Rheology J., 28(3), 2016 209
respects (Cheng and Evans, 1965):
• It controls the severity of the fluid's shear-thinning
behavior. This can easily be shown by integrating Eqs.
(2) and (3) in simple (steady) shear, which renders: μeq
= [1+α/(1+(b/a) )], where α is the viscosity ratio.
This equation clearly shows that, at equilibrium, a
Moore fluid is a shear-thinning fluid with its degree of
shear-thinning being controlled by the ratio, b/a. That
is, an increase in the b/a ratio means a more severe
shear-thinning behavior.
• It controls the degree of the fluid's time-dependency
behavior. This can readily be demonstrated by integrat-
ing Eqs. (2) and (3) with the result being: μ(t) = μeq −
(μeq − μ0)exp(−t /Λ), where in this equation Λ = 1(1+(b/
a) ) is the so-called decay time. Fluids having larger Λ
(or smaller b/a ratio) are more likely to exhibit time-
dependent (thixotropic) effects − in a sense, the decay
time is equivalent to the relaxation time for elastic flu-
ids.
Finally, we would like to mention that the Moore model
has been found to well represent the rheology of ceramic
pastes (Moore, 1959). It has also been found to be a good
model to represent the thixotropy of drilling muds (Teh-
rani, 2008). Still, it must be stressed that there are many
thixotropic fluid systems which might require a more
robust rheological model such as Dullaert-Mewis model
(Dullaert and Mewis, 2006), or de Souza Mendes model
(de Souza Mendes, 2009) in order to describe their rhe-
ology. Despite its limitations, the Moore model is still
widely used in theoretical/numerical studies (thanks to its
simplicity) for elucidating the role played by a fluid's
thixotropy in any given fluid mechanics problem (see, for
example, Derksen and Prashant, 2009). And, this is the
main reason this rheological model has been adopted for
the present study. The pioneering nature of the present
work, and also the fact that this fluid model excludes com-
plications which might arise through a liquid's elasticity
and/or viscoplasticity, makes the Moore model a good
candidate for fundamental studies. That is to say that, we
are not concerned with any specific thixotropic fluid sys-
tem although in our numerical simulations we try to choose
parameter setting based on data available for industrially-
important fluids such as waxy crude oils. Now, before pro-
ceeding with solving the governing equations we try to
make all pertinent parameters dimensionless. To that end,
we substitute,
(4)
where the subscript “L” refers to the liquid. In the above
relationships ρL is the liquid's density and D0 is the initial
bubble diameter. In dimensionless form, the governing
equations become,
, (5)
(6)
where Re is the Reynolds number and Bo is the Bond
number defined by,
, (7)
(8)
In addition to the Reynolds and Bond numbers, we can
introduce another dimensionless number by simply divid-
ing the destruction-to-rebuild ratio, b/a, by the flow's char-
acteristic time, . The dimensionless number so-
obtained will be called the thixotropy number; it will be
shown by “b/a” (i.e., with no asterisk) in the rest of the
work. Our focus in this work is to investigate the effect of
the thixotropy number on the bubble's characteristics
during its rise. Therefore, all in all, we have three import-
ant dimensionless numbers in our fluid mechanics prob-
lem: Bo, Re, and b/a. Since the effect of Re and Bo have
already been addressed in published works, we will focus
mostly on the effect of the thixotropy number, b/a. Still,
limited results will also be presented as to the Reynolds
number effect. Equations (5)-(8) together with equation
constitute the equations governing the motion
of a single bubble rising in a Moore fluid. (It should be
noted that the kinetic equation, Eq. (3), is already dimen-
sionless.) There is no simple analytical solution in close
sight for this set of differential equations. Thus, we look
for a numerical solution. In the sections to follows, the
asterisk ( * ) above dimensionless parameters will be dropped,
for convenience.
3. Numerical Method
In the present work, we have decided to rely on the
smoothed-particle hydrodynamics (SPH) method for sim-
ulating the shape and speed of a gas bubble rising in a
Moore fluid (Monaghan, 1988; Liu and Liu, 2003). Our
interest in this numerical method stems partly from our
success in simulating a similar problem for shear-thinning
and viscoelastic fluids (Vahabi and Sadeghy, 2013; 2014).
Further interest in this numerical method can be attributed
to its ease of handling the boundary conditions in free-sur-
face flows. Smoothed-particle hydrodynamics is a fully-
Lagrangian mesh-less technique in which particles serve
μ∞
γ·
γ·
x*
= x
D0
------, ∇*
= ∇
D0
1–--------, t
*=
t
D0/g--------------, V
*
= V
gD0
-------------, p*=
p
gρLD0
---------------, ρ*=
ρρL
-----,
γ·*=
γ·
g/D0
--------------, μ*=
μμL,0
--------, κ*=
κ
D0
1–--------, δ*
= δ
D0
1–--------, a
*=
a
g/D0
--------------, τ˜
*=
τ˜
μL,0 g/D0
-----------------------
dρ*
dt*
-------- + ρ* ∇*V
*
⋅( ) = 0
ρ*dV
*
dt*
--------- = ∇*
– p*+
1
Re------∇* τ
˜
*⋅1
Bo-------– κ*δ*
n + ρ* δ*
Re = ρLD0 gD0
μL,0
-------------------------
Bo = ρLgD0
2
σ---------------
D0/g
dx*/dt
* = V
*
Kayvan Sadeghy and Mohammad Vahabi
210 Korea-Australia Rheology J., 28(3), 2016
as entities conveying fluid properties. At any other point
in the computational domain, field variables are computed
by averaging (or smoothing) particle values over the region
of interest. Although it should be conceded that SPH can
be more expensive than Eulerian methods, its flexibility in
handling complex geometries and free-surface flows (for
both Newtonian and non-Newtonian fluids alike) makes it
an attractive computational method. In our recent publi-
cations (Vahabi and Sadeghy, 2013; 2014) this numerical
method has been described in great details, and so it will
not be repeated here. The code had to be slightly modified
to handle the Moore fluid though (Vahabi, 2015). Our
SPH code so-developed could easily recover Newtonian
data obtained by Hysing et al. (2009) for the two-dimen-
sional bubbles rising in Newtonian fluids. They proposed
a particular configuration as a benchmark problem to eval-
uate the performance of a finite-element/level-set in-house
code (FEM/LES) with two commercial CFD codes: Fluent
and Comsol. The computational domain proposed by them
for this benchmark problem, showing also the Cartesian
coordinate system, has been shown in Fig. 1. This figure
also shows the boundary conditions used for the simula-
tions (i.e., no-slip and no-penetration at the top and bottom
walls, and free-slip and no-penetration at the side walls).
The gas bubble is circular in shape and planar having a
dimensionless radius of 0.25; it is placed at the location
0.5, 0.5, as shown in Fig. 1. Fig. 2 shows a comparison
between our results and the results reported by Hysing et
al. (2009) for Re = 35 and Bo = 125. We have relied on
80×160 particles for these set of simulations for Newto-
nian fluids − this grid might not be adequate for the Moore
fluid; see below. As can be seen in Fig. 2, the two sets of
results are virtually the same. Evidently, our SPH code is
doing a nice job as far as the bubble’s shape is concerned
in a Newtonian fluid. This is found to be also true for the
bubble's center-of-mass location and its rise velocity
(Vahabi, 2015).
4. Results and Discussions
Having verified the code with Newtonian results, we are
now ready to present our new results in which we address
the effects of a fluid’s thixotropic behavior on the shape
and velocity of a rising bubble. We are going to rely on
the same boundary and initial conditions as used by Hys-
ing et al. (2009). However, the domain's size was increased
(see Fig. 1) from those adopted by Hysing et al. (2009)
when it was realized that, in the Moore fluid, the bubble
moves too fast. We only present a summary of the results
− the reader is referred to Vahabi (2015) for more results.
As mentioned in the introductory section, the main objec-
tive of the work is to investigate the role played by the
fluid's thixotropy number, b/a, on the bubble motion. For
most industrial fluids, this ratio is larger than one. For
example, for a commercial waxy crude oil produced in the
Middle East, based on rheological measurements, it was
estimated to be roughly 7 (Salehi-Shabestari et al., 2016).
There are also waxy crude oils in other parts of the world
having a b/a close to 100 (Mendes et al., 2015). It is also
known that for certain thixotropic polymeric solutions
(namely, Laponite/CMC blend) this ratio is smaller than
one (Escudier et al., 1995). For this reason, we have
decided to cover a wide range of the b/a ratio (say, in the
range of 0.1 to 100). As to the properties of the gas, we
assume that its density and viscosity are fixed at: ρg = 1
kg/m3, μg = 1.8×10−5 Pa·s (Hysing et al., 2009). The den-
sity of the liquid is set at ρL = 1000 kg/m3, but two dif-
ferent viscosity ratios, , equal to 100 and 1000 are
used for the simulation which correspond to two different
Reynolds number of 1.2 and 120. The surface tension is
μo/μ∞
Fig. 1. Initial configuration and the boundary conditions adopted
from Hysing et al. (2009) as our computational domain.
Fig. 2. (Color online) Code-verification using FEM-LS data
from Hysing et al. (2009) for Newtonian liquids (Re = 35,
Bo = 125, t = 3).
The effect of thixotropy on a rising gas bubble: A numerical study
Korea-Australia Rheology J., 28(3), 2016 211
also set at 0.002 N/m, which amounts to a fixed Bonds
number of Bo = 54. As mentioned above the bubble's ini-
tial dimensionless diameter is set at 0.5, as used in Hysing
et al. (2009). To check grid-independency, we have tried
different number of particles in the x- and y-directions.
Fig. 3a shows the effect of the number of particles on the
shape of a bubble rising in a Newtonian fluid, as studied
by Hysing et al. (2009) which was. And, in Fig. 3b its
effect on the center-of-mass of a bubble rising in a Moore
fluid has been shown (Re = 1.2, Bo = 54). These results
show that a grid size of 150×250 can ensure grid-inde-
pendent results for both fluids. To be on the safe side, in
all simulations to be reported shortly a grid comprising
180×300 particles is used, where the first number refers to
the x-direction with the second number referring to the y-
direction. Having checked grid-independency, we proceed
with presenting our new results in which the effect of the
thixotropy parameter in the Moore model, b/a, is investi-
gated on the characteristics of a rising bubble.
4.1. Effect of the thixotropy number, b/aFig. 4 shows the effect of the (dimensionless) thixotropy
number, b/a, on the bubble’s center-of-mass position obtained
at Re = 1.2. This figure also includes results obtained for
the corresponding Newtonian fluid (i.e., a fluid having the
same zero-shear viscosity, which is 0.5 Pa·s for this case).
As can be seen in this figure, the bubble moves faster in
the thixotropic fluid as compared with its Newtonian
counterpart. Interestingly, an increase in the b/a ratio is
seen to increase the rise velocity at any given time. The
difference between thixotropic and Newtonian velocity is
also increased when the b/a ratio is increased.
To interpret the above results we can resort to the vis-
cosity field around the bubble. The fact is that, a rising
bubble destroys structures while it is moving upward. This
reduces the fluid's viscosity in the vicinity of the bubble
which facilitates bubble's upward motion. The viscosity
field around the bubble shows that it is indeed surrounded
by a fluid layer having a viscosity close to its infinite-
shear viscosity (see Fig. 5a). The structural parameter fol-
lows the same trend because in a Moore fluid it is pro-
portional to the viscosity of the fluid − which is why we
have not plotted it here. So, the prediction that the bubble
should move faster in a thixotropic fluid can be argued to
be a general effect. Similarly, the prediction that by an
increase in b/a, the bubble moves faster is not surprising
either because, as mentioned earlier, Moore fluid becomes
more shear-thinning when this ratio is increased. The vis-
cosity field shown in Fig. 5a should translate into a veloc-
ity field of a similar nature, as can be seen in Fig. 5b. That
is, regions having the lowest viscosity should correspond
to regions with the highest velocity, and vice versa. Of
course, far upstream from the bubble (i.e., close to the
Fig. 3. (Color online) Effect of the grid size on the characteristics
of a rising bubble: (a) bubble's shape at t = 4 in a Newtonian
fluid, (b) time-evolution of bubble’s center-of-mass in a Moore
fluid.
Fig. 4. Effect of the thixotropy number, b/a, on the vertical posi-
tion of the center-of-mass for a rising bubble obtained at Re =
1.2, and t = 4: (a) b/a = 5; (b) b/a = 10.
Kayvan Sadeghy and Mohammad Vahabi
212 Korea-Australia Rheology J., 28(3), 2016
ceiling) the viscosity is highest and the velocity is lowest,
as can be seen in Fig. 5.
Fig. 6 shows a plot of the three stress components for a
rising bubble obtained at t = 4 and Re = 1.2. This figure
shows that the maximum stress is occurring adjacent to
the bubble, as expected. But, what is more striking is the
notion that there are regions of a negative shear and nor-
mal stress on the surface of the bubble. The prediction that
the two normal stresses in this figure (τxx, τyy) have a left/
right symmetry means that the bubble experience no side-
force during its journey; that is, it always remains in the
same course. But, there is up/down asymmetry and, at first
sight, it appears that the two normal stress might be
involved in the bubble's motion. But, when they are super-
imposed on each other, we see that they cancel each other
out. This is not surprising realizing the fact a Moore fluid
is an inelastic fluid, and so (like Newtonian fluids), the
first normal-stress-difference (i.e., the difference between
τxx and τyy) should be zero on all boundaries where the no-
slip condition is imposed. So, it is evident that it is the
shear stress and pressure distributions which are the main
cause of the bubble's deformation.
Fig. 7 shows how the shape of the bubble changes when
the thixotropy number, b/a, is increased starting from b/a
= 0.1. As can be seen in this figure, this ratio has a sig-
nificant effect on the bubble’s shape and velocity. This fig-
ure shows that at b/a = 0.1 the bubble moves very slowly.
This can be attributed to the fact that, for small b/a ratio
the structure breakdown is very weak so that the viscosity
of the fluid in the vicinity of the bubble is close to its
(large) zero-shear viscosity. On the other hand, since for
small b/a ratios the bubble needs a longer time to reach its
new equilibrium viscosity, therefore shear-thinning is not
so strong to alter the bubble's shape (see Fig. 7). Indeed,
Fig. 5. (Color online) Viscosity and velocity fields around a bubble rising in the Moore fluid (Re = 1.2, b/a = 10, t = 4).
Fig. 6. (Color online) Stress contours for a bubble rising in the Moore fluid (Re = 1.2, b/a = 10, t = 4).
The effect of thixotropy on a rising gas bubble: A numerical study
Korea-Australia Rheology J., 28(3), 2016 213
the effect of b/a become more evident when it is suffi-
ciently large. For example, at b/a = 100 we are witnessing
a significant change in the shape of bubble, which also
moves quite fast. To explain these results, it should be
noted that at very large b/a ratios, the fluid is basically
behaving like a shear-thinning fluid, and shear-thinning
facilitates bubble motion. Lowering b/a weakens this effect.
This notion can best be seen in Fig. 8 which shows results
at b/a = 10 instead of b/a = 100.
4.2. Effect of the reynolds number, ReIn the above simulations we have mostly focused on the
effect of the b/a ratio on the results. The simulations were
carried out at a Reynolds number of 1.2 which is typical
of the real world applications. To see the effect of the
Reynolds number, we have also carried out simulations at
Re = 120. Fig. 9 shows results for the center-of-mass of
the bubble as a function of time. In this figure results for
Fig. 7. The effect of the thixotropy number, b/a, on the shape of the bubble (Re = 1.2 and Bo = 54).
Fig. 8. Bubble’s shape during its rise in the Moore fluid (Re = 1.2, b/a = 10).
Fig. 9. A comparison between bubble rise in the Moore fluid and
in a Newtonian fluid (Re = 120, b/a = 10, t = 4).
Kayvan Sadeghy and Mohammad Vahabi
214 Korea-Australia Rheology J., 28(3), 2016
its Newtonian counterpart (having a viscosity equal to
0.005 Pa·s) have also been included for comparison pur-
poses. As can be seen in this figure, there is not much to
separate Newtonian results from the Moore results at this
Reynolds number. This prediction looks peculiar at first
sight. But, when we look at the shape of the bubble in Fig.
10, we realize that the bubble has lost its integrity at this
Reynolds number. That is to say that, smaller bubbles are
formed at the expense of the main (big) bubble. Formation
of such satellite bubbles are more evident in Fig. 11. Such
tiny bubbles, move upward with a smaller velocity as
compared with the main bubble. In practice, their net
effect is such that the center-of-mass of the whole system
moves with a velocity closer to its Newtonian counterpart
(see Fig. 9).
A comparison between the bubble's rise velocity at Re =
120 and Re = 1.2 (at the same b/a = 10) reveals that the
bubble moves much faster when the Reynolds number is
increased. The viscosity field in Fig. 10 (left plot) shows
that the bubble is completely surrounded by a low-viscos-
ity region, unlike the case for Re = 1.2 (see Fig. 5). The
formation of a wake is also evident in the velocity field in
this figure (right plot). Formation of the wake can better
be seen in Fig. 11.
Fig. 12 shows the time-evolution of the bubble’s shape
during its rise in a Moore fluid obtained at b/a = 10 for
two different Reynolds numbers of 1.2 and 120. The bub-
ble is seen to move faster when the Reynolds number is
increased. This is quite surprising realizing the fact that
the bubble shape is more flattened at Re = 120 inferring a
larger drag. But, if we look at the definition of the Reyn-
olds number (see Eq. (5)) we realize that an increase in the
Reynolds number is tantamount to saying that the zero-
shear viscosity of the fluid has been dropped − because
Fig. 10. (Color online) Viscosity and velocity fields around a bubble rising in the Moore fluid (Re = 120, b/a = 10, t = 4).
Fig. 11. Bubble’s shape during its rise in the Moore fluid (Re = 120, b/a = 10).
The effect of thixotropy on a rising gas bubble: A numerical study
Korea-Australia Rheology J., 28(3), 2016 215
the other parameters are fixed. Therefore, bubble's initial
acceleration and eventual upward motion is facilitated by
an increase in the Reynolds number, as can be seen in
Fig. 12.
5. Concluding Remarks
In this study, the motion and deformation of a single gas
bubble rising in a thixotropic fluid obeying the Moore
model was numerically simulated using SPH method. It
was found that the destruction-to-rebuild parameter (b/a)
in the Moore model could significantly affect the bubble's
dynamics provided that it is sufficiently large. An increase
in the b/a ratio was also shown to boost the bubble's
upward motion. The effect was attributed to a drop in the
fluid’s viscosity adjacent to the bubble which is intensified
when this ratio is increased (i.e., when the destruction of
microstructures is intensified). Since for large values of
the b/a ratio Moore fluid's shear-thinning prevails over its
thixotropic behavior, the bubble behaved as though it was
moving in a shear-thinning fluid. To address the effect of
thixotropy more clearly (at the expense of shear-thinning),
simulations were carried out at sufficiently small b/a ratios
such that the time needed by the fluid for its viscosity
change to occur increases. It was found that the change in
the bubble's shape is minimal at short times, and it moves
upward with a small rise velocity. This behaviour was
attributed to the fact that by lowering the b/a ratio, the
rebuild mechanism becomes more important so that it can
somewhat compensate the structure breakdown. There-
fore, within the time scale of the flow, the bubble behaved
as though it was moving in a Newtonian fluid having a
viscosity equal to the zero-shear viscosity of the Moore
fluid. An increase in the Reynolds number was found to
allow the bubble to rise faster even though its shape was
more flattened. This trend was attributed to the notion that
an increase in the Reynolds number means a decrease in
the fluid's zero-shear viscosity − an effect which facilitates
the bubble's upward motion. All in all, it is concluded that
it is the shear-thinning behavior of the Moore model (not
its time-dependent behavior) which significantly controls
the bubble's upward motion. In other words, in competi-
tion with shear-thinning, thixotropy plays a secondary role
in the bubble rise problem, as is often found to be the case
(see, for example, Escudier et al., 1995). A major problem
with the Moore model is that its b/a ratio simultaneously
controls shear-thinning and thixotropy, and this makes
interpretation of the results quite difficult. Future works
should preferably resort to rheological models which sep-
arate thixotropy from other non-Newtonian effects. Thixo-
tropic models such as Dullaert-Mewis (Dullaert and Mewis,
2006), or de Souza Mendes (de Souza Mendes, 2009)
appear to be a better choice rather than the Moore model
for fundamental studies in this area.
Acknowledgement
The authors wish to express their sincerest thanks to Iran
National Science Foundation (INSF) for supporting this
work under contract number 95815139. Special thanks are
also due to the respectful reviewers for their constructive
and encouraging comments.
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