The effect of thixotropy on a rising gas bubble: A ... · PDF fileIt is also known that in Newtonian fluids bubbles deform from a spherical shape to an oblate ellipsoidal shape and

© 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

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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


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

*



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).



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.



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 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).



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 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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