A review on rising bubble dynamics in viscosity-stratified fluids

Kirti Chandra Sahu∗

Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Sangareddy 502 285, Telangana, India

Systems with a bubble rising in a fluid, which has a variation of viscosity in space and timecan be found in various natural phenomena and industrial applications, including food processing,oil extraction, waste processing and biochemical reactors, to name a few. A review of the aspectsstudied in the literature on this phenomenon, the gaps that exist and the direction for furthernumerical and experimental studies to address these gaps is presented.

INTRODUCTION

The fluid dynamics of a gas bubble rising due to buoy-ancy in a surrounding liquid (as shown in Fig. 1) hasbeen studied from many centuries ago (the first refer-ence in this subject goes back to Leonardo Da Vinci inthe 1500s). It continues to be a problem of interest eventoday due to its relevance in many industrial and nat-ural phenomena (see e.g. [1–3]). This includes aerosoltransfer from sea, oxygen dissolution in lakes due to rainand electrification of atmosphere by sea bubbles, in bub-ble column reactors, in microfluidics, in the petroleumindustry, for the flow of foams and suspensions and incarbon sequestration, to name a few. In many of theseprocesses, viscosity of the fluid surrounding the bubbleis not constant, but varies with space and time. Thisviscosity variation in turn greatly affects the rising dy-namics of a gas bubble as compared to the ideal case ofconstant viscosity and density of the surrounding fluid.The latter case (ideal case) has been frequently studiedin the past (see for instance [4]), and we hereafter termedit as “standard system”. However, in real situations, dueto the inherent presence of temperature variation and/orconcentration gradient of species in the above-mentionedprocesses, the viscosity of the surrounding fluid, as wellas the surface tension at the interface separating the flu-ids, can vary drastically in space and time. For example,in the ice-cream and lotion industries, by the addition ofonly 2% carboxymethylcellulose (CMC) the viscosity ofwater or milk is made to increase by almost three ordersof magnitude, while keeping the density constant [5]. Thepresence of temperature and concentration gradients alsocreates a variation of the surface-tension normal and tan-gential to the interface separating the fluids, which leadsto another class of complex flows in the vicinity of the in-terface, commonly known as Marangoni flows [6–8]. Yetanother class of fluids are non-Newtonian fluids, wherethe viscosity depends on shear rate and/or yield stress,which in turn can be a function of space and time in adynamically changing system [9].

The study of dynamics of an air bubble rising in liquidsis difficult both numerically and experimentally, due tothe high contrasts in fluid properties, dynamically chang-ing interface between the fluids, presence of interfacialtension, etc. The time and length scales associated with

this complex phenomenon, particularly at the time oftopological change and break-up are very small. Thiscreates extra difficulties in numerical simulations and ex-periments. Thus, a great variety of numerical methodshave been proposed, which range from boundary-fittedgrids [10, 11], to the level-set method [12, 13], the VOFmethod [14], diffuse-interface method [15], coupled level-set and volume-of-fluid method [16], and hybrid schemesof the lattice-Boltzmann and the finite difference method[17]. Also in the recent years, with the development ofpowerful high-speed cameras and sophisticated experi-mental facilities, several researchers continue to performexperiments in this area (see e.g. [18, 19]). The firstsystematic experiment of a rising air bubble in liquidswas conducted by Bhaga & Weber [20]. As this review islimited to summarise the physical phenomena associatedwith the rising of an air bubble in viscosity-stratified flu-ids, the reader is referred to the abovementioned papersfor the details on various numerical methods and exper-imental techniques used to study this complex problem.

In this review, the fluid dynamics of a gaseous bub-ble inside viscosity-stratified fluids is discussed. In orderto keep the system simple and isolate the influence ofviscosity stratification only, the density of the fluids areassumed to be constant and the flow is assumed to beincompressible. Also this review is limited to the stud-ies of single bubble and two bubbles only. There arealso several studies in the literature, which consideredthe dynamics of many bubbles simultaneously (e.g. stud-ies associated with bubble-column reactors). The studiesinvolving many bubbles are not discussed in the presentreview.

Brief discussion on “standard” systems (constantviscosity of the surrounding fluid)

Before discussing what happens in viscosity-stratifiedsystems, it is useful to understand the bubble rise phe-nomena in the “standard” systems. Here, we will alsodefine some dimensionless parameters, which will beused throughout this review. In the dimensionless for-mulation, the rising bubble problem is completely de-scribed by four non-dimensional parameters, namely, theGalilei number Ga(≡ ρBV R/µB), the Eotvos number

2

FIG. 1. Schematic diagram of a bubble (fluid ‘B’) rising insidea viscosity stratified medium (fluid ‘A’) under the action ofbuoyancy. The gravity is acting in the negative z-direction.

Eo(≡ ρBgR2/σ), the density ratio ρr ≡ ρA/ρB , and the

viscosity ratio µr ≡ µA/µB , wherein σ is surface tension,µA, ρA and µB , ρB are the viscosity and density of fluids‘A’ and ‘B’, respectively (as shown as Fig. 1). V =

√gR,

wherein g is the acceleration due to gravity and R is theequivalent radius of the bubble. Another dimensionlessnumber, the Morton number, Mo ≡ Eo3/Ga4, that de-pends on fluid properties only, but not on the radius ofthe bubble, will also be used sometime in the followingsections.

FIG. 2. Behaviour of single bubble rising in quiescent liq-uid in a “standard” system. The red dash-dotted line is theMo = 10−3 line, which separates the regions of skirted bubble(region II) and oscillatory bubble (region III). This figure istaken from Tripathi et al. [4].

A number of numerical and experimental studies havebeen conducted on single bubble rising in quiescent liquidin the “standard” systems (see for instance [20, 25, 28–32]). Most of these earlier studies investigated termi-nal velocity, deformation and path instability associatedwith the rising bubble inside another fluid. Recently,Tripathi et al. [4] revisited this problem by conducting

Eo

0.1

0.223

0.707

2.236

7.071

Ga10 0.05-5 -4 -3 -310 10 5x10 0.01 0.02 0.04

Break-up

A

BC

D

Break-up Break-up

Break-up

FIG. 3. The terminal shapes of the bubble obtained from thepresent simulations in Ga−Eo space for the “standard” sys-tem. The rest of the parameter values are ρr = 103, µ0 = 102.This figure is taken from Premlata et al. [22]. A similar fig-ure was also given by Tsamopoulos et al. [23], who conductedsteady state simulations.

three-dimensional numerical simulations, and identifiedregimes of starkly distinct behaviours in the Galilei (Ga)and Eotvos (Eo) numbers plane (as shown in Fig. 2).They found that for low Ga and Eo (region I) the bub-ble maintains azimuthal symmetry. In region II (low Gaand high Eo), the bubble forms skirt, and in region III(for high Ga and low Eo), the bubble moves in a spiralor a zigzag path. The vortex shedding is observed in thewake of region III bubbles. For high values of Ga and Eo,the bubble breaks to form satellite bubbles (region IV)or undergoes topological change to form toroidal shape(region V). However, it is to be noted here that the phasediagram presented by Tripathi et al. [4] (shown in Fig.2) corresponds to an initially spherical gaseous bubblein water, and a different initial condition may alter theboundaries separating these regions. It is an importantpoint because creating a spherical shape bubble initiallyis a difficult task in experiments, particularly for largebubbles. In another study, Tsamopoulos et al. [23] pro-vided a library of terminal shapes of a bubble rising in aliquid in the Ga-Eo plane for a “standard” system (seeFig. 3) by performing time-independent axisymmetricnumerical simulations.

Another important aspect is the influence of shapechange on the drag and lift forces acting on a risingbubble. Hadamard [24] reported that the drag force ob-served in spherical bubbles are different from that of solidspheres due to the internal flow in bubbles. Later, Taylor& Acrivos [25] analytically derived the first order correc-tion to the drag force of a slightly deformed spheroidalbubble in the creeping flow limit. For bubbles rising non-axisymmetrically (region III bubbles), a sideways motioncan ensue. The force responsible for this asymmetric mo-tion is the lift force. Both the lift and the drag forces arehighly dependent on the shape of the rising bubble. Sev-eral researchers [26, 27] have studied the variations of

3

these forces as a function of aspect ratio of the bubble.

GOVERNING EQUATIONS

An initially spherical air bubble (designated by fluid‘B’) of initial radius R, having constant viscosity µB ris-ing under the action of buoyancy in another fluid (des-ignated by fluid ‘A’) is considered, as shown in Fig. 1.Both the fluids are considered to be incompressible. Inthe Cartesian coordinates system (x, y, z), the dimension-less governing equations of mass and momentum conser-vation are given by

∇ · u = 0, (1)

ρ

[∂u

∂t+ u · ∇u

]= −∇p+

1

Ga∇ ·

[µ(∇u +∇uT )

]+

δ

Eon∇ · n− ρ~ez. (2)

In the framework of volume-of-fluid (VoF) method, theequation for the volume fraction, c, of the fluid ‘A’, whosevalues are 0 and 1 for the air and liquid phases, respec-tively, is given by

∂c

∂t+ u · ∇c = 0. (3)

Here u(u, v, w) represents velocity field, wherein u, v andw are the velocity components in the x, y and z direc-tions, respectively; p denotes the pressure field; δ is theDirac delta function; κ = ∇ · n is the curvature, n isthe unit normal to the interface pointing towards fluid‘A’, ~ez represents the unit vector in the vertically up-ward direction, and σ is the interfacial tension coefficientof the liquid-gas interface. The above equations are non-dimensionalised by using the initial radius of bubble, R,and√gR as the length and velocity scales, respectively.

The dimensionless density ρ is given by:

ρ = (1− c)ρr + c, (4)

and the viscosity field is given by:

µ = (1− c)µr + c. (5)

Although governing equations associated with a VoFmethod are given above, there are several other numeri-cal approaches, such as level-set method [12, 13], diffuse-interface method [15], coupled level-set and volume-of-fluid method [16], hybrid schemes of the lattice-Boltzmann and the finite difference method [17], etc arealso frequently used to study such interfacial problems.Each of the above-mentioned method has certain advan-tages and disadvantages, and is selected based on param-eter range considered and/or the problem at hand.

DISCUSSION

As discussed in the introduction, the objective of thiswork is to present the dynamics of a bubble in viscosity-stratified media. Three situations are considered, namely(i) variation of viscosity of the surrounding fluid in spacedue to some species, (ii) viscosity-stratification of the sur-rounding fluid due to the presence of temperature gradi-ent, and (iii) viscosity variations due the non-Newtoniannature of the fluid surrounding the bubble. These arediscussed one-by-one in the following subsections.

Viscosity variation of the surrounding fluid in space

Temperature and concentration variations lead to con-tinuous viscosity-stratifications. In such systems, the ris-ing bubble dynamics is not only affected by viscosity-stratification, but also greatly influenced by the varia-tions of surface tension across and along the interfaceseparating the fluids. There are several situations wherethe viscosity stratification is inherent, i.e., not due to thepresence of temperature and concentration gradients. Inmany applications [9], immiscible fluid-layers of almostthe same density, but significant viscosity jump are ob-served. A gaseous bubble rising through an interface sep-arating two immiscible liquids of different viscosities havebeen studied in the past by several researchers [33–35].As the bubble crosses the interface, the bottom fluid mi-grates along the wake of the bubble to the upper fluid,having a different viscosity, which in turn changes thedynamics of the bubble as compared to the “standard”system.

Below, we discuss the rising bubble dynamics in a sim-pler system, wherein the viscosity of the surroundingfluid continuously increases in the vertical direction inthe absence of Marangoni stresses, and other complicatedphysics discussed above. Consider a system, where theviscosity of the surrounding fluid increases linearly in thevertical direction, z, such that

µA = µ0(a1 + a2z), (6)

where a1 and a2 are constants and µ0 is the viscosityof surrounding fluid at a reference location, z = zi. Inthis configuration, as the viscosity of the surroundingfluid is varying, the viscosity ratio can be defined asµr0 ≡ µ0/µB . Now, let us contrast the dynamics of abubble in this viscosity stratified medium with that in a“standard” system. The parameter values considered forthis comparison are µr0 = 10−2, ρr = 10−3, Ga = 2.236and Eo = 0.04. This set of parameter values correspondsto point A in Fig. 3, which is taken from Tsamopouloset al. [23].

In Fig. 4(a) and (b), the temporal evolution of bubbleshapes are presented for Ga = 2.236 and Eo = 0.04 for

4

(a) (b)

(a) (b)

FIG. 4. Time evolution of bubble shapes for Ga =2.236, Eo = 0.04: (a) constant viscosity system (a1 = 1, a2 =0), (b) linearly increasing viscosity (a1 = 0.2, a2 = 0.2). Thisfigure is taken from Premlata et al. [22].

(a) (b)

FIG. 5. Streamlines at t = 40 for (a) constant viscosity sys-tem, and (b) linearly increasing viscosity. The rest of theparameter values the same as those used to generate Fig. 4.This figure is taken from Premlata et al. [22].

a standard system and a medium with linearly increas-ing viscosity, respectively. When we compare the shapesof the bubble at different dimensionless times, it can beseen that for the case of the linearly increasing viscos-ity medium (Fig. 4(b)) an elongated skirt (longer thanthat appears for the constant viscosity case (Fig. 4a)) isformed. This is unexpected due to the following reason.As the bubble migrates in the vertical direction the lo-

cal Ga decreases due to an increase in the local viscosityof the surrounding fluid. In Fig. 3, we can see that forEo = 0.04, with a decrease in Ga the bubble changes itsshape from a skirted (at Ga = 2.236) to a dimpled ellip-soidal (at Ga = 0.1). By lowering the value of Ga furtherwe will get a spherical shape bubble (not shown). Thisobservation is for the “standard” system. The dynamicsis different in the viscosity-stratified system, wherein asthe bubble rises in the upward direction, the less viscousfluid from the bottom part of the domain advects alongwith the recirculation region (inside the skirted region).This leads to a continuous increase in the viscosity con-trast between the inside and outside regions separatedby the skirt. The resultant stresses due to this differen-tial viscosity on both sides of the skirt forces it to curlinwards. This separation of fluids due to the skirt allowsfast recirculation to occur in the fluid captured insidethe wake of the bubble, while simultaneously allowing aslow flow outside of the skirt. This creates a contrast inthe inertia across the skirt. The streamlines at t = 40are plotted in Fig. 5(a) and (b) for the systems withconstant viscosity and linearly increasing viscosity, re-spectively. It can be seen that for the constant viscositysystem there are two recirculation zones in the wake re-gion of the bubble, whereas only one recirculation zoneappears in the in wake region (enclosed by the skirt) ofthe bubble in linearly increasing viscosity medium. Theappearance of two recirculation zones in constant viscos-ity case prevents the closing of the skirt, which can beseen in the linearly increasing viscosity medium. It is alsoto be noted here that the drag force is more in case of thebubble having a longer skirt (in case of linearly increas-ing fluid) as compared to that observed in the constantviscosity medium. This phenomenon may be beneficialto certain processes where a less viscosity fluid is requiredto be carried through a highly viscous fluid with the helpof a carrier bubble. More discussion on this subject canbe found in Premlata et al. [22].

Viscosity variation due to temperature gradient

The pioneering work of Young et al. [6] was the firstto investigate the migration of bubbles due to thermalgradient. They investigated a system where the temper-ature of the surrounding viscous fluid decreases in thevertically upward direction, and found that small bub-bles move in the downward direction, even though buoy-ancy acts in the upward direction. In this situation, thethermocapillary force and buoyancy force act in the op-posite direction. In case of a small bubble the latterforce dominates the former one, whereas for bigger bub-bles buoyancy wins, which causes the bubble to migratein the upward direction. They also provided a theoreti-cal description assuming spherical bubble and creepingflow conditions, which predicted the terminal velocity

5

0 2 4 6 8 10

t

10

11

12

13

14

15

zCG

Isothermal

Linear

Self-rewetting

FIG. 6. Migration of bubble in a isothermal system, linear(common) fluid and self-rewetting fluid for Ga = 10, Eo =10−2, ρr = 10−3, µr = 10−2, the temperature gradient, Γ =0.1 and thermal diffusivity ratio, αr = 0.04. For isothermalsystem: M1 = 0 and M2 = 0, for linear: M1 = 0.4 andM2 = 0, and for self-rewetting fluid: M1 = 0.4 and M2 = 0.2.This figure is taken from Tripathi et al. [8].

obtained in their experiment. Since then several the-oretical analyses were performed to study thermocapil-lary migration of a bubble in the limit of both small andlarge Reynolds numbers, but neglecting buoyancy [36–40]. Later, Merritt et al. [41] numerically investigatedthe dynamics of a bubble under the combined action ofbuoyancy and thermocapillarity forces. Zhang et al. [42]also performed a theoretical analysis of a rising bubble forsmall Marangoni numbers under the influence of gravity.

The surface tension of common fluids, such as air, wa-ter, and various oils, decreases almost linearly with in-crease in temperature. Here, these fluids are referred as“linear” fluids. All the above-mentioned investigationsconsidered the bubble dynamics in common/linear fluids.However, there is another class of fluid that exhibit a non-monotonic dependence of the surface tension on temper-ature. These fluids are known as “self-rewetting” fluids,which are non-azeotropic, high carbon alcohol solutions[43–47]. These fluids were first studied by Vochten andPetre [43] who observed the occurrence of a minimum insurface tension with temperature in high carbon alcoholsolutions. Later, Abe et al. [48] named them as “self-rewetting” fluids. Recently, Tripathi et al. [8] studied thebuoyancy-driven rise of a bubble inside a tube imposinga constant temperature gradient along the wall and con-taining a liquid, whose surface tension has a quadraticdependence on temperature given as:

σ = σ0[1−M1T +M2T

2], (7)

where σ0 is the reference value of surface tension at

the reference temperature, T0. M1

(− 1σ0

dσdT |T0

)and

M2

(1σ0

d2σdT 2 |T0

)are Marangoni numbers associated with

the linear and quadratic terms, respectively. For isother-

mal systems: M1 = M2 = 0, for common or linear fluids:M1 is non-zero, but M2 is zero; and for self-rewettingfluids: both M1 and M2 are non-zero.

The temporal variations of the centre of gravity, zCG,of a rising bubble for three different cases: the isothermalcase, and the cases of a simple linear fluid, and a self-rewetting fluid are shown in Fig. 6. The temperature ofthe tube is maintained with a linear temperature profileof constant gradient Γ > 0. It can be seen that theterminal velocity is higher for the non-isothermal case ascompared to the isothermal system due to the presence ofMarangoni stresses driving liquid towards the cold regionof the tube and thereby enhancing the upward motionof the bubble. For the self-rewetting fluid in the non-isothermal case, the bubble reaches a constant speed fora certain time duration (rising phase), which is followedby downward motion (motion reversal). The bubble getsarrested eventually. This dynamics might be of interestto researchers working in microfluidics and multiphasemicroreactors.

Viscosity variation due to the non-Newtonian natureof the surrounding fluid

Another instance in which one could experienceviscosity-stratification is when the surrounding fluid isa non-Newtonian fluid. In this situation, viscosity candepend on the shear and its history [49]. The dynam-ics of a rising bubble in surrounding fluids that exhibit anon-Newtonian nature is important in many engineeringapplications, including food processing, oil extraction,waste processing and biochemical reactors. A range ofnon-Newtonian fluids can be considered, such as power-law and Bingham plastic fluids, whose behaviour is morecomplex than relatively simple fluids (such as water, sil-icone oil, etc). However, this review is restricted to onlybubble rise inside viscoplastic and shear-thinning fluidsas discussed below.

Viscoplastic fluid

Yield stress fluids or viscoplastic materials behave likeboth fluid and solid. If the stress is beyond a criticalvalue, known as yield stress, these materials flow likeliquids when subjected to stress, but behave as a solidbelow this critical level of stress. An extensive reviewon yield stress fluids can be found in [50, 51]. Bing-ham [52] was the first to propose a constitutive law todescribe yield stress fluids. Few years later, Herscheland Bulkley [53] extended this law to include the shear-thinning/thickening behaviour. According to this model,the material can be yielded (unyielded) if the stress ishigher (lower) than the yield stress of the material. Itis to be noted here that, at the boundary separating the

6Bubble rise and deformation in Newtonian and viscoplastic fluids 145

Ar

Bo (e) Bn = 0.19

10

0

1

50

500

5000

(0.000, 0.000)

(0.005, 1.2×10–4) (0.006, 4.2×10–4)

(0.000, 0.000) (0.000, 0.000) (0.000, 0.000)

(0.250, 0.006) (0.325, 0.021) (0.361, 0.052) (0.367, 0.067)

(2.488, 0.062) (3.206, 0.205) (3.548, 0.205) (3.607, 0.651)

(29.09, 1.692) (31.40, 3.943) (31.77, 5.048)

(0.007, 0.001) (0.007, 0.001)

20 40 50

Figure 10. Map of bubble shapes in a Bingham fluid as a function of the Bond andArchimedes numbers. Underneath each figure we give the corresponding Reynolds and Webernumbers (Re, We): (a) Bn = 0.01, (b) Bn = 0.05, (c) Bn = 0.1, (d) Bn =0.14 and (e) Bn = 0.19.Unyielded material is shown black, and always when it arises in contact with the bubble, butaway from the bubble only when it is close enough, Bn ! 0.14.

shapes they observed resembled an inverted teardrop. We tried to reproduce theseshapes numerically, first using the Papanastasiou model and then using the Herschel–Bulkley model which is more appropriate for the Carbopol solutions used in theseexperiments, by assuming either shapes closer to the experimental ones as initialbubble shapes to start the Newton–Raphson iterations or higher Bn. Our iterationsnever converged to such shapes, but to the shapes given in figure 10(a, b). We couldattribute the inverted teardrop shape to a number of reasons: (i) Carbopol solutionshave a small elasticity which may be important at the rear of the bubble where slowerflow takes place and closer to the axis of symmetry the flow is elongational. It is well-known that bubbles assume inverted teardrop shapes in viscoelastic fluids: Astarista& Apuzzo (1965), Pilz & Benn (2007), Malaga & Rallison (2007). (ii) Another reasoncould be that Carbopol is thixotropic, which could introduce phenomena that cannotbe predicted by viscoplastic models, Gueslin et al. (2006). (iii) A third reason could bethat, in the experiments by Dubash & Frigaard (2007), the bubbles were rising in atube with diameter not much larger than a typical bubble diameter. In such a narrowtube the fluid must flow downwards closer to the bubble surface giving it a prolateshape. If this deformation is large enough for a liquid drop in creeping flow, Koh &Leal (1989) have shown that the drop shape becomes time-dependent, forming a tailthat may constantly elongate and even break. They have also shown that the largerthe initial deformation, the smaller the capillary number required for this instability

FIG. 7. The steady state shapes and unyielded surfaces ofa bubble in Bond number and Archimedes number space forBingham number Bn(≡ τ/ρAgR) = 0.19. τ is the yield stressof the material. The black surfaces represents the unyieldedregions. The numbers written below each figures are Reynoldsand Weber numbers, respectively. This figure is taken fromTsamopoulos et al. [23].

yielded and unyielded regions, the model becomes sin-gular, where the viscosity of the fluid has an extremelysharp jump. Frigaard & Nouar [54] proposed a sim-pler way to overcome this singularity by introducing a‘regularization’ parameter. The regularised models wereused several researchers to model the viscosity of non-Newtonian fluids [23, 55, 56].

The motion of air bubbles in viscoplastic materials hasbeen studied by many research groups in the past. As-tarita & Apuzzo [57] was the first to conduct an exper-imental study on rising bubbles in viscoplastic materi-als (by considering Carbopol solutions). By varying theconcentration of Carbopol in the solution (thereby vary-ing the yield stress of the material), they investigatedthe shapes and rise velocities of the bubble. Since then,several researchers (e.g. [19, 58]) have experimentally in-vestigated the rising bubble phenomenon in viscoplasticmaterials.

Several researchers also studied this problem by con-ducting theoretical analysis (see e.g. [59]). Recently,Tsamopoulos et al. [23] performed a detailed time-independent numerical study of a bubble rise phenomena,using the regularized Papanastasiou model [60]. Theypresented a library of bubble and yield surface shapes

for a wide range of dimensionless parameters, taking intoaccount the effects of inertia, surface tension and gravity.The shapes and unyielded surfaces of a bubble rising ina Bingham fluid as a function of Bond (≡ ρAgR2/σ) andArchimedes (≡ ρ2AgR3/µ2

A) numbers are given in Fig. 7.They found that when the yield stress of the surround-ing Bingham plastic fluid is large the bubble does notmove at all. This work was followed by the study ofDimakopoulos et al. [61] who used the augmented La-grangian method to obtain a more accurate estimationof the stopping conditions than Tsamopoulos et al. [23].It is to be noted here that Tsamopoulos et al. [23] werenot able to calculate steady shapes for a certain set of pa-rameters, which is probably an indication that the flowmay have become time-dependent in these conditions.Potapov et al. [62] and Singh & Denn [63] investigatedthe bubble dynamics in viscoplastic fluids through time-dependent simulations in the creeping flow conditions.Recently, Tripathi et al. [56] performed unsteady simu-lations in the inertial regime of the buoyancy-driven riseof an air bubble inside an infinitely extended viscoplasticmedium by considering the regularized Herschel-Bulkleymodel for the surrounding fluid, which is given by

µ =τ0

Π + ε+ µ0 (Π + ε)

n−1, (8)

where τ0 and n are the yield stress and flow index, re-spectively. ε is a small regularization parameter, and

µ0 is the fluid consistency; Π = (EijEij)1/2

is the sec-ond invariant of the strain rate tensor, wherein Eij ≡12 (∂ui/∂xj + ∂uj/∂xi). They found that in the presenceof inertia and in the case of weak surface tension the bub-ble does not reach a steady state and the dynamics maybecome complex for sufficiently high yield stress of thematerial.

Shear-thinning fluid

The peculiar physics associated with rising bubble inshear-thinning fluid due to the viscosity variation nearthe bubble surface has been a subject of research of sev-eral researchers [64, 65]. An extensive review on thissubject is given by Chhabra [66, 67]. Various numericalmethods, such as volume-of-fluid (VoF) [68, 69], level set[70] and lattice Boltzmann [71] have been developed toimprove the accuracy of the numerical procedure to inves-tigate these problems. Recently, Zhang et al. [18] exper-imentally and numerically investigated the dynamics ofa rising bubble in a quiescent shear-thinning fluids, suchas solutions of carboxymethyl cellulose (CMC), sodiumhydroxyl-ethyl cellulose (HEC) and xanthan gum. Theyused the Carreau rheological model to describe the vis-cosity of the surrounding fluids. They found that theflow patterns around the bubble in shear-thinning fluidsare very different from the one observed when the sur-

7

560 L. Zhang et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 555–567

Eq. (7) is fitted to the experimental data, which is represented inFig. 4 by full lines. The other solutions studied are not plotted forbetter visualization. All the solutions exhibited a shear-thinningbehaviour. The parameter values of the Carreau model and otherphysical properties under test conditions are listed in Table 2.

In this study, the terminal rising velocity of air bubbles in 5 non-Newtonian liquids with shear-thinning property are determined

and these bubbles are simulated using the present numerical pro-gram. Table 3 shows the numerical and experimental results for asingle bubble rising freely in pure shear-thinning non-Newtoniansolutions. The calculated terminal velocities are in close agreementwith the present experimental values with the relative deviationbelow 10%. The satisfactory agreement indicates the validity ofnumerical procedure for the numerical simulation of a single bub-

Fig. 5. Viscosity distribution and shape of a bubble rising in solution. (a) System 1: Newtonian fluid, ReM = 2.24, !0 = 0.5 Pa s. (b) System 2: Carreau shear-thinning fluid,ReM = 6.65, !0 = 0.5 Pa s. (c) System 3: Carreau shear-thinning fluid, ReM = 30.69, !0 = 0.5 Pa s. (d) System 4: Carreau shear-thinning fluid, ReM = 48.8, !0 = 0.5 Pa s. (e) System 5:Carreau shear-thinning fluid, ReM = 55.5, !0 = 0.1 Pa s. (f) System 6: Carreau shear-thinning fluid, ReM = 80.85, !0 = 0.5 Pa s.

FIG. 8. A typical distribution of viscosity and shape of thebubble in Carreau shear-thinning fluid. This figure is takenfrom Zhang et al. [18].

rounding liquid is a Newtonian fluid. They also foundthe viscosity in the wake region is large as compared tothe other part of the surrounding fluid, which leads tothese differences in the flow pattern and deformation ofthe bubble as compared to those in the Newtonian case.The viscosity variation along with the shape of the bubbleshowing the above mentioned features is presented in Fig.8. There are also few studies (see e.g. [72, 73]), whichinvestigated the dynamics of a pair of bubbles aligned indifferent ways inside non-Newtonian fluids. They foundthat the irregularity of the bubble shape (deviation fromthe spherical shape) increases with an increase in theshear thinning tendency. This effect was found to besignificant for flow index n < 0.5.

CONCLUDING REMARKS

In this review, the dynamics of bubble rise in viscosity-stratified media is discussed. Many peculiar phenom-ena observed in such systems are presented, which arefound to be very different from those observed whenthe surrounding fluid has a constant viscosity. Threeconfigurations, wherein viscosity-stratifications are dueto the presence of a species, temperature gradient andnon-Newtonian surrounding fluids, are considered. Al-though, dynamics of rising bubble in constant viscosityfluids has a long history, the dynamics in viscosity strati-fied media has not been investigated to that detail due tothe requirement of powerful computers and sophisticatedexperimental techniques. The numerical simulations ofsuch complex phenomena involving high density and vis-cosity ratios along with other physics are extremely dif-ficult and computationally very expensive. Due to the

improvement of advanced scientific computing resourcesand the development of sophisticated experimental tech-niques, it is possible to study such complex phenomenato greater depth at present.

ACKNOWLEDGEMENTS

The author gratefully acknowledges Prof. Karri Badar-inath of IIT Hyderabad for reading the manuscript andproviding valuable suggestions. I also thank my PhD stu-dent A. R. Premlata for her help in re-plotting some ofthe results presented in this review.

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