Rising Bubbles and Falling Drops

Rising bubbles and falling drops

Manoj Kumar Tripathi

A Thesis Submitted to

Indian Institute of Technology Hyderabad

in Partial Fulﬁllment of the Requirements for

the Degree of Doctor of Philosophy

Department of Chemical Engineering

Indian Institute of Technology Hyderabad

February 2015 Declaration

I declare that this written submission represents my ideas in my own words, and where ideas or words of others have been included, I have adequately cited and referenced the original sources. I also declare that I have adhered to all principles of academic honesty and integrity and have not misrepresented or fabricated or falsiﬁed any idea/data/fact/source in my submission. I understand that any violation of the above will be a cause for disciplinary action by the Institute and can also evoke penal action from the sources that have thus not been properly cited, or from whom proper permission has not been taken when needed.

————————– (Signature)

————————— ( Manoj Kumar Tripathi)

—————————– (Roll No.) Approval Sheet

This Thesis entitled Rising bubbles and falling drops by Manoj Kumar Tripathi is approved for the degree of Doctor of Philosophy from IIT Hyderabad

————————– (———-) Examiner Dept. of Chem Eng IITM

————————– (———-) Examiner Dept. Math IITH

————————– (Dr. Kirti Chandra Sahu) Adviser Dept. of Chem Eng IITH

————————– (Dr. Rama Govindarajan) Co-Adviser Tata Institute of Fundamental Research Center for Interdisciplinary Sciences

————————– (———) Chairman Dept. of Mech Eng IITH Acknowledgements

Thanks to the inspirations which affected my choices and others’ actions to bring me where I am. Thanks to all the wonderful people I have come in contact with, starting from my parents, my brother and my sister. I will always be grateful to my mother and father for the sacrifices they have made for me. I could never imagine writing a PhD thesis without their hard work. I have been very lucky to get good teachers who taught me many things including the things that were outside school curricula. Also, I have been lucky to come to the Indian Institute of Technology Hyderabad and stay here for a PhD, as these have probably been the most defining years for me. I have been blessed with really good association for which I am very grateful. Ashwani, Chhavikant, Priyank and Varun, who were practically my roommates, entertained and pulled legs of each other, sang weird stuff on the tune of famous songs, made diaries for counting cuss words uttered by us, and had philosophical discussions among many other things. My colleagues, Prasanna didi and Ashima, who had their tables next to mine were the people I talked to about many things, took help in plotting, helped in scripting among other useless (or was it?) chit-chat. All of this made my PhD seem so smooth and memorable. I am very grateful to have associated with Prof. Rama Govindarajan and Prof. Kirti Sahu. Thanks, Rama Madam, for allowing me to be your student and to teach me many important things just by being yourself. Thank you, Sahu sir, for pushing me when I got lazy. Thanks to Professor Mahesh Panchagnula for inviting me to his lab to conduct experiments on bubbles and drops. This experience was like a crash-course in experimental methods for me, and the discussions with Prof. Panchagnula, his students and other lab staff have been very beneficial. Special thanks to Stephane Popinet and others for developing such a wonderfulfluidflow solver - gerris, and other open-source community members who submitted important patches to the code and gave their inputs for the development of this code.

iv Dedication

To my parents who made me able to write this.

v Abstract

The fascinating behaviour of bubbles and drops rising or falling under gravity, even without the presence of any impurities or other forces (such as electric, magnetic and marangoni forces), is still a subject of active research. Let alone a unified description of the dynamics of bubbles and drops, a full description of a single bubble/drop is out of our reach, as of now. The thin skirted bubbles, for instance, may rise axisymmetrically or may have travelling waves in azimuthal or vertical direction; may or may not remain axisymmetric; may eject satellite bubbles, or they may form wrinkles in their skirt. The length scales may vary across 3 or more orders of magnitude. A rising bubble may change its topology to become a toroidal bubble and become unstable to break into smaller bubbles, which may further break into even smaller bubbles. Bubbles which attain a terminal shape and velocity may change theirfinal behaviour depending on the initial conditions of release. Ellipsoidal bubbles, released axisymmetrically, may often take a zigzag or a spiral path as they rise. On the other hand, drops have a completely different dynamics. Drops have been studied due to their importance in atomization, rain drop size distribution, emulsification and many other problems of industrial importance. Apart from the low Reynolds number regime and density ratios close to 1, any literature seldom compares bubbles and drops because of the inherent difference in their dynamics. The reason for this difference has been investigated in thefirst part of this thesis. We show that a bubble can be designed to behave like a drop in the Stokesflow limit when the density of the drop is less than 1.2 times that of the outerfluid. It has been shown that Hadamard’s exact solution for zero Reynolds number yields a better condition for equivalence between a bubble and a drop than the Boussinesq condition. Scaling relationships have been derived for density ratios close to unity for equivalence at large inertia. Numerical simulations predict a similar equivalence for large inertia as well. For density ratios far from unity, bubbles and drops are very different. Axisymmetric numerical simulations show that the vorticity tends to concentrate in lighterfluid, which manifests into a totally different dynamics for bubbles and drops. This is the reason for thin trailing end of the drops and thick base of bubbles, which result in a peripheral breakup of drops, but a central breakup of bubbles at large inertia and low surface tension. The three dimensional nature of the bubbles and drops has been studied next. We present the results of one of the largest numerical study of three-dimensional rising bubbles and falling drops. We show that as the size of the bubble is increased, the dynamics goes through three abrupt transitions from one known class of shapes to another. A small bubble will attain an axially symmetric equilibrium shape dictated by gravity and surface tension, and travel vertically upwards at a terminal velocity thereafter. A bubble larger than afirst critical size loses axial symmetry. We show that this can happen in two ways. Beyond the next critical size, it breaks up into a spherical cap and many satellite bubbles, and remarkably, the cap regains axial symmetry. Finally, a large bubble will prefer not to break up initially, but will change topologically to become an annular doughnut-like structure, which is perfectly axisymmetric. A central result of this work is to characterise the bubble motion according to their mode of asymmetry, and mode of breakup. Some preliminary results of three-dimensional drop simulations show that the effect of density ratio is to increase the inertia of the drop which changes the way a drop breaks up. The effect of viscosity ratio was found to delay the breakup of a drop. Also, this study confirms that a drop breaks up from the sides while a bubble breaks up from the center for high inertia and low surface tension. Next, we examined the buoyancy-driven rise of a bubble inside an infinite viscoplastic medium,

vi assuming axial symmetry. Our results indicate that in the presence of inertia and in the case of weak surface tension the bubble does not reach a steady state and the dynamics may become complex for sufficiently high yield stress of the material. Past researchers had assumed the motion to be steady or in the creepingflow regime, whereas we show that for low surface tension and large yield stresses, the bubble exhibits a periodic motion along with oscillations in bubble shape. These oscillations are explained by the periodic formation and destruction of an unyielded ring around the bubble. Another physics often encountered in bubble/drop motion is that of heat transfer. A curious case is that of self-rewettingfluids which have been reported to increase the heat transfer rate significantly in heat-pipes. Rising bubble in a self-rewettingfluid with a temperature gradient imposed on the container walls has been studied. To account for the non-monotonicity of surface tension we consider a quadratic dependence on temperature. We examine the Stokesflow limitfirst and derive conditions under which the motion of a spherical bubble can be arrested in self-rewetting fluids even for positive temperature gradients. Our results indicate that for self-rewettingfluids, the bubble motion departs considerably from the behaviour of ordinaryfluids and the dynamics may become complex as the bubble crosses the position of minimum surface tension. Under certain circumstances, motion reversal and a terminal location is observed. The terminal location has been found to agree well with the analytical result obtained from the Stokes solution. Also, a taylor bubble is formed when the confinement is increased, thus implying a higher heat transfer rate to the gas slug inside the tube. Finally, the effect of evaporation in ambient conditions was examined. To this end, a phase- change model has been incorporated to gerris (open sourcefluidflow solver) in order to handle the complex phenomena occurring at the interface. We found that the vapour is generated more on the regions of the interface with relatively high curvature, and the vapour generation increases with breakup of the drop. Furthermore, a competition between volatility and the dynamics governs the vapour generation in the wake region of the drop. This is an ongoing work, and only a few results have been presented.

vii List of Publications

Journal Papers (Published/Accepted) 1. “Dynamics of an initially spherical gas bubble rising in a quiescent liquid (2015)”, M. K. Tripathi, K. C. Sahu and R. Govindarajan, Nature Communications,6, 6268.. 2. “Non-isothermal bubble rise: non-monotonic dependence of surface tension on temperature (2015)”, M. Tripathi, K. C. Sahu, G. Karapetsas, K. Seﬁane and O. K. Matar, Journal of Fluid Mechanics, 763, 82-108. 3. “Why a falling drop does not in general behave like a rising bubble (2014)”, M. Tripathi, K. C. Sahu and R. Govindarajan, Scientiﬁc Reports (Nature Publishing Group),4, 4771. 4. “Bubble rise dynamics in a viscoplastic material”, M. K. Tripathi, K. C. Sahu, G. Karapetsas and O. K. Matar, Journal of Non-Newtonian Fluid Mechanics, accepted - 2015.

Conference Proceeding 5. “Evaporating falling drops”, M. K. Tripathi and K. C. Sahu, IUTAM Symposium on multiphase flows with phase change: Challenges and opportunities, 8-11 December 2014, in Hyderabad, India. Journal Papers (submitted/under preparation) 6. “Bubble rise dynamics in viscosity stratified medium”, Premlata A. R., M. K. Tripathi and K. C. Sahu, submitted to Physics of Fluids. 7. “Solutal marangoni effects on an octanoic acid drop rising in water”, K. Swaminathan, M. K. Tripathi, K. C. Sahu, M. V. Panchagnula and R. Govindarajan, under preparation. 8. “Effect of evaporation on falling drop dynamics”, M. K. Tripathi, K. C. Sahu and R. Govin- darajan, under preparation. 9. “Stability of double-diffusive displacementflow in three-dimensions”, K. Bhagat, M. K. Tri- pathi and K. C. Sahu, under preparation. 10. “Bubble dynamics in a pressure driven wavy-walled channel”, H. Konda, M. K. Tripathi and K. C. Sahu, under preparation.

viii Contents

Declaration ...... ii Approval Sheet ...... iii Acknowledgements ...... iv Abstract ...... vi

Nomenclature x

1 Introduction and previous work 1 1.1 Background and motivation ...... 1 1.2 Literature review ...... 4 1.3 Outline of the thesis ...... 11 1.4 Future work ...... 12

2 Formulation and numerical methods 13 2.1 Formulation ...... 13 2.2 Numerical methods ...... 16 2.2.1 Diffuse-interface method ...... 17 2.2.2 Volume offluid method: Gerris ...... 18 2.3 Validation ...... 18 2.3.1 Grid convergence test ...... 18 2.3.2 Effect of domain size ...... 19 2.3.3 Comparison with numerical simulations ...... 21 2.3.4 Comparison with the experimental result of Bhaga & Weber [1] ...... 21 2.3.5 Comparison with analytical results ...... 23 2.4 Effect of regularization parameter ...... 25

3 Bubbles and drops: Similarities and differences 28 3.1 Introduction ...... 28 3.2 In Hadamardflow regime ...... 28 3.3 Bigger bubbles and drops ...... 34 3.4 Differences in bubble and drop dynamics ...... 37 3.5 Before breakup ...... 44 3.5.1 Effects of viscosity ...... 46 3.5.2 Drop breakup ...... 46 3.6 Summary ...... 47

ix 4 Three dimensional bubble and drop motion 49 4.1 Introduction ...... 49 4.2 Bubbles ...... 50 4.2.1 Regimes of diﬀerent behaviours ...... 50 4.2.2 Path instability and shape asymmetry ...... 53 4.2.3 Breakup regimes ...... 57 4.2.4 Upward motion ...... 59 4.3 Determination of the behaviour type ...... 60 4.3.1 Shape analysis ...... 60 4.3.2 Energy analysis ...... 61 4.4 Drops ...... 61 4.5 Conclusions ...... 63

5 Bubble rise in a Bingham plastic 66 5.1 Introduction ...... 66 5.2 Formulation ...... 67 5.3 Results ...... 68 5.3.1 Discussion ...... 70 5.4 Conclusions ...... 74

6 Non-isothermal bubble rise 78 6.1 Eﬀect of temperature gradients ...... 78 6.2 Formulation ...... 78 6.3 Analytical results: Stokesﬂow limit ...... 81 6.4 Numerical results ...... 86 6.5 Concluding remarks ...... 97

7 Evaporating falling drop 98 7.1 Introduction ...... 98 7.2 Formulation ...... 98 7.2.1 Evaporation model ...... 99 7.2.2 Model implementation in gerris...... 100 7.3 Results: Evaporating falling drops ...... 101 7.4 Future work ...... 103

8 Conclusions 104

References 106

x List of Figures

1.1 Figure showing a variety of bubbles and drops observed in experiments. (a) A train of air bubbles rising in water for a constantﬂow rate of air in the nozzle; (b) single octanoic-acid bubble in distilled water exhibiting a spiralling motion (images at diﬀerent times merged into a single image). These experiments were performed in collaboration with Prof. Mahesh Panchagnula in his lab at IIT Madras. (c) A falling water drop, from Edgerton’s book [2]; (d) a falling water drop breaking in a bag- breakup mode, courtesy E. Villermaux [3]...... 2

1.2 Visible internal circulation in a glycerine drop falling in castor oil (from the experimental study by Spells [4]). The parameter values corresponding to this experiment

are: Ga=0.792, Bo=0.1,ρ r = 1.3 andµ r = 1.24. This was theﬁrst published evidence of the internal circulation in falling drops...... 3

1.3 Density and viscosity ratios of about 1650 pairs ofﬂuids. Blue (open) and red (ﬁlled) symbols represent liquid-liquid and liquid-gas systems, respectively. It shows that the density and viscosity ratios range across 8 and 10 orders of magnitude, respectively. 4

2.1 Schematic diagram of the simulation domains considered to solve (a) axisymmetric (dot-dashed line represents the axis of symmetry), and (b) three-dimensional rising bubble problem. Bubble size is not to scale. Similar domains are considered for falling drop problem with inverted gravity. The domain is considered to have a square base of size,L in three-dimensions and a circular base of diameter,L in cylindrical coordinates. The outer and innerﬂuids are designated by ‘o’ and ‘i’, respectively. The height of the domain,H is chosen according to the expected dynamics of the bubble/drop...... 14

2.2 Eﬀect of grid reﬁnement on the shape of the bubble at (a)t = 4, and (b)t = 7 for 4 6 Ga=3.09442, Bo = 29,ρ = 7.4734 10 − andµ = 8.1536 10 − . The solid r × r × and dot-dashed lines correspond to the results obtained usingΔx=Δz=0.015 and 0.029, respectively...... 19

2.3 Grid convergence test. The shapes of the bubble for two diﬀerent grid sizes att=3 3 2 is shown. The parameter values are Ga = 70.7,Bo = 200,ρ r = 10− andµ r = 10− . The smallest grid sizes in panels (a) and (b) are about 0.029 and 0.015, respectively. The grid reﬁnement criteria used here are based on the vorticity magnitude and the

gradient of volume fraction (ca)...... 19

xi 2.4 Eﬀect of domain size on upward velocity,w of a bubble exhibiting a spiralling motion. The dashed and solid lines represent domains of base width,L = 30 and 60, respectively. The dimensionless parameters used for the simulations are: Ga = 100, 3 2 Bo=0.5,ρ r = 10− andµ r = 10− ...... 20 2.5 Eﬀect of domain size on the bubble shape att = 4, andt = 7 (left to right) for 4 6 Ga=3.09442,Bo = 29,ρ = 7.4734 10 − andµ = 8.1536 10 − . The solid and r × r × dot-dashed lines correspond to computational domains 8 24 and 16 48, respectively. × × The results are generated using square grid ofΔx=Δz=0.015...... 20 2.6 Comparison of the shape of the bubble obtained from our simulation (shown by solid red line) with those from the level-set simulations of Sussman & Smereka [5] (dashed line) at various times: (a)t = 0, (b)t=0.8, (c)t=1.6 and (d)t=2.4. The

parameter values are Ga = 100,Bo = 200,ρ r = 0.001 andµ r = 0.01. The transition to toroidal bubble (topological change) is observed att=1.6, which matches exactly with the result of Sussman & Smereka [5]...... 21 2.7 Variation of upward velocity of center of gravity of the drop with time for Ga = 219.09,

Bo = 240,ρ r = 1.15 andµ r = 1.1506. The dashed line is the result due to Han & Tryggvason [6] and the solid line is the result of the present simulation. Thefigure is plotted till breakup. It could be seen that the results match to a very good accuracy, however small deviations can be seen which may be attributed to the differences in the interface tracking/capturing methods in the two simulations...... 22 2.8 Comparison of the shape of the bubble obtained from the present diffuse interface simulation (shown by red line) with that of Bhaga and Weber [1]. The parameter 4 6 values are Ga = 3.09442, Bo = 29,ρ = 7.4734 10 − andµ = 8.1536 10 − . r × r × The dimple is not clearly visible in the experimental result because it is hidden by the periphery of the bubble. Also, the apparent dimple seen from the side view is different from the actual dimple because of the refraction due to the curved bubble surface...... 22 2.9 Comparison of terminal shape of the bubble with Bhaga & Weber [1] (left to right): Ga=2.316, Bo = 29, Ga=3.094, Bo = 29, Ga=4.935, Bo = 29, and Ga= 10.901, Bo = 84.75. The results in the bottom row are obtained from the present three-dimensional simulations. The results from the corresponding axisymmetric simulations are shown by red lines in the top row...... 23 2.10 Comparison of streamline pattern in the wake of the bubble with Bhaga & Weber [1] for the following dimensionless parameters: (a) Ga=7.9, Bo = 17, (b) Ga = 9, Bo = 21, (c) Ga = 12.6,Bo = 17, (d) Ga = 17.8,Bo = 27, (e) Ga = 21.9,Bo = 17, 3 and (f) Ga = 33.2,Bo = 11. The rest of the parameter values areρ = 7.747 10 − r × 2 andµ r = 10− . The results on the left hand side and right hand side of each panel are from the present simulations and Bhaga & Weber’s [1] experiments, respectively. 24 2.11 Streamlines obtained from the analytical result for Hadamardflow (Re 0) in a → spherical bubble (left hand side), and volume offluid simulation with gerris (right hand side) for a domain of half-width 16R. The dimensionless parameters are: Ga= 3 2 0.1,Bo=0.1,ρ r = 10− andµ r = 10− ...... 25

xii 2.12 Comparison of present numerical result with Hadamard-Rybczynski [7] theory. The terminal velocity agrees well for a domain size of 30 30 120 and for the parameter × × 3 2 values: Ga=0.1, Bo=0.1,ρ r = 10− andµ r = 10− . The center of the vortex is well predicted by our numerical simulation...... 25 2.13 The unyielded region in the non-Newtonianﬂuid (shown in black) at time,t = 10 for diﬀerent values of the regularized parameter,�: (a)�=0.01, (b)�=0.001, (c)

�=0.0001. The rest of the parameter values are Ga = 70.71,Bn = 14.213,µ r = 0.01,

ρr = 0.001,m = 1 andBo = 30. The unyielded regions for�=0.001 and 0.0001 are visually indistinguishable...... 26

2.14 (a) Temporal variation of the center of gravity (zCG), (b) the aspect ratio (h/w) of the bubble for diﬀerent values of the regularization parameter,�. The rest of the parameter values are the same as those used to generate Fig. 2.13...... 26 2.15 The bubble shape (shown by red line) and unyielded region in the non-Newtonian ﬂuid (shown in black) at time,t = 2 for (a) regularised model, (b) Papanastasiou’s model. The rest of the parameter values are�=0.001, Ga = 70.71, Bn = 14.213,

µr = 0.01,ρ r = 0.001,m = 1 and Bo = 30. The aspect ratios of the bubble in (a) and (b) are the same (h/w=1.018)...... 27

3.1 Figure taken from the poster presented on Prof. Roddam Narasimha’s 80th birthday.

Rising bubble and falling drop for parameter values: (a) Ga = 50, Bo = 29,ρ r = 4 6 7.47 10 − andµ = 8.15 10 − , and (b) Ga = 30,Bo = 29,ρ = 10 andµ = 10. 29 × r × r r 3.2 Theoretical streamlines in a spherical bubble for the Hadamardﬂow(Re << 1). The stagnation ring (center of the spherical vortex) lies at a distance of 1/√2 from the axis of symmetry...... 30 3.3 A sketch showing a spherical body falling under gravity and the forces acting on it,

wherez represents the vertical coordinate, andF b,F g andF D denote the gravitational, buoyancy and drag forces, respectively...... 30

3.4 Vertical location of the center of gravity as a function of time for a drop (ρr = 1.214

andµ r = 76), an equivalent bubble based on Boussinesq approximation (ρr = 0.786

andµ r = 76) and an equivalent bubble based on conditions (3.21) and (3.22) (ρr = 4 0.85 andµ = 0.1). The rest of the parameters are Ga = 6 andBo=5 10 − . The r × bubble designed using the Hadamard’s solution is shown to be better than the one derived using the often employed Boussinesq condition...... 33

3.5 (a) Evolution of bubble shape with time forρ r = 0.9,µ r = 0.5, Ga = 50,Bo = 50; (b)

evolution of drop shape with time forρ r = 1.125,µ r = 0.625, Ga = 50,Bo = 50. The direction of gravity has been inverted for drop to compare the respective shapes with those of the bubble. Even for high Ga andBo, the dynamics can be made similar if density ratios are close to unity...... 37

3.6 Dynamics in the absence of gravity: (a)evolution of bubble shape with time forρ r =

0.9,µ r = 0.5,Re = 50,Bo = 50, (b) evolution of drop shape with time forρ r = 1.125,

µr = 0.625, Re = 50,Bo = 50. The initial shape of both drop and bubble was kept

spherical and the initial velocity given to theﬂuid blobs isU 0 = 1 for both. The shapes of bubble and drop tend to be similar for density ratios close to unity. . . . . 38

xiii 3.7 Dynamics in the absence of gravity: (a) evolution of bubble shape with time for

ρr = 0.52,µ r = 0.05, Re = 50, Bo = 50, (b) evolution of drop shape with time for

ρr = 13,µ r = 1.25,Re = 50,Bo = 50. The initial shape of both drop and bubble was

kept spherical and the same initial velocityU 0 given to bothﬂuid blobs. The bubble regains a spherical shape, whereas the drop breaks up in the bag-breakup mode. . . 39

3.8 Evolution of (a) bubble shape with time forρ r = 0.9,µ r/ρr = 0.56. (b) drop shape

with time forρ r = 1.125,µ r/ρr = 0.56. (c) drop shape with time forρ r = 1.125, with viscosity obtained from Eq. (3.21). The direction of gravity has been inverted for the drop in order to compare the respective shapes with those of the bubble. In all three simulations, Ga = 50,Bo = 50, and the initial shape was spherical...... 40 3.9 Evolution of (a) bubble and (b) drop shapes with time, when densities of outer and innerfluid are significantly different. As before, for the drop (b), the direction of gravity has been inverted. In both simulations Ga = 50 and Bo = 10. The other

parameters for the bubble system areρ r = 0.5263 andµ r = 0.01, while for the drop

systemρ r = 10 andµ r = 0.19. Note the shear breakup of the drop at a later time. Shown in color is the residual vorticity [8]...... 41 3.10 Streamlines in the vicinity of a bubble fort = 1, 2, 3 and 4 for parameter values

Ga = 50, Bo = 10,ρ r = 0.5263 andµ r = 0.01. The bubble is shown in grey and a red outline. The circulation can be seen lying inside the bubble, which does not allow the bubble to thin out at its base...... 42 3.11 Streamlines in the vicinity of a drop fort = 1, 2, 3 and 4 for parameter values Ga = 50,

Bo = 10,ρ r = 10 andµ r = 0.19. The bubble is shown in grey and a red outline. The direction of gravity has been inverted to compare the shapes with those in Fig. 3.10. The circulation is seen to move out of the drop, making the drop to thin out at its trailing end...... 42 3.12 Evolution of (a) bubble and (b) drop (gravity reversed) shapes with time. Parameters

for both bubble and drop systems are: Ga = 100,Bo = 50 andµ r = 10. The density

ratio for the bubble and drop areρ r = 0.52 andρ r = 13 respectively, based on Eq. (3.22). Thefigure shows that the density, rather than viscosity, decides the location of vortical structures, which results in altogether different deformation in bubbles and drops...... 43 3.13 Variation of dimple distance versus time for different Bond numbers for Ga = 50, 4 6 ρ = 7.4734 10 − ,µ = 8.5136 10 − . The tendency of a bubble to break from the r × r × center is evident. However, a bubble may form a skirt for intermediate Bond numbers (Bo = 15), which may lead to breakup or shape oscillations in certain cases. . . . . 45 3.14 Streamlines in and around the bubble at time,t = 1, 1.5, 2.0 and 2.5 respectively, 4 6 for Ga = 50, Bo = 29,ρ = 7.4734 10 − andµ = 8.5136 10 − . The shape r × r × of the bubble is plotted in red. The toroidal vortex inside the bubble maintains the thickness of its base as the liquid jet penetrates the remaining airfilm at the top. . . 45 3.15 Streamlines in and around the bubble at time,t=2.5, 5, 7, 9 and 11 respectively, 4 6 for Ga = 50, Bo = 15,ρ = 7.4734 10 − andµ = 8.5136 10 − . Three toroidal r × r × vortices form inside and outside the bubble which compete with the surface tension force to make the bubble shape oscillate...... 45

xiv 3.16 Variation of dimple distance versus time for diﬀerent Gallilei numbers forBo = 8,ρ r = 4 6 7.4734 10 − ,µ = 8.5136 10 − . The bubble shapes are shown at corresponding × r × times for Ga = 5 (top) and 125 (bottom). The shape oscillations ensue after a threshold in outerﬂuid’s viscosity i.e. Ga...... 46 4 3.17 Variation of dimple distanceD versus time forBo = 29,ρ = 7.4734 10 − ,µ = d r × r 6 8.5136 10 − . Bubble shapes are shown for non-oscillating (top, black), oscillating × (blue) and breaking (bottom, black) bubbles...... 47 3.18 Streamlines in and around the drop at time,t=4.5, 6 and 7.5, respectively (from

left to right), for Ga = 50,Bo = 5,ρ r = 10 andµ r = 10. The circulation zones form outside the drop, as observed in Fig. 3.9...... 47

3.19 Variation of break-up time with Bond number for Ga = 50,ρ r = 10 andµ r = 10. A typical bag breakup mode is shown in thisﬁgure. Shapes of the drop just before breakup are shown for various Bond numbers...... 48

4.1 Different regimes of bubble shape and behaviour. The different regions are: axisymmetric (circle), asymmetric (solid triangle) and breakup (square). The axisymmetric regime is called region I. The two colors within the asymmetric regime represent non-oscillatory region II (shown in green), and oscillatory region III (blue) dynamics. The two colors within the breakup regime represent the peripheral breakup region IV (light yellow), and the central breakup region V (darker yellow). The red dash-dotted 3 line is theMo = 10 − line, above which oscillatory motion is not observed in experiments [1, 9]. Typical bubble shapes in each region are shown. In this and similar figures below, the bubble shapes have been made translucent to enable the reader to get a view of the internal shape...... 51 4.2 Dynamics expected for bubbles in different liquids. Constant Morton number lines, each corresponding to a different liquid, are overlayed on the phase-plot to demonstrate that our transitions can be easily encountered and tested in commonly found liquids. The initial radius of the air bubble increases from left to right on a given line. Circles, triangles and squares represent air bubbles of 1 mm, 5 mm and 20 mm radii, respectively...... 52 4.3 Agreement and contrast between present and previous results for differentflow regimes. Comparison between the onset of asymmetric bubble motion obtained in the numerical stability analysis of Cano-Lozano et al. [10] (solid black line), and the present boundary between regions I and II. Also given in thisfigure arefive different conditions (diamond symbols) studied by Baltussen et al. [11]. The dynamics they obtain are as follows: A - Spherical, B - Ellipsoidal, C - Boundary between skirted and ellipsoidal, D - Wobbling and E - Peripheral breakup. The correspondence between present results and [11] is excellent. Grace et al. [12] obtained spherical bubbles below the solid blue line shown...... 53 4.4 Dynamics and shapes of region III bubbles: trajectory of the bubble centroid for (a) Ga = 70.7, Bo = 10, and (b) Ga = 100, Bo = 4, and (c) shape evolution of bubble corresponding to the latter case. In panel (c), the radial distance of the center of

gravity (rs) of the bubble measured in the horizontal plane from the original location is shown below the shapes at each time...... 54

xv 4.5 Differences between two dimensional and three dimensional bubble shapes: (a) A region III bubble att = 20 for Ga = 100 andBo=0.5, (b) att = 30 for Ga = 100 andBo = 4, again in reign III, and (c) a region IV bubble att = 5 for Ga = 70.71 andBo = 20. The second row shows the side view of the three-dimensional shapes of bubbles rotated by 90 degrees about thex = 0 axis with respect to the top row. . . 55 4.6 Characteristics of a region III bubble of Ga = 100 and Bo=0.5. (a) Oscillating upward velocity, with different behaviour at early and late times, (b) trajectory of the bubble centroid. The two regions corresponding to two different behaviours in the rise velocity correspond to the inline oscillations and zig-zagging motion. . . . . 55 4.7 Region III bubble corresponding to that shown in Fig. 4.6 (Ga = 100 andBo=0.5).

(a) Iso-surfaces of the vorticity component in thez direction at timet = 15 (ω z = 0.0007) and 26 (ω = 0.006), (b) The evolution of the shape of the bubble. The ± z ± radial distance of the center of gravity (rs) of the bubble measured in the horizontal plane from the original location is shown below the shapes at each time...... 56 4.8 Time evolution of bubbles exhibiting a peripheral and a central breakup. Three- dimensional and cross-sectional views of the bubble at various times (from bottom to top the dimensionless time is 1, 2, 4 and 5). (a) region IV, a bubble breaking into a spherical cap and several small satellite bubbles, Ga = 70.7 and Bo = 20, and (b) region V, a bubble changing in topology from dimpled ellipsoidal to toroidal, Ga = 70.7 andBo = 200...... 57 4.9 A new breakup mode in region IV for Ga = 500 and Bo = 1. Bubble shapes are shown at dimensionless times (from left to right)t = 2, 4, 6, 7, 8, 9 and 9.1). . . . . 57 4.10 Comparison of our 3D results with those of Bonometti & Magnaudet [13] for bubble breakup. The light yellow and dark yellow colours represent the regions for peripheral and central breakup. The corresponding data points from the present numerical simulation are shown as blue and black squares, respectively...... 58 4.11 Rise velocity for bubbles having markedly different dynamics. (a) region I: axisymmetric (Ga = 10, Bo = 1) (b) region II: skirted (Ga = 10, Bo = 200), (c) region III: zigzagging (Ga = 70.7, Bo = 1), (d) region IV: offset breaking up (Ga = 70.7, Bo = 20) and (e) region V: centrally breaking up bubble (Ga = 70.7,Bo = 200). In addition to the upward velocity, the in-plane components are unsteady too in regions III to V...... 60 4.12 Variation of dimensionless terminal velocity withBo for different Ga. The terminal velocity tends to decrease with decreasing surface tension because of the increased drag on the bubble...... 61 4.13 Variation of the sum of kinetic and surface energies (TE) for (a) Bo = 20, and (b) Ga = 100. The peak in energy corresponds to the boundaries of the regions shown in Fig. 4.1...... 62

4.14 Time evolution of drops for diﬀerent values of density ratios (ρr) for parameter values:

Ga = 40,Bo = 5 andµ r = 10...... 63 4.15 A large liquid drop violently breaking up while falling in the air at timest = 4 and 5

(from left to right) for parameter values: Ga = 40,Bo = 5,ρ r = 1000 andm = 10 . . 64

xvi 4.16 Time evolution of drops for diﬀerent values of viscosity ratios (µr) for parameter

values: Ga = 40,Bo = 5 andρ r = 10...... 65

5.1 Schematic diagram of a bubble ofﬂuid ‘B’ rising inside a Binghamﬂuid ‘A’ under

the action of buoyancy. The bubble is placed atz=z i; the value ofH,L andz i are taken to be 20R, 48R, and 10.5R, respectively. Initially the aspect ratio of the bubble, h/w is 1, whereinh andw are the maximum height and width of the bubble. 68 5.2 The shape of the bubble along with the mesh att=1.5 are shown for (a)finer and (b) coarser grids. Adaptive grid refinement has been used in the interfacial and yielded regions. The smallest mesh size in thefiner and coarser grids are 0.015 and 0.0625, respectively. Note that thefiner grid has been used to generate the results presented in the subsequentfigures. The parameter values are Ga = 70.71, Bn = 14.213,

µr = 0.01,ρ r = 0.001,m = 1 andBo = 30. The aspect ratios of the bubble obtained using theﬁner and courser grids are 1.002 and 1.003, respectively...... 69 5.3 (a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for

diﬀerent values ofBn. The parameter values are Ga=7.071,µ r = 0.01,ρ r = 0.001, m = 1 andBo=10...... 71 5.4 (a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for

different values ofµ r. The parameter values are Ga=7.071,Bn=0.99,ρ r = 0.001, m = 1 andBo=10...... 71 5.5 The evolution of the shape of the bubble (shown by red lines) and the unyielded region in the non-Newtonianfluid (shown in black) for different values of Bingham number. The results of the Newtonian case are shown for the comparison purpose. The rest of the parameter values are the same as those used to generate Fig. 5.3...... 73 5.6 Contour plots for the radial (right) and axial (left) velocity components for (a)Bn=0 att = 6 (Newtonian case), (b)Bn=0.354 att = 6, (c)Bn=0.99 att = 20 and (d) Bn=1.34 att = 20. In each panel the shape of the bubble is shown by red line. The rest of the parameter values are the same as those used to generate Fig. 5.3. . . . . 74 5.7 (a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for different values ofBo. The rest of the parameter values areRe = 70.71,Bn = 14.213,

µr = 0.01,ρ r = 0.001, andm=1...... 75 5.8 The evolution of the shape of the bubble (shown by red lines) and the unyielded regions in the Binghamﬂuid (shown in black) for diﬀerent values ofBo. The rest of the parameter values are the same as those used to generate Fig. 5.7...... 76 5.9 Contour plots for the radial (right) and axial (left) velocity components for (a)Bo=1 att = 6, (b)Bo = 1 att=8.5, (c)Bo = 30 att = 6 and (d)Bo = 30 att=8.5. In each panel the shape of the bubble is shown by red line. The rest of the parameter values are the same as those used to generate Fig. 5.7...... 77

6.1 Schematic diagram of a bubble moving inside a Newtonianﬂuid under the action of

buoyancy. The initial location of the bubble is atz=z i; unless speciﬁed, the value

ofH,L andz i are 6R, 48R, and 10.5R, respectively. The acceleration due to gravity, g, acts in the negativez direction...... 79

xvii 6.2 Variation of the liquid-gas surface tension along the wall of the tube forΓ=0.1 and

various values ofM 1 andM 2...... 80 6.3 Temporal variation of the center of gravity of the bubble for the parameter values 2 3 2 Ga = 10,Bo = 10 − ,ρ r = 10− ,µ r = 10− ,Γ=0.1 andα r = 0.04. The plots for the

isothermal (M1 = 0 andM 2 = 0), linear (M1 = 0.4 andM 2 = 0) and self-rewetting

(M1 = 0.4 andM 2 = 0.2) cases are shown in theﬁgure. The horizontal dotted line indicates the prediction of Eq. (6.51) for the self-rewetting case...... 86 6.4 (a) The terminal velocity of the center of gravity of the bubble along with the aspect

ratio for diﬀerent values ofM 1 forM 2 = 0; (b) temporal variation of the center of

gravity of the bubble forM 2 =M 1/2; (c) variation of the time at whichz CG reaches

its maximum for different values ofM 1. The rest of the parameter values are Ga = 10, 2 3 2 Bo = 10− ,ρ r = 10− ,µ r = 10− ,Γ=0.1 andα r = 0.04. The numerical predictions of Eq. (6.51) are shown by thefilled square symbols on the right vertical axis. . . . . 88 2 6.5 Effect of Ga on the temporal evolution of the bubble centre of gravity forBo = 10 − , 3 2 ρr = 10− ,µ r = 10− ,M 1 = 0.2,M 2 = 0.1,Γ=0.1 andα r = 0.04. The prediction of Eq. (6.51) is shown by the dotted line...... 89 6.6 Effect ofBo on bubble motion for (a) Ga = 10 and (b) Ga = 5; effect ofBo on the

(c,d) length of the bubble,l B, (e,f) aspect ratio of the bubble,A r for Ga = 5. The 3 2 rest of the parameters valuesρ r = 10− ,µ r = 10− ,M 1 = 0.4,M 2 = 0.2,Γ=0.1

andα r = 0.04...... 90 6.7 Evolution of bubble shape (blue line), streamlines (lines with arrows), and tempera- 2 ture contours (shown in color) with time for (a) Bo = 10 and (b) Bo = 10− . The

initial location of the bubble,z i = 10. The inset at the bottom represents the colormap for the temperature contours. The rest of the parameter values are Ga = 10, 3 2 ρr = 10− ,µ r = 10− ,M 1 = 0.4,M 2 = 0.2,Γ=0.1 andα r = 0.04...... 92 6.8 The eﬀect of initial location of the bubble on the temporal evolution of the center of 2 3 gravity,z CG. The rest of the parameter values are Ga = 10,Bo = 10 − ,ρ r = 10− , 2 µr = 10− ,M 1 = 0.2,M 2 = 0.1,Γ=0.1 andα r = 0.04. The prediction of Eq. (6.51) is shown by the dotted line...... 93

6.9 (a) Evolution of the length of the bubble,l B for two values of Bo when he initial

location of the bubblez i = 8. (b) The eﬀects of initial location of the bubble on elongation of the bubble forBo = 100. The radius of the tube,H=2.5. The rest of 3 2 the parameters are Ga = 10,ρ r = 10− ,µ r = 10− ,M 1 = 0.4,M 2 = 0.2,Γ=0.1 and

αr = 0.04...... 93 6.10 Evolution of bubble shape (blue line), streamlines (lines with arrows), and temperature contours (shown in color) with time for (a) Bo = 10 and (b) Bo = 100, and

H=2.5. The initial location of the bubblez i = 8. The inset at the bottom represents the colormap for the temperature contours. The rest of the parameters are Ga = 10, 3 2 ρr = 10− ,µ r = 10− ,M 1 = 0.4,M 2 = 0.2,Γ=0.1 andα r = 0.04...... 94

6.11 Evolution of (a) the length of the bubble,l B, (b) the location of center of gravity, in

a tube havingH=2.1. The initial location of the bubblez i = 8. The rest of the 3 2 parameters are Ga = 5,ρ r = 10− ,µ r = 10− . The non-isothermal curve is plotted

forΓ=0.1 andα r = 0.04...... 95

xviii 6.12 Evolution of bubble shape with time for (a) isothermal case, and (b)M 1 = 1.8,M2 = 0.9 (temperature contours shown in color). The inset at the bottom represents the colormap for the temperature contours. The rest of the parameters are the same as those used to generate Fig. 6.11...... 96

7.1 Vapour mass source calculated only in the interfacial cells. Normal to the interface (yellow, dashed line) and its components (yellow, solid lines) are shown...... 100 7.2 Drop shape and vapour volume fraction contours with minimum and maximum levels 3 as 0 and 10− , for a water drop falling in air at time,t = 1, 3, 4 and 5 (from left

to right). The other parameters are: Ga = 500, Bo=0.025,ρ rb = 1000,ρ rv = 0.9, µ = 55,µ = 0.7,Pe = 200,λ = 26,λ = 1.0,c = 4,c = 2, = 0.2, rb rv rb rv p,rb p,rv M T Tc = 293K, andT h = 343K...... 101 7.3 Drop shape and vapour volume fraction contours with minimum and maximum levels 3 as 0 and 3 10 − , for a chloroform drop falling in air at time,t = 1, 3, 5 and 7 (from × left to right). The other parameters are: Ga = 100,Bo=0.1,ρ rb = 1480,ρ rv = 0.9,

µrb = 281.2,µ rv = 0.7,Pe = 230,λ rb = 6,λ rv = 1.0,c p,rb = 1.05,c p,rv = 2, = 0.2,T = 293K, andT = 343K...... 102 MT c h 7.4 Drop shape and vapour volume fraction contours with minimum and maximum levels 3 as 0 and 3 10 − , for a chloroform drop falling in air at time,t = 1, 3, 5 and 7 (from × left to right). The other parameters are: Ga = 100,Bo=0.1,ρ rb = 1480,ρ rv = 0.9,

µrb = 281.2,µ rv = 0.7,Pe = 230,λ rb = 6,λ rv = 1.0,c p,rb = 1.05,c p,rv = 2, = 0.2,T = 293K, andT = 343K...... 102 MT c h

xix List of Tables

2.1 Frequently used dimensionless groups relevant to the present work...... 17 2.2 Comparison of the terminal velocities by Joseph [14] and the present work for the parameter values: (a) Ga = 50, and (b) Ga = 100. The rest of the parameter values

areBo = 10,ρ r = 0.001 andµ r = 0.01...... 24

xx Chapter 1

Introduction and previous work

1.1 Background and motivation

Blobs of afluid in anotherfluid are commonly known as bubbles and drops. Since, there is no strict definition of a bubble or a drop, it would be helpful for this work to begin by defining a bubble as a blob offluid having lower density (ρ r < 1) than its surrounding medium and a drop as afluid blob surrounded by a lower densityfluid (ρ r > 1), whereρ r is the ratio of innerfluid density to the surroundingfluid density. Most of theflows in nature involve multiple phases, which may get disconnected from their respective streams to form blobs of a phase dispersed in another phase. Bubbles and drops may form as a result of encapsulation of afluid in anotherfluid, for example exhaled air by sea creatures, vapour bubbles in boiling water, molten glass globules in air at a glass marble manufacturing industry, air in molten glass in glass-blowing factories, fuel droplets from a fuel injector, cloudsfloating in air, and bubbles formed by active galactic nuclei which rise due to buoyancy [15]. The length scale for bubble motion may vary from micrometers to kiloparsecs (1 kiloparsec 3.0857 10 19m), and the time scales may range from nanoseconds [16] to ≈ × a million years [17]. The earliest documented mention of a study of bubble motion has been found in a manuscript, Codex Leicester, by Leonardo Da Vinci, discovered by Prosperetti [18]. Da Vinci reported the paradoxical spiral motion of bubbles when released axisymmetrically from bottom of a containerfilled with water. This is now known as path instability. A few pictures of rising bubbles and falling drops are shown in Fig. 1.1, and a few animations and movies of rising bubbles are also available in the supplementary material of [19]. Bubble dynamics is of huge importance in heat and mass transfer processes, in natural phenomena like aerosol transfer from sea, oxygen dissolution in lakes due to rain and electrification of atmosphere by sea bubbles [20], in bubble column reactors, in petroleum industries, for theflow of foams and suspensions and in carbon sequestration [21], to name just a few. An important property of bubbles and drops is the internal circulation, which enhances mixing which results in greater heat and mass transfer. Fig. 1.2 shows the internal circulation within a glycerine drop released in castor oil [4]. The internal circulation insidefluid bubbles/drops is responsible for a reduction in drag which causes them to move faster than solid ones. Furthermore, this circulation affects theflowfield in the wake which differentiates afluid bubble/drop from a solid one. A large part of this thesis is contained in our published papers [19, 22–24] . A bubble or drop is commonly influenced by gravitational, surface tension, and viscous forces,

1 (a) (b)

Figure 1.1: Figure showing a variety of bubbles and drops observed in experiments. (a) A train of air bubbles rising in water for a constantflow rate of air in the nozzle; (b) single octanoic-acid bubble in distilled water exhibiting a spiralling motion (images at different times merged into a single image). These experiments were performed in collaboration with Prof. Mahesh Panchagnula in his lab at IIT Madras. (c) A falling water drop, from Edgerton’s book [2]; (d) a falling water drop breaking in a bag-breakup mode, courtesy E. Villermaux [3]. although there may be electric, magnetic and other forces depending on the types offluids and their environment. The interplay of these forces results in different bubble and drop behaviours, depending on bubble/drop size, density and viscosity of thefluids involved. Even without the consideration of surfactant, thermal, magnetic, miscibility effects and so on, the parameter space consists of (R,ρ i,ρ o, µi, µo,σ, g), wherein the parameters are radius of a volume equivalent sphere, density of the innerfluid, density of the outerfluid, viscosity of the innerfluid, viscosity of the outer fluid, interfacial tension at bubble/drop interface, and the acceleration due to gravity, respectively. This makes it difficult to perform a parametric study of the problem. As a result, in a number of studies, a few parameters are considered to be negligible and the dynamics is studied with respect to one or two of these parameters. However, by applying Buckingham-π theorem, wefind that the number of parameters required to describe the system can be brought down to only four dimensionless numbers instead of seven dimensional ones. These four dimensionless numbers can be chosen as the

2 Figure 1.2: Visible internal circulation in a glycerine drop falling in castor oil (from the experimental study by Spells [4]). The parameter values corresponding to this experiment are: Ga=0.792, Bo=0.1,ρ r = 1.3 andµ r = 1.24. This was theﬁrst published evidence of the internal circulation in falling drops.

Gallilei number Ga( ρ R√gR/µ ), the Bond numberBo( ρgR 2/σ), the density ratioρ ( ρ /ρ ), ≡ o o ≡ r ≡ i o and the viscosity ratioµ ( µ /µ ). Eötvösnumber, which has the same definition as the Bond r ≡ i o number, is also commonly used in bubble literature, however in this thesis we have used Bond number to represent this dimensionless quantity. It is to be noted that a number of experimental and numerical works [1, 25] also employ other dimensionless numbers in their studies, such as the Reynolds numberRe( ρ V R/µ ), Weber numberWe( ρ ρ V 2/R) and Morton numberMo( ≡ o o ≡| o − i| ≡ gµ4 ρ ρ /ρ2σ3). Although, these numbers are better indicators of the ratios of various forces in the o| o − i| o system, the rise velocityV is not known a-priori. Therefore these dimensionless numbers (dependent on the rise velocityV ) do not provide one with a set of conditions which could be controlled before performing the experiment/simulation. Furthermore, a Morton number defined asBo 3/Ga4 yields straight lines of constant Morton numbers on a log-log plot of Ga versusBo, which makes it easy for researchers to study the dynamics of bubbles/drops with respect to thefluid properties. In the past several decades, thousands of published works have attempted tofit various regimes of bubble motion into simple models. The number of parameters, the nonlinearity and the fully three-dimensional nature of the problem makes it vast and daunting. The viscosity and density ratios of nearly 1650 pairs offluids used in industries and households are presented in Fig. 1.3. The red circles on the left and right hand sides of theρ r = 1 line represent high density contrast bubbles (air in liquid) and drops (liquid in air), respectively. These bubbles and drops with a high contrast in their densities are very different from each other in their behaviour, however thefluid pairs marked with blue circles (liquid in liquid systems) may behave in a similar fashion even for high inertia. A popular shape regime chart for low density and viscosity ratiofluid blobs has been presented by Clift et al. [25]. It should be noted that inclusion of temperature or concentration (of some species like sugar) often changes the viscosity drastically. Such effects along with the consideration of non-Newtonian behaviour of otherfluids push the boundaries of, or give new dimension to Fig. 1.3. The behaviour regimes of these different bubbles and drops spread across decades of density and viscosity ratios is one of the objectives of the present work. As stated above, bubbles and drops have been a subject of active research for more than a century, and in all probability a lot longer (for example see the representative review articles [26–28]); there are yet many unsolved problems, which are the subject of recent research (see e.g. [10, 29–34]). Appealing introductions to the complexity associated with bubble and drop phenomena can also be found in Refs. [35,36]. A vast majority of the earlier experimental and theoretical studies have had

3 Figure 1.3: Density and viscosity ratios of about 1650 pairs offluids. Blue (open) and red (filled) symbols represent liquid-liquid and liquid-gas systems, respectively. It shows that the density and viscosity ratios range across 8 and 10 orders of magnitude, respectively. one of the following goals (i) to obtain the rise velocity (ii) to evaluate the path instability (iii) to understand bubblyflows, (iv) to make quantitative estimates for particular industrial applications, and (v) to derive models for estimating different bubble parameters. Most of these restrict themselves to only a few Ga and Bo. Our study, in contrast, is focussed on the dynamics of a single bubble. Starting from the initial condition of a spherical stationary bubble, we are interested in delineating the physics that can happen. We cover a range of several decades in the relevant parameters. The review of the research work conducted in the related areas is presented below. For readability this has been classified into few sub-areas: (i) axisymmetric bubble and drop motion, (ii) bubble rise in non-Newtonian media, (iii) bubble rise in non-isothermal media, (iv) three-dimensional bubble and drop dynamics and (v) phase change of liquid drops. A brief literature is presented next.

1.2 Literature review

Bubbles and drops: Similarities and diﬀerences

Bubbles and drops have often been studied separately, for instance see [1, 33, 37–42] for bubbles and [4, 43–47] for drops. However, there is also a considerable amount of literature which discusses both together, e.g. [7, 25, 48–51]. The parameter space for this problem is very large as mentioned above. Therefore bubble/drop dynamics has been investigated in limiting conditions, for instance Taylor [52] derived the Oseen’s approximation for small inertia and deformation and showed that the bubbles and drops deform diﬀerently. An introduction to the complexity associated with bubble and drop phenomena can be found in [28]. When the bubble or drop is tiny, it merely assumes a spherical shape, attains a terminal velocity, and moves up or down, respectively, under the action of gravity. An empirical formula for the terminal velocity of small air bubbles was found by Allen [53] in 1900. Later, two independent

4 studies by Hadamard [7] and Rybczynski [54] led to thefirst solution for the terminal velocity and pressure inside and outside of a slowly movingfluid sphere in anotherfluid of different density and viscosity. The spherical vortex solution due to Hill [55] has been a keystone for most of the analytical studies on the subject. Later studies [52] showed that at low Reynolds (a measure of the ratio of inertial to the viscous forces) and Weber numbers (a measure of the ratio of inertial to the surface tension forces), drops and bubbles of same size behave practically the same way as each other, both displaying an oblate ellipsoidal shape. Bigger drops and bubbles are different. A comparison of bubble and drop literature will reveal that in the typical scenario, bubbles dimple in the centre [1, 5], while drops more often attain a cup-like shape [6, 34]. This difference means that drops and bubbles which break up would do so differently. The dimples of breaking bubbles run deep and pinch off at the centre to create a doughnut shaped bubble, which will then further break-up, while drops will more often pinch at their extremities. A general tendency of a drop is toflatten into a thinfilm which is unlike a bubble. There could be several other modes of breakup (see [6]) like shear and bag breakup for drops falling under gravity and catastrophic breakup at high speeds. Flow pastfluid blobs has not been studied for the completeρ µ phase plane. Researchers r − r interested in cloud physics [56–58] have studied the hydrodynamic behaviour of water drops in air in great detail. It was already established by Spells [4] and others [59, 60] that the internal circulation shown by Hadamard-Rybczynski formula really does exit. Pruppacher & Beard [61], and Le Clair et al. [62] found the surface velocity and thus the strength of internal circulation to quantify this phenomenon. They found the maximum surface velocity to be aboutw T /25, wherein wT is the terminal velocity of the drop. Thus it was concluded that the Hadamard-Rybczynski formula under-predicts the strength of internal circulation. This motivated Le Clair et al. [62] to consider the boundary layer effects in the vicinity of the drop. However the modification did not work well above Re 0.5 due to a wrong assumption in their theory. They assumed the boundary ≈ 0.5 layer thickness variation to be same as that for a rigid sphere, i.e.δ Re − , which does not agree ∝ with the experimentalfindings. An interesting review on this subject has been presented in a book by Pruppacher et al. [63].

Three dimensional dynamics

While most earlier computational studies have been axisymmetric or two-dimensional, several three- dimensional simulations have been done as well, see e.g. [11, 64–71]. A remarkable set of papers [65, 66, 72–74] study bubblyflows in which the interaction between theflow and a large number of bubbles is studied. In particular, turbulentflows can be significantly affected by bubbliness. These studies typically used one or two sets of Bond number and Galilei number. There have also been several studies in which the computational techniques needed to resolve this complicated problem have been perfected [11, 67–69, 71, 75]. Furthermore, [76] reported a numerical technique which combines volume offluid and level-set methods and limits the interface to three computational cells. It is remarkable in its relative simplicity in the extension from two to three-dimensions. This problem has attracted a large number of experimental studies as well, see e.g. [41, 77]. A library of bubble shapes is available, including skirted, spherical cap, and oscillatory and non- oscillatory oblate ellipsoidal. Approximate boundaries between the regimes where each shape is displayed are available in [1,25] for unbroken bubbles. In experiments on larger bubbles, the shapes at

5 release are designed to be far from spherical. Secondly, experiments which give a detailed description of theflowfield are few, and accurate shape measurements are seldom available. An important point is that bubble shapes and dynamics are significantly dependent on initial conditions at release, which are difficult to control in experiments. One of the objectives of our work is to standardise the initial conditions, a luxury not easily available to experimenters! A curious phenomenon, the path instability, has been the subject of a host of experimental [9,42,78,79], numerical [10,80] and analytical [81,82] studies. This is the name given to the tendency of the bubble, under certain conditions, to adopt a spiral or zigzagging path rather than a straight one. After Prosperetti (2004) discovered it in the books of Leonardo Da Vinci, he termed the path instability as Leonardo’s paradox, since it was not known then why an initially axisymmetric bubble would take up a spiral or zigzag path. We will demonstrate that path and shape-symmetry are intimately connected, but only the former has been measured experimentally. It is not easy to measure the evolving bubble shape [79] andflowfield accurately in this highly three-dimensional regime. Most of the workers embarking on this studyfind it satisfactory to investigate the effect of initial bubble diameter on the rising dynamics, and no experimental investigations are available to our knowledge which study the effects of just the surface tension or viscosity of water on the bubble rise. We mention one study [77] here where the effect of surfactant concentration on the oscillatory motion of bubbles is evaluated. The path instability of bubbles was obtained by numerical stability analysis of afixed axisymmetric bubble shape by Cano-Lozano et al., and Magnaudet & Mougin [10,80]. An interesting numerical study due to Gaudlitz & Adams [83] shows hairpin vortices in the wake of an initially zigzagging bubble.

Bubble rise in non-Newtonian media

The motion of droplets influids that exhibit yield stress is important in many engineering applications, including food processing, oil extraction, waste processing and biochemical reactors. Yield stressfluids or viscoplastic materialsflow like liquids when subjected to stress beyond some critical value, the so-called yield stress, but behave as a solid below this critical level of stress; detailed review on yield stressfluids can be found in the publications by Bird et al., and Barnes [84, 85]. As a result the gravity-driven bubble rise in a viscoplastic material is not always possible as in the case of Newtonianfluids but occurs only if buoyancy is sufficient to overcome the material’s yield stress [86, 87]; the situation is also similar for the case of a settling drop or solid particle [88]. Thefirst constitutive law proposed to describe this material behavior is the Bingham model [89] which was later extended by Herschel & Bulkley [90] to take into account the effects of shear-thinning (or thickening). According to this model the material can be in two possible states; it can be either yielded or unyielded, depending on the level of stress it experiences. As the common boundary of the two distinct regions the so-called yield surface is approached, the exact Bingham model becomes singular. In simpleflows this singularity does not generate a problem, but, in more complexflows the discontinuous behaviour of the Bingham model may pose significant difficulties due to the fact that in most cases the yield surface is not known a priori but must be determined as part of the solution. Nevertheless, there are examples of successful analysis of two-dimensionalflows using this model at the expense of relatively complicated numerical algorithms [88, 91–93]. A simpler way to overcome these difficulties is to modify the Bingham constitutive equation in order to produce a non-singular constitutive law, by introducing a ‘regularization’ parameter [94]. This method has

6 been used with success by several researchers in the past [87,95–98] and when used with caution can give signiﬁcant insight in the behaviour of viscoplastic materials.

The motion of air bubbles in viscoplastic materials has attracted the interest of many research groups in the past. Thefirst reported experimental study on rising bubbles in a viscoplastic material was done by Astarita & Apuzzo [99] who reported bubble shapes and velocities in Carbopol solutions. They observed that curves of bubble velocity vs bubble volume for viscoplastic liquids had an abrupt change in slope at a critical value of bubble volume that depended on the concentration of Carbopol in the solution, i.e. the yield stress of the material. Many years later, Terasaka & Tsuge [100] used xanthan gum and Carbopol solutions to examine the formation of bubbles at a nozzle and derived an approximate model for bubble growth. Dubash & Frigaard [101] also performed experiments with Carbopol solutions and were able to comfirm the observations of Astarita & Apuzzo [99] on the existence of a critical bubble radius required to set it in motion and noted that the entrapment conditions are affected significantly by surface tension. It is also noteworthy that the observed bubble shapes inside a vertical pipe were different from [99] exhibiting a cusped tail, resembling much the inverted teardrop shapes often found inside a viscoelastic medium [99, 102, 103]. Similar bubble shapes have been found in the experimental studies by Sikorski et al. [104] and Mougin et al. [105], using Carbopol solutions of different concentrations. The latter authors also studied the significant role of internal trapped stresses within a Carbopol gel on the trajectory and shape of the bubbles; theirfindings were in agreement with an earlier study presented by Piau [106].

From a theoretical point of view, Bhavaraju et al. [107] performed a perturbation analysis in the limit of small yield stress for a spherical air bubble. Stein & Buggish [108] were interested on the mobilization of bubbles by setting an oscillating external pressure and provided analytical solutions along with some experimental data; the latter suggested that larger bubbles tend to rise faster than smaller bubbles at similar amplitudes. Dubash & Frigaard [86] employed a variational method to estimate the conditions under which bubbles should remain static. These estimations, however, were characterized as conservative, in the sense that they provide a sufficient but not necessary condition. A detailed numerical study of the steady bubble rise, using the regularized Papanastasiou model [109], has been performed by Tsamopoulos et al. [87]. These authors presented mappings of bubble and yield surface shapes for a wide range of dimensionless parameters, taking into account the effects of inertia, surface tension and gravity. Moreover, they were able to evaluate the conditions for bubble entrapment. Their work was followed by the study of Dimakopoulos et al. [93] who used the augmented Lagrangian method to obtain a more accurate estimation of the stopping conditions. It was shown that the critical Bingham number, Bn, does not depend on the Archimedes number in accordance with Tsamopoulos et al. [87], but depends non-monotonically on surface tension. We should note that in both studies the shape of the bubble near critical conditions could not reproduce the inverted teardrop shapes seen in experiments [101, 104, 105] and raised questions whether this is due to elasticity, thixotropy or wall effects. Besides the steady solutions it is also interesting to investigate the bubble dynamics through time-dependent simulations. This was done by Potapov et al. [110] and Singh & Denn [111] using the VOF method and the level-set method, respectively. Singh & Denn [111] considered creepingflow conditions and performed simulations for single and multiple bubbles. It was shown that multiple bubbles and droplets can move inside the viscoplastic material under conditions that a single bubble or droplet with similar properties would have been trapped unable to overcome the yield stress. Potapov et al. [110] also studied the case of a single or two

7 interacting bubbles but also took into account the eﬀect of inertia, albeit for a low Reynolds number. For the parameter range that they have used the single bubble always reached a quasi-steady state. We should note at this point that for some cases (e.g. for high values of the Archimedes number) Tsamopoulos et al. [87] were not able to calculate steady shapes which is probably an indication that theﬂow may become time-dependent.

Bubble rise in non-isothermal systems

The variation in temperature of a liquid-gas interface results in the formation of surface tension gradients which induce tangential stresses, known as Marangoni stresses, drivingflow in the vicinity of the interface. This mechanism is always present in non-isothermal interfacialflows and can be important in a great variety of technological applications. A characteristic problem where thermal Marangoni stresses play a significant role is the thermocapillary migration of drops and bubbles. Much of the work in thisfield has been reviewed by Subramanian and co-workers [112, 113]. Thefirst reported study on the thermal migration of bubbles can be found in the pioneering work of Young et al. [114]. These authors conducted experiments on air bubbles in a viscousfluid heated from below and showed that under the effect of the induced Marangoni stresses small bubbles move downwards, whereas larger bubbles move in the opposite direction as buoyancy overcomes the effect of thermocapillarity. Young et al. [114] also provided a theoretical description of the bubble motion assuming a spherical shape and creepingflow conditions and were able to derive an analytical expression for the terminal velocity. Following this work, a series of theoretical analyses took into account the effect of convective heat transfer in the limit of both small and large Reynolds numbers [115–119]. Balasubramaniam & Chai [120] showed that the solution of Young et al. [114] is an exact solution of the momentum equation for arbitrary Reynolds number, provided that convective heat transfer is negligible. These authors also calculated the small deformations of a drop from a spherical shape. The main motivation for the aforementioned studies came from microgravity applications and buoyancy was considered to be negligible. The effect of combined action of buoyancy and thermocapillarity was studied by Merritt et al. [121] employing numerical simulations. Balasubramaniam [122] presented an asymptotic analysis in the limiting case of large Reynolds and Marangoni numbers, including the buoyant contribution as well as a temperature varying viscosity. It was shown that the steady migration velocity, at leading order, is a linear combination of the velocity for purely thermocapillary motion and the buoyancy-driven rising velocity. Later, Zhang et al. [123] performed a theoretical analysis for small Marangoni numbers under the effect of gravity and showed that inclusion of inertia is crucial in the development of an asymptotic solution for the temperaturefield. The asymptotic analysis presented by these authors is based on the assumption of afinite velocity and cannot be used for the case of a stationary bubble. The latter problem has to be analyzed separately as was done by Balasubramaniam & Subramaninan [124]. The solution of this problem is complicated by the presence of a singularity failing to satisfy the far-field condition. Yariv & Shusser [125] introduced an exponentially small artificial bubble velocity as a regularization parameter to account for the inability of the asymptotic expansion to satisfy the condition of exact bubble equilibrium. They were able to evaluate the correction for the hydrodynamic force exerted on the bubble including convective heat transfer; this correction was shown to be independent of the regularization parameter.

8 A great variety of numerical methods have been proposed in order to take into account the effect of surface deformation. These range from boundary-fitted grids [126, 127], to the level-set method [128,129], the VOF method [130], diffuse-interface methods [131] and hybrid schemes of the Lattice-Boltzmann and thefinite difference method [132]. It was shown by Chen & Lee [126] that surface deformation of gas bubbles reduces considerably their terminal velocity. The same effect was found also in the case of viscous drops by Haj-Hariri et al. [128]. Later, Welch [127] demonstrated that as the capillary number increases and the bubble deformation becomes important, the bubbles do not reach a steady state terminal velocity. As was shown by Hermann et al. [133], the assumption of quasi-steady-state is not valid also for large Marangoni numbers. The latterfinding was very recently confirmed by Wu & Hu [134, 135]. Most of the studies mentioned above concern the motion of a single bubble or drop in an unconfined medium. Acrivos et al. [136] studied systems of multiple drops in the creepingflow limit and showed that the drops do not interact. However, when inertial effects are included it was shown by Nas & Tryggvason [137] and Nas et al. [138] that there are strong interactions between the droplets. The thermocapillary interaction between spherical drops in the creepingflow limit was discussed by several authors [139, 140]. In the vicinity of a solid wall, the drop migration velocity is affected by the hydrodynamic resistance due to the presence of the wall as well as by the thermal interaction between the wall and the drop. Meyyappan & Subramanian [141] examined the motion of a gas bubble close to a rigid surface with an imposed far-field temperature gradient and found that the surface exerts weaker influence in the case of parallel motion than in the case of motion normal to it. Keh et al. [142] investigated the motion of a spherical drop between two parallel plane walls and found that the wall effect could speed up or slow down the droplet depending on the thermal conductivity of the droplet and the imposed boundary conditions at the wall. Chen et al. [143] considered the case of a spherical drop and studied the thermocapillary migration inside an insulated tube with an imposed axial temperature gradient. They found that the migration velocity in the tube never exceeds the value in an infinite medium due to the hydrodynamic retarding forces that are being developed. Very recently, Mahesri et al. [144] extended the work of Chen et al. [143] to take into account the effect of interfacial deformation. It was found that as in the case of the spherical drop the migration velocity of the confined drop is always lower than that of an unbounded drop. Brady et al. [145] presented numerical simulations of a droplet inside a rectangular box and showed that for low Marangoni numbers the drop rapidly settles to a quasi steady state whereas for high Marangoni numbers the initial conditions affect significantly the behaviour of the droplet. In the case of severe confinement inside a tube, the drop can become quite long. Such a case was studied by Hasan & Balasubramaniam [146] and Wilson [147] who focused their attention on the thinfilm region away from the drop ends, and were able to derive a relation between the migration velocity and thefilm thickness. Later, Mazouchi & Homsy [148,149] used lubrication theory to determine the liquidfilm thickness and migration velocity for the case of a cylindrical and polygonal tube, respectively. It is well known that the surface tension of commonfluids, such as air, water, and various oils, decreases linearly with increasing temperature; all of the above mentioned studies have considered suchfluids. In this thesis, we are interested in the thermocapillary migration of a deformable bubble inside a cylindrical tubefilled with liquids that exhibit a non-monotonic dependence of the surface tension on temperature. In particular, these so-called “self-rewetting”fluids [150–154], which are

9 non-azeotropic, high carbon alcohol solutions, have parabolic surface tension-temperature curves with well-defined minima; the parabolicity of these curves increases with alcohol concentration. Thesefluids werefirst studied by Vochten & petre [150] who observed the occurrence of the minimum in surface tension with temperature in high carbon alcohol solutions. Petre & Azouni [151] carried out experiments that involved imposing a temperature gradient on the surface of alcohol aqueous solutions, and used talc particles to demonstrate the unusual behaviour of thesefluids. Experimental work on thesefluids was also carried out under reduced-gravity conditions by Limbourgfontaine et al. [152]. The term “self-rewetting” was coined by Abe et al. [155] who studied the thermophysical properties of dilute aqueous solution of high carbon alcohols. Due to thermocapillary stresses, and the shape of the surface tension-temperature curve, thefluids studied spread “self-rewet” by spreading spontaneously towards the hot regions, thereby preventing dry-out of hot surfaces and enhancing the rate of heat transfer. Due to the abovementioned properties, “self-rewetting”fluids were shown to be associated with substantially higher critical heatfluxes in heat pipes compared to water [156–158]. Savino et al. [153] illustrated the anomalous behaviour of self-rewettingfluids by performing experiments to visualise the behaviour of vapour slugs inside wickless heat pipes made of pyrex borosilicate glass capillaries. They found that the size of the slugs was considerably smaller than that associated withfluids such as water. More recently, work on self-rewettingfluids was extended to microgravity conditions for space applications on the International Space Station. Savino et al. [154], and Hu et al. [159] demonstrated that the use of thesefluids within micro oscillating heat pipes led to an increase in the efficiency of these devices. In a slightly different context, it was very recently shown that the presence of a minimum in surface tension can also have a significant impact on the dynamics of the flow giving rise to very interesting phenomena such as the thermally induced “superspreading” [160].

Phase change in falling drops

Multiphaseflows with phase-change are ubiquitous and have several industrial applications, for instance, energy generation, manufacturing, and combustion. Phase change in interfacialflows may occur due to chemical reaction, evaporation, melting, etc. In the past, several researchers have discussed chemically reactingflows [161,162], which is not the subject of present discussion. In this case, when the temperature and pressure are favourable for the reaction to occur, a change in phase can take place and mass of a new species (product of chemical reaction) increases at the expense of existing ones (reactants). In the present study, we discuss the dynamics of a blob of a heavierfluid falling under the action of gravity inside a lighterfluid initially kept at a higher temperature, and undergoing evaporation. If the static pressure at the interface of both thefluids is less (more) than the saturation vapour pressure at the given temperature, evaporation (condensation) can occur at the interface. Due to the relevance in many industrial applications, such as spray combustion,film evaporation and boiling, and naturally occurring phenomena in oceans and clouds, several investigations on evaporation/condensation have been conducted [163–165]. The phase change phenomena occurring during evaporation/condensation depend on several factors, including the environmental conditions. Thus, computationally, it is an extremely difficult problem, and most of the previous studies are experimental in nature. Recently, with the advent of computationalfluid dynamics due to the development of powerful

10 supercomputers, many researchers have tried to incorporate the phase-change models to their multi- phaseflow solvers [166–168]. Although accurate in describing the interface, the method of Esmaeeli & Tryggvason [167] is computationally expensive because of the explicit interface tracking and usage of linked lists for describing the elements on the interface. Workers in this area also have tried using one-fluid approaches, like volume-of-fluid (VOF), diffuse-interface and level-set methods to compute phase-change phenomena (see e.g Schlottke & Weigand [168]). In spite of the above-mentioned work, the effect of viscosity and density ratios with temperature dependentfluid properties have not been investigated on falling drop undergoing evaporation.

Recently, the dynamics of bubbles (ρr < 1) and drops (ρr > 1) has been studied by Tripathi et al.

1.3 Outline of the thesis

The present work is an attempt to study some aspects of the abovementioned phenomena. Many a times, no clear question was there to start with, but the questions emerged as we observed the unexcpected and expected solutions of the Navier-Stokes equations, which are bubbles and drops. The next chapter (Chapter 2) discusses the general formulation and numerical methods common to the entire work. In most of the axisymmetric simulations, we have employed a bespokefinite- volume diffuse-interface code and compared it with the results obtained from the volume-of-fluid code. Gerris alone has been used in all of the three-dimensional simulations because of its adaptive mesh refinement feature. Moreover, extensive validations have been presented in this chapter. In chapter 3, the similarities and differences between a rising bubble and a falling drop are investigated theoretically and numerically. We also investigate the exclusive behaviour that bubbles and drops exhibit. Scaling relationships and numerical simulations show a bubble-drop equivalence for moderate inertia and surface tension, so long as the density ratio of the drop to its surroundings is close to unity. When this ratio is far from unity, the drop and the bubble are very different. We show that this is due to the tendency for vorticity to be concentrated in the lighterfluid, i.e. within the bubble but outside the drop. As the Galilei and Bond numbers are increased, a bubble displays under-damped shape oscillations, whereas beyond critical values of these numbers, over-damped behaviour resulting in break-up takes place. The different circulation patterns result in thin and cup-like drops but bubbles thick at their base. These shapes are then prone to break-up at the sides and centre, respectively. In chapter 4, we present the results of one of the largest numerical study of three-dimensional rising bubbles and falling drops. Herein, we study bubbles rising due to buoyancy in a far denser and more viscousfluid. We show that as the size of the bubble is increased, the dynamics goes through three abrupt transitions from one known class of shapes to another. A small bubble will attain an axially symmetric equilibrium shape dictated by gravity and surface tension, and travel vertically upwards at a terminal velocity thereafter. A bubble larger than afirst critical size loses axial symmetry. We show that this can happen in two ways. Beyond the next critical size, it breaks up into a spherical cap and many satellite bubbles, and remarkably, the cap regains axial symmetry. Finally, a large bubble will prefer not to break up initially, but will change topologically to become an annular doughnut-like structure, which is perfectly axisymmetric. Chapter 5 presents a study of the buoyancy-driven rise of a bubble inside an infinite viscoplastic medium, assuming axial symmetry. To account for the viscoplasticity, we consider the regularized

11 Herschel-Bulkley model. We employ the Volume-of-Fluid method to follow the deforming bubble along the domain. Our results indicate that in the presence of inertia and in the case of weak surface tension the bubble does not reach a steady state and the dynamics may become complex for sufficiently high yield stress of the material. Rising bubble in a self-rewettingfluid with a temperature gradient imposed on the container walls has been studied in Chapter 6. To account for the non-monotonicity of surface tension we consider a quadratic dependence on temperature. We examine the Stokesflow limitfirst and derive conditions under which the motion of a spherical bubble can be arrested in self-rewettingfluids even for positive temperature gradients. We then employ a diffuse-interface method [169] to follow the deforming bubble along the domain in the presence of inertial contributions. Our results indicate that for self-rewettingfluids, the bubble motion departs considerably from the behaviour of ordinary fluids and the dynamics may become complex as the bubble crosses the position of minimum surface tension. As will be shown below, under certain conditions, the motion of the bubble can be reversed, and then arrested, or the bubble can become elongated significantly. Finally, a preliminary study of the effect of volatility of liquids have been presented for the case of falling evaporating drops in Chapter 7. To this end, the open-source code, gerris, created by Popinet [170] is used and a phase-change model, similar to that employed by Schlottke & Weigand [168], is incorporated to gerris in order to handle the complex phenomena occurring at the interface. We found that the vapour is generated more on the regions of the interface with relatively high curvature, and the vapour generation increases with breakup of the drop. Furthermore, a competition between volatility and the dynamics governs the vapour generation in the wake region of the drop. This is an ongoing work, and only few of the results are presented in this Chapter.

1.4 Future work

As mentioned before, a large number of studies have focused upon the dynamics of single drops and bubbles from theoretical, experimental and numerical analyses. With the growth of computing speed, researchers [66] have developed techniques to simulate hundreds of bubbles simultaneously in aflow. Complex problems such as these are completely three dimensional and it is very difficult to visualize theflowfield experimentally. Moreover,flows involving thermal gradients, non-Newtonian fluids, evaporation, moving contact lines, surfactants and other complexities change the dynamics drastically. Such dynamics is often not possible or challenging to study theoretically or numerically, while being of great importance to the industries and natural physics. Apart from this, the simple coalescence and breakup of drops and bubbles is less understood and the studies are mostly experimental. To gain a better understanding of these processes, better numerical techniques have to be devised. The present work acknowledges the three-dimensional nature of bubble and drop motion, and attempts to incorporate various complexities into the dynamics of a single bubble/drop. This work could be naturally extended to include multiple bubbles/drops, contact lines and other additional physics mentioned above.

12 Chapter 2

Formulation and numerical methods

The bubble and drop dynamics has been investigated numerically and theoretically. In this chapter we describe the numerical methods employed, and their validations. Theﬂow is assumed to be incompressible and in the case of air bubbles in liquids, the height of rise of air bubble is assumed to be small enough to cause any change in the bubble volume. Axisymmetric and three-dimensional simulations have been carried out in this work, the domains of computation for which are depicted in Fig. 2.1. The formulation and validation present in this chapter are contained in our published works [19, 22–24] .

2.1 Formulation

The equations of mass, momentum and energy conservation which govern theﬂow can be respectively written as:

(ρu) = ˙m , (2.1) ∇· − v

∂u ρ +u u = p+ µ( u+ u T ) +F +F , (2.2) ∂t ·∇ −∇ ∇· ∇ ∇ b s � � � � ∂(ρc T) p + (ρc uT)= (λ T) Δh ˙m, (2.3) ∂t ∇· p ∇· ∇ − v v whereinu,p andT denote the velocity, pressure, and temperatureﬁelds of theﬂuid, respectively;t

represents time;F b andF s are the additional body and surface forces, respectively. Here,˙mv is the mass source term per unit volume, non-zero only at the interface. The sign convention is such that

a positive˙m v is associated with evaporation and a negative˙m v with the condensation. Note that

˙mv is set to zero when there is no phase change, therefore the source term has been considered only in Chapter 7. In addition to this, the advection-diﬀusion equation for the vapour volume fraction is as follows

13 (a) (b)

Figure 2.1: Schematic diagram of the simulation domains considered to solve (a) axisymmetric (dot-dashed line represents the axis of symmetry), and (b) three-dimensional rising bubble problem. Bubble size is not to scale. Similar domains are considered for falling drop problem with inverted gravity. The domain is considered to have a square base of size,L in three-dimensions and a circular base of diameter,L in cylindrical coordinates. The outer and innerﬂuids are designated by ‘o’ and ‘i’, respectively. The height of the domain,H is chosen according to the expected dynamics of the bubble/drop.

∂cv ˙mv + (c vu) = ( av cv) + , (2.4) ∂t ∇· ∇· D ∇ ρv

In Eqs. (2.1)–(2.4),ρ,µ andc v are the density, viscosity and the volume fraction of the vapour, respectively; represents binary diffusion coefficient of the gas mixture. The specific latent heat of D av vaporization, specific heat and thermal conductivity are denoted byΔh v,c p andλ, respectively. In addition to this, for the non-isothermal cases i.e. in Chapters 6 and 7, the hot and cold temperatures are designated byT h andT c, respectively. The body and surface forces in Eq. (2.2), in the absence of electric, magnetic, or any other forces except gravitational and surface tension forces, can be written as

F = ρg�e , (2.5) b − z and F =δ[σκˆn+ σ], (2.6) s ∇ s wherein,δ(= c ) is the dirac distribution function, ˆn is the unit normal to the interface,κ is |∇ b|

14 the local curvature of the interface separating the two phases,�ez is the unit vector in the vertically upward direction, andσ is the coeﬃcient of surface tension. For non-isothermal systems (discussed in Chapters 6 and 7), the surface tension coeﬃcient is assumed to be a quadratic function [153] of

temperature (where a choice ofβ 2 = 0 yields a linear dependence of surface tension on temperature)

σ=σ β (T T ) +β (T T )2, (2.7) 0 − 1 − c 2 − c

dσ 1 d2σ whereβ 1 andβ 2 2 . ≡ − dT |Tc ≡ 2 dT |Tc The parametersβ 1 andβ 2 are set to zero for isothermal systems Furthermore, a one-ﬂuid approach is followed and only 6 equations (Eqs. (2.1)–(2.4)) are solved for all theﬂuid phases. There-

fore, thefluid properties are defined to be dependent of a colour function (volume fraction) (c a) which varies from 0 to 1 in a thin region between the two phases for a given pair offluids. The advection equation for the colour function is discussed in the next section. Density (ρ), viscosity (µ) and thermal conductivity (λ) are volume averaged between the two phases, whereas the specific

heat capacity (cp) is mass averaged as follows

ρ = (1 c )ρ + (c c )ρ +c ρ , (2.8) − a i a − v o v v µ=G (T )(1 c )µ +G (T)[(c c )µ +µ c ], (2.9) i − a i o a − v o v v λ = (1 c )λ + (c c )λ +c λ , (2.10) − a i a − v o v v (c c )c ρ +c c ρ a− v p,o o p,v v v (1 c a)cp,iρi +c a (c c )c ρ +ρ c ρo c = − a− v p,o o v v , (2.11) p � ρ �

whereinρ o,ρ i andρ v are the densities of outerfluid (fluid ‘o’), innerfluid (fluid ‘i’), and pure water

vapour, respectively;µ o,µ i, andµ v are the viscosities of theseﬂuids, respectively. Similarly the

thermal conductivities and speciﬁc heat capacities for theseﬂuids are denoted byλ o,λ i, andλ v, and

cp,o,c p,i, andc p,v, respectively. The functionsG i(T ) andG o(T ) allow the viscosity to be temperature dependent. If the inner and outerﬂuids are liquid and gas, respectively, the Reynolds’ [171] and Sutherland’s [172] models are used to express the variation in viscosity of the two phases with temperature T T c T − T Gi(T)=e − h− c , (2.12) � � T Tc 3/2 G (T)=1+ − . (2.13) o T T � h − c � The following scalings are employed in order to render the governing equations dimensionless:

R (x, y, z)=R( x, y, z),t= t,(u, v, w)=V( u, v, w), p=ρ V 2 p, V o

µ= µµ ,ρ= ρρ �, c� =� c c ,λ=� λλ ,σ= σσ� ,T� =� T(T T )� +T , o o p p p,o o 0 h − c c σ σ δ � �0 � 0 � � ˙ � β1 = M1,β 2 = 2 M2,˙m v = mvρA g/R,δ= , (2.14) Th T c (Th T c) R − − � � � � where the tildes designate dimensionless quantities, the velocity� scale isV= √gR, andσ 0 is the

surface tension at the liquid gas interface at reference temperatureT c. After dropping tildes from

15 all dimensionless terms, the non-dimensional governing equations are given by

(ρu) = ˙m , (2.15) ∇· − v ∂u 1 ρ +u u = p+ µ( u+ u T ) ρ�e +F , (2.16) ∂t ·∇ −∇ Ga∇· ∇ ∇ − z s � � � � ∂(ρcpT) 1 ˙mv + (uρc pT)= (λ T) , (2.17) ∂t ∇· GaP r ∇· ∇ − Ja

∂c 1 ˙m v + (c u) = c + v , (2.18) ∂t ∇· v ∇· P e∇ v ρ � � rv where Ga( ρ V R/µ ),Pe( VR/ ),Ja( c (T T )/Δh ),Pr( ρ c /λ ) andF denote ≡ o o ≡ D av ≡ p,o h − c v ≡ o p,o o s the Galilei number, the Peclet number, the Jackob number, the Prandtl number and the dimensionless surface tension force, respectively. The dimensionlessﬂuid properties can be written as,

ρ = (1 c )ρ + [(c c ) +ρ c ], (2.19) − a ro a − v rv v T 3/2 µ=e − (1 c )µ + 1 +T [(c c ) +µ c ], (2.20) − a ro a − v rv v � � λ = (1 c )λ + [(c c ) +λ c ], (2.21) − a ro a − v rv v (c c )+c c ρ a− v p,rv v rv (1 c a)cp,roρro +c a (c c )+ρ c c = − a− v rv v , (2.22) p ρ� �

whereρ ro =ρ i/ρo,ρ rv =ρ v/ρo,µ ro =µ i/µo,µ rv =µ v/µo,λ ro =λ i/λo,λ rv =λ v/λo,c p,ro =

cp,i/cp,o andc p,rv =c p,v/cp,o, respectively. The dimensionless surface tension force,F s becomes

δ F = [σκ�n+ σ], (2.23) s Bo ∇s

whereBo( ρ gR2/σ ) is the Bond number. To close the problem, we deﬁne the initial and boundary ≡ o 0 conditions now. The relevant initial and boundary conditions used in this work are discussed in each chapter for the respective problems. A list of dimensionless groups frequently used in bubble and drop studies relevant to our work are listed below. Some of the dimensionless numbers listed in the table will be discussed later.

2.2 Numerical methods

In this work, we have used two numerical techniques, namely difuse interface method and volume offluid method to capture the interface separating the pairs offluids. Both the methods and their validation have been presented below. Volume offluid and diffuse interface methods, both, belong to the class of interface capturing methods, which means that the interface is not explicitly tracked, but is reconstructed by means of a colour function. Thus thefluid properties are smeared in a region containing a few computational cells. Volume offluid method prevents the smearing of the interface by reconstructing the interface at every time step, and thus providing a sharp interface. These two methods are different in how the colour function is advected (or diffused), thus determining how the interface is defined. These will be described briefly in the following text.

16 ρr (ρro) ρi/ρo Density ratio of inner to outerfluid µr (µro) µi/µo Viscosity ratio of inner to outerfluid λr (λro) λi/λo Thermal conductivity ratio of inner to outerfluid cp,ro (cp,r) cp,i/cp,o Specific heat capacity ratio of inner to outerfluid ρrv ρv/ρo Density ratio of vapour to outerfluid µrv µv/µo Viscosity ratio of vapour to outerfluid λrv λv/λo Thermal conductivity ratio of vapour to outerfluid cp,rv cp,v/cp,o Specific heat capacity ratio of vapour to outerfluid 3/2 1/2 Ga ρoR g /µo Gallilei number 2 Bo ρogR /σ Bond number 4 3 Mo gµo/ρoσ Morton number P r cp,oµo/λo Prandtl number P e /R√gR Peclet number for diffusion of vapour in dry air Dav Re ρoUR/µo Reynolds number based on velocity,U 2 W e ρoU R/σ Weber number based on velocity,U Bn τ0R/µ0√gR Bingham number n 1 m (g/R) −2 Fluid consistency for Binghamfluid dσ M1 ΔT/σ0 First derivative of surface tension coefficient − dT Tc 1 d2σ 2 M2 �2 dT 2 � ΔT /σ0 Second derivative of surface tension coefficient Tc � � Table 2.1: Frequently used dimensionless groups relevant to the present work.

2.2.1 Diﬀuse-interface method

In the diﬀuse-interface framework, the interface is captured by tracking the volume fraction (colour

function) of the outerfluid,c a. A thin region where the twofluid may mix is defined at the interface, such that the advection-diffusion equation for the volume fraction of the outerfluid becomes

∂ca 1 + (uc a) (M φ) = 0, (2.24) ∂t ∇· − P ed ∇· ∇

1 wherePe d is a very large number of the order of� − , wherein� is of the order of grid size;M is the mobility defined asc (1 c );φ is the chemical potential of thefluid system defined as the a − a change in free energy with respect toc a. Additionally, the dimensionless force per unit volumeF s (excluding the tangential force) in the Navier-Stokes equation (Eq. (2.16)) is obtained as,

φ c F = ∇ a . (2.25) s Bo

The pressure and the volume fraction of the outerfluid are defined at the cell-centres, and the velocity components are defined at the cell faces, respectively. In our code afifth order weighted-essentially- non-oscillatory (WENO), and central difference schemes are used to discretize the advective and diffusive terms appearing in the advection-diffusion equation for the volume-fraction, respectively. In order to achieve second-order accuracy, the Adams-Bashforth and the Crank-Nicholson methods are used to discretize the advective and dissipation terms in Eq. (2.16), respectively. The implementation is similar to that discussed in the work of Ding et al. [169].

17 2.2.2 Volume ofﬂuid method: Gerris

In the volume ofﬂuid (VOF) framework, the surface tension force is included as a force per unit volume in the Navier-Stokes equation in the following manner

δ F = (κσˆn+ σ). (2.26) s Bo ∇s

The curvature,κ is calculated using a generalized height-function method implemented in gerris.

The volume fraction of the outerﬂuid (c a) is advected with the localﬂuid velocity as follows:

∂c a +u c = 0. (2.27) ∂t ·∇ a

A piecewise linear interface calculation (PLIC) is employed to reconstruct a sharp interface at every time-step from the volume fraction data. This avoids smearing of the interface due to the numerical diffusion. We have used an open-sourceflow solver, gerris, based on VOF framework. Gerris uses a generalized height-function method for calculating the curvature of the interface, thus improving the accuracy of the surface tension force calculation for the VOF methods. Level set and front-tracking methods used to be considered as the state-of-the-art for highfidelity interfacialflow simulations, but with the implementation of height-functions [173], VOF methods are getting the recognition as state- of-the-art, again. Moreover, gerris uses a balanced force algorithm for inclusion of surface-tension force in the Navier-Stokes equations, which combined with the height-function implementation reduces the amplitude of spurious velocity to machine error (i.e. lowest possible error achievable on a computer calculation). Another feature of gerris, the dynamic adaptive mesh refinement, allows one to cluster the grid more in the desired regions dynamically, thus saving the computation time remarkably.

2.3 Validation

A number of tests have been performed to check the accuracy of gerris [174] and the diffuse-interface method [169]. We present below a few validation cases relevant to the bubble and drop dynamics. First, we check the domain and grid dependence of results for our numerical methods. The results have been obtained for both, diffuse interface method and volume offluid method, however only one set of validation exercises (for gerris) are shown here. The results for diffuse-interface method have been found to compare well with the results obtained using gerris, however the computation cost was several times that for gerris.

2.3.1 Grid convergence test

We start with a test against the discretization errors, such that the results would be independent of the grid size used. The shape of the bubble att = 4 and 7 for two diﬀerent grids (Δx=Δz=0.029 and 0.015) in a computational domain of size 16 48 are shown in Fig. 2.2. It is found that a × square grid withΔx=Δz=0.029 is enough to get results to within 0.1% accuracy. A similar test is presented for three-dimensional bubble in Fig. 2.3. In Fig. 2.3 we show bubble shapes obtained using two diﬀerent grids. It reveals that grid convergence is achieved for simulations having the

18 smallest grid less than 0.029. Thus all the three-dimensional simulations have been conducted using this grid size.

(a) (b)

Figure 2.2: Eﬀect of grid reﬁnement on the shape of the bubble at (a)t = 4, and (b)t = 7 for 4 6 Ga=3.09442,Bo = 29,ρ r = 7.4734 10 − andµ r = 8.1536 10 − . The solid and dot-dashed lines correspond to the results obtained using× Δx=Δz=0.015 and× 0.029, respectively.

(a) (b)

Figure 2.3: Grid convergence test. The shapes of the bubble for two diﬀerent grid sizes att = 3 is 3 2 shown. The parameter values are Ga = 70.7, Bo = 200,ρ r = 10− andµ r = 10− . The smallest grid sizes in panels (a) and (b) are about 0.029 and 0.015, respectively. The grid reﬁnement criteria used here are based on the vorticity magnitude and the gradient of volume fraction (ca).

2.3.2 Eﬀect of domain size

The effect of domain size is investigated in Fig. 2.4, where the axial rise velocity of the spiralling bubbles is plotted versus time for two different values of dimensionless base width,L. It is found that doubling the size of the lateral cross-section of the domain (i.e doublingL) has a negligible effect (less than 0.2% on the rise velocity as shown in Fig. 2.4) on theflow dynamics. Although the results are somewhat different for the two domain sizes at later times for these parameter values, the results do not differ significantly so as to change the shape regime. Thus, a domain ofL = 30 is found to be satisfactory for a qualitative study of shape regimes of bubbles. We also found that the

19 shapes of the bubble for both the cases are close to each other. In addition to this, it is to be noted here that [9] and [34] consideredL to be about 30 in their experimental and numerical studies of the path instability of rising bubble, and fragmentation process of a falling drop, respectively. Thus a computational domain 30 30 120 is used for three-dimensional simulations of bubbles and drops. × × All the dimensions are scaled with the radius of the bubble.

2.5

1.5 w L 1 30 60 0.5

0 0 3 6 9 12 15 t

Figure 2.4: Eﬀect of domain size on upward velocity,w of a bubble exhibiting a spiralling motion. The dashed and solid lines represent domains of base width,L = 30 and 60, respectively. The 3 dimensionless parameters used for the simulations are: Ga = 100, Bo=0.5,ρ r = 10− and 2 µr = 10− .

For axisymmetric simulations, a half-domain size of 16 48 was found to be sufficient to simulate × bubbles and drops in an unconfined medium for the parameter values considered in the present work i.e. for Ga > 2. Fig. 2.5 shows the results for two different domain sizes for the parameter values: 4 6 Ga=3.09442,Bo = 29,ρ = 7.4734 10 − andµ = 8.1536 10 − . It is concluded that a domain r × r × size of 16 48 can be used to simulate bubble rise in an unbounded surroundingfluid medium. ×

Figure 2.5: Eﬀect of domain size on the bubble shape att = 4, andt = 7 (left to right) for 4 6 Ga=3.09442,Bo = 29,ρ r = 7.4734 10 − andµ r = 8.1536 10 − . The solid and dot-dashed lines correspond to computational domains× 8 24 and 16 48, respectively.× The results are generated using square grid ofΔx=Δz=0.015. × ×

20 (a) (b)

Figure 2.6: Comparison of the shape of the bubble obtained from our simulation (shown by solid red line) with those from the level-set simulations of Sussman & Smereka [5] (dashed line) at various times: (a)t = 0, (b)t=0.8, (c)t=1.6 and (d)t=2.4. The parameter values are Ga = 100, Bo = 200,ρ r = 0.001 andµ r = 0.01. The transition to toroidal bubble (topological change) is observed att=1.6, which matches exactly with the result of Sussman & Smereka [5].

2.3.3 Comparison with numerical simulations

Comparison with Sussman & Smereka [5]

In order to validate our code, in Fig. 2.6 we compare our results obtained for Ga = 100,Bo = 200,

ρr = 0.001 andµ r = 0.01 with those of Sussman & Smereka [5], who studied theﬂuid dynamics of rising bubble with topology change in the framework of a level-set approach. The dashed lines on the left hand side of each panel are the results from Sussman & Smereka [5], whereas the present results are plotted by solid red lines on the right hand side of the panels. It can be seen that the topology changes observed in our simulation agree excellently with the results of Sussman & Smereka [5].

Comparison with Han & Tryggvason [6]

The simulation domain was taken to be the same as that of Han & Tryggvason [6], i.e. 10 30 and × the grid size was taken to beΔx=Δz=0.015 for the parameter values stated in Fig. 2.7. The dimensionless time at which the drop breaks up (t 25.0) is in agreement with the that of [6]. br ≈ The oscillations in velocity are well replicated too.

2.3.4 Comparison with the experimental result of Bhaga & Weber [1]

In Fig. 2.8, we compare the shape of the bubble obtained from our simulation (shown by the red line) with the corresponding results given in the experiment of Bhaga & Weber [1] (shown by the gray scale picture). The parameter values used to generate thisﬁgure are Ga=3.09442,Bo = 29, 4 6 ρ = 7.4734 10 − andµ = 8.1536 10 − , which are the values at which the experimental shape r × r × is presented by Bhaga & Weber [1], after suitable transformation, as follows:

1 Bo3 4 Bo Ga= BW , and Bo= BW , (2.28) 64Mo 4 � BW �

21 Figure 2.7: Variation of upward velocity of center of gravity of the drop with time for Ga = 219.09, Bo = 240,ρ r = 1.15 andµ r = 1.1506. The dashed line is the result due to Han & Tryggvason [6] and the solid line is the result of the present simulation. Theﬁgure is plotted till breakup. It could be seen that the results match to a very good accuracy, however small deviations can be seen which may be attributed to the diﬀerences in the interface tracking/capturing methods in the two simulations.

4 3 2 whereMo BW =gµ o/ρoσ ,Bo BW = 4gR ρo/σ, where the subscriptBW refers to Bhaga & Weber [1]. It can be seen that the shape of the bubble obtained from our numerical simulation is in qualitative agreement with the experimentally obtained bubble of Bhaga & Weber [1]. Note that a dimple at the bottom of the bubble (if one exists) in the experiment will not be visible in this photograph, and would appear as a horizontal edge at the bottom.

Figure 2.8: Comparison of the shape of the bubble obtained from the present diﬀuse interface simulation (shown by red line) with that of Bhaga and Weber [1]. The parameter values are Ga = 4 6 3.09442, Bo = 29,ρ r = 7.4734 10 − andµ r = 8.1536 10 − . The dimple is not clearly visible in the experimental result because× it is hidden by the periphery× of the bubble. Also, the apparent dimple seen from the side view is diﬀerent from the actual dimple because of the refraction due to the curved bubble surface.

Next we validated the volume offluid code (gerris) by comparing the results obtained using it with the experimental results of Bhaga & Weber [1] in Fig. 2.9 for different values of Ga andBo. Furthermore, the streamline pattern in the wake of the bubble for different Ga andBo is also plotted

22 in Fig. 2.10. It can be seen that the results are in good agreement.

Figure 2.9: Comparison of terminal shape of the bubble with Bhaga & Weber [1] (left to right): Ga=2.316, Bo = 29, Ga=3.094, Bo = 29, Ga=4.935, Bo = 29, and Ga = 10.901, Bo = 84.75. The results in the bottom row are obtained from the present three-dimensional simulations. The results from the corresponding axisymmetric simulations are shown by red lines in the top row.

It is also noted here that this solver has been extensively validated with theoretical results for capillary instability of a cylindrical liquid coloumn [175], and linear instability theory for a two phase mixing layer and a two-dimensional drop in a shearﬂow [174]. More test cases are available at the webpage for gerris test-suite (http: //gfs.sourceforge.net/tests/tests/).

2.3.5 Comparison with analytical results

The next two comparisons are not so much for validation, since the analytical results are for idealised limits, but a demonstration that the analytical results are valid in a range of parameters lying near the idealized limits.

In the Hadamard [7] limit

By balancing the drag force with the weight of the bubble/drop, and neglecting the inertial and surface tension forces, Rybczinsky [54] and Hadamard [7] analytically derived an expression for terminal velocity (famously known as Hadamard-Rybczinsky equation), which is given by

2 R2g(r 1)ρ 1 +m V = − 1 . (2.29) t 3 µ 2 + 3m 1 � � Thus the dimensionless terminal velocity can be written as:

2Ga(1 r) 1 +m V = − . (2.30) t 3 2 + 3m � � � 3 2 In an example simulation with the parameters Ga=0.1, Bo=0.1,ρ r = 10− andµ r = 10− , we found that the terminal velocity isV = 0.0329 for a domain of size 16 48 and larger. The t × corresponding dimensionless velocity obtained using Eq. (2.30) is 0.0331. In Fig. 2.11 and 2.12, we plot streamlines obtained from the Hadamard solution and our numerical simulations for Ga=0.1. The qualitative similarity is apparent. The small discrepancies may be attributed to the fact that at Re 0 an inﬁnitely large computational domain is required for accurate solutions, and also to →

23 Figure 2.10: Comparison of streamline pattern in the wake of the bubble with Bhaga & Weber [1] for the following dimensionless parameters: (a) Ga=7.9, Bo = 17, (b) Ga = 9, Bo = 21, (c) Ga = 12.6,Bo = 17, (d) Ga = 17.8,Bo = 27, (e) Ga = 21.9,Bo = 17, and (f) Ga = 33.2,Bo = 11. 3 2 The rest of the parameter values areρ r = 7.747 10 − andµ r = 10− . The results on the left hand side and right hand side of each panel are from× the present simulations and Bhaga & Weber’s [1] experiments, respectively.

small deviations from a spherical shape at ourﬁnite surface tension values, whereas the analytical result assumes a perfectly spherical bubble.

Comparison with potentialﬂow solution

In Table 2.2 we compare the terminal velocities obtained from our numerical simulations for Ga = 50 and 100 with those obtained from the analytical solution of [14], who studied a rising spherical cap bubble in the potentialﬂow regime. The other parameter values are Bo = 10,ρ r = 0.001 and

µr = 0.01. In this parameter range, the computationally obtained bubble resembles a spherical cap. It can be seen that the potentialﬂow assumption is able to predict the terminal velocity qualitatively.

Cases Joseph [14] Present work a 0.864 0.883 b 0.882 0.906

Table 2.2: Comparison of the terminal velocities by Joseph [14] and the present work for the parameter values: (a) Ga = 50, and (b) Ga = 100. The rest of the parameter values are Bo = 10, ρr = 0.001 andµ r = 0.01.

24 Figure 2.11: Streamlines obtained from the analytical result for Hadamardﬂow (Re 0) in a spherical bubble (left hand side), and volume ofﬂuid simulation with gerris (right hand→ side) for a 3 domain of half-width 16R. The dimensionless parameters are: Ga=0.1,Bo=0.1,ρ r = 10− and 2 µr = 10− .

0.04

0.035

0.03

w 0.025 Present study 0.02 Theory (Hadamard, 1911)

0.015

0.01 0 2 4 6 8 10 t

Figure 2.12: Comparison of present numerical result with Hadamard-Rybczynski [7] theory. The terminal velocity agrees well for a domain size of 30 30 120 and for the parameter values: Ga=0.1, 3 2 × × Bo=0.1,ρ r = 10− andµ r = 10− . The center of the vortex is well predicted by our numerical simulation.

2.4 Eﬀect of regularization parameter

Rising bubble in a viscoplastic medium has been studied in Chapter 5. The outerﬂuid is modelled as a regularized Herschel-Bulkelyﬂuid as follows:

τ0 n 1 µ = +µ (Π+�) − , (2.31) o Π+� 0 whereτ 0 andn are the yield stress andﬂow index, respectively,� is a small regularization parameter, 1/2 andµ 0 is theﬂuid consistency;Π=(E ij Eij ) is the second invariant of the strain rate tensor, whereinE 1 (∂u /∂x +∂u /∂x ). ij ≡ 2 i j j i In the dimensionless form, Bn n 1 µ = +m(Π+�) − , (2.32) o Π+�

25 n 1 where Bn τ R/µ V is the Bingham number, andm=(V/R) − , whereinV (= √gR) is the ≡ 0 0 velocity scale. The characteristic scales and non-dimensionalization is explained in Chapter 5. After 3 careful evaluation, we have chosen the value of� down to 10 − in order to neither affect the yield surface by overly increasing� nor produce numerical instabilities or stiff equations by decreasing it further; similar values for� have been used earlier by Singh and Denn [111]. The effect of regularization parameter is shown in Figs 2.13 and 2.14. The height and width of the bubble are denoted byh andw, respectively (see Fig. 5.1). Also, a comparison with Papanastasiou’s model, presented in Fig. 2.15, shows a good match with the regularized model. Hence the present regularized model is employed to simulate a rising bubble inside a viscoplasticfluid in Chapter 5.

(a) (b) (c)

Figure 2.13: The unyielded region in the non-Newtonianﬂuid (shown in black) at time,t = 10 for diﬀerent values of the regularized parameter,�: (a)�=0.01, (b)�=0.001, (c)�=0.0001. The rest of the parameter values are Ga = 70.71,Bn = 14.213,µ r = 0.01,ρ r = 0.001,m = 1 andBo = 30. The unyielded regions for�=0.001 and 0.0001 are visually indistinguishable.

(a) (b) 5 1.6 ε ε 4 0.01 1.4 0.01 0.001 0.001 3 0.0001 0.0001 z 1.2 CG h/w 2 1

1 0.8 0 0 5 10 15 20 0 5 10 15 20 t t

Figure 2.14: (a) Temporal variation of the center of gravity (zCG), (b) the aspect ratio (h/w) of the bubble for diﬀerent values of the regularization parameter,�. The rest of the parameter values are the same as those used to generate Fig. 2.13.

26 (a) (b)

Figure 2.15: The bubble shape (shown by red line) and unyielded region in the non-Newtonianﬂuid (shown in black) at time,t = 2 for (a) regularised model, (b) Papanastasiou’s model. The rest of the parameter values are�=0.001, Ga = 70.71, Bn = 14.213,µ r = 0.01,ρ r = 0.001,m = 1 and Bo = 30. The aspect ratios of the bubble in (a) and (b) are the same (h/w=1.018).

27 Chapter 3

Bubbles and drops: Similarities and diﬀerences

3.1 Introduction

Bubbles and drops are often studied separately. To our knowledge, no published work discusses the reason for the differences between a bubble and a drop in their motion, deformation or the manner of their breakup. In afluid mechanics symposium, “Fluids Days 2014”, organized on the 80 th birthday of Prof. Roddam Narasimha, I presented a poster containing Fig. 3.1 which attracted attention from other researchers. Some people looked confused and they had no clue as to why a bubble should behave different from a drop if the density and viscosity were inverted. Prof. Garry Brown, among others, mused about why should a drop be different from a bubble. Some of them, including Prof. Roddam Narasimha, suggested that if I tried to adjust the viscosities or other dimensionless numbers in some way, I could get an equivalence between the bubble and the drop motion. However, experts on the subject (Prof. K. R. Sreenivasan) seemed to know that a bubble and a drop could never behave the same because of the basic violation of the dynamical similarity inherent in the problem. A series of very natural questions arose in this regard: Can we derive a general theory for the motion of both bubbles and drops? Can a rising bubble be designed to behave as the mirror image of a falling drop? If yes, under what conditions? If not, what are the fundamental differences between a bubble and a drop that differentiate them? This chapter deals with an analytical as well as numerical analysis of bubbles and drops in a unified sense for thefirst part, and the differences and some interesting dynamics is dealt with in the latter part.

3.2 In Hadamardﬂow regime

The motion of a sphericalfluid bubble/drop in a different quiescentfluid in creepingflow regime is termed as Hadamardflow, or Hadamard-Rybczynskiflow in the honour of Hadamard [7] and Rybczynski [54] who determined theflowfield in this kind offlow for thefirst time in the year 1911, independently. A spherical vortex solution which has the similar streamfunction solution The analytical derivation of the stream function for suchflow can be found in standard textbooks such

28 (a) (b)

Figure 3.1: Figure taken from the poster presented on Prof. Roddam Narasimha’s 80th birthday. 4 Rising bubble and falling drop for parameter values: (a) Ga = 50,Bo = 29,ρ r = 7.47 10 − and 6 × µ = 8.15 10 − , and (b) Ga = 30,Bo = 29,ρ = 10 andµ = 10. r × r r

as those authored by Batchelor [176] or Leal [177]. The theory assumes that the shape of the bubble/drop is maintained to be spherical as a result of a balancing surface tension force, which could be possible for a very high surface tension force as compared to inertial or viscous forces. This implies that in the dimensionless parameter space, we are concerned with very low Gallilei and Bond numbers i.e. Ga << 1 and Bo << 1. The dimensionless stream function for Hadamard’sﬂow inside theﬂuid sphere is given by: x2(1 cos 2 θ)(1 x 2) ψ = − − , (3.1) iH − 4(1 +m) and that for the surroundings is given by

1 cos 2 θ 3m+2 m ψ = − 2x2 x+ , (3.2) oH 4 − m+1 x(1 +m) � � wherex andθ are the radial distance and the polar angle in a spherical coordinate system having its origin at the center of the spherical bubble/drop.

The bubble/drop reaches a terminal velocity when the drag force is balanced by the buoyant weight of the bubble/drop. Also known as Hadamard-Rybczinsky equation, the terminal rise velocity is given by 2 2 ρoR g(ρr 1) 1 +µ r V ∗ = − , (3.3) t 3 µ 2 + 3µ o � r � or, in the dimensionless form, where velocity is scaled with √gR, the terminal velocity can be written as V 2 1 +µ V = t∗ = Ga(ρ 1) r . (3.4) t √gR 3 r − 2 + 3µ � r �

29 Figure 3.2: Theoretical streamlines in a spherical bubble for the Hadamardﬂow(Re << 1). The stagnation ring (center of the spherical vortex) lies at a distance of 1/√2 from the axis of symmetry.

From Eq. (3.4), it seems possible to design a bubble which is a mirror image of a drop, only by modifying the viscosity ratio (µr). It should be noted again that a bubble is defined as afluid blob for whichρ r < 1 and a drop is afluid blob for whichρ r > 1. This implies that for a given bubble, the only unknown is the viscosity ratio of the drop, which can be determined from the following relation:

1 +m 1 +m (r 1) d = (1 r ) b . (3.5) d − 2 + 3m − b 2 + 3m � d � � b � Strictly in the zero Reynolds number limit, Ga can also be different for a bubble and a drop, thus yielding infinite solutions for the equivalence of a bubble and a drop. But for afinite value of it, the Gallilei number has to remainfixed in order to have the sameflow dynamics for the outerfluid in both the systems i.e. bubble and the corresponding drop). Thus, only a part of the solution has been arrived at with this analysis. Moreover, we have not found the range of relevant parameters for which a rising bubble may behave similar to a falling drop. Next, it is investigated if a similar acceleration phase can be obtained for a bubble and a drop and the conditions required to do so.

Figure 3.3: A sketch showing a spherical body falling under gravity and the forces acting on it, wherez represents the vertical coordinate, andF b,F g andF D denote the gravitational, buoyancy and drag forces, respectively.

Consider a spherical immiscible mass (solid orﬂuid) in a quiescentﬂuid falling under the action

30 of gravity as shown in Fig. 3.3. According to the Newton’s second law of motion,

F F +F =m a , (3.6) B − g D d d

wherein,F B,F g andF D are the forces exerted on the body due to buoyancy, gravity and viscous

drag. The mass of the falling body i.e. drop is denoted bym d. Here, we have neglected the added mass and Basset history forces to keep the solution simple. More accurate solutions can be obtained by including these terms. Additionally, the acceleration of the center of gravity of the drop

is assumed to bea d. Eq. (3.6) can be written in terms ofﬂow and material properties as

4 4 4 πR3ρ g πR3ρ g f(m)µ Rv+O(v 2) = πR3ρ a . (3.7) 3 o − 3 i − o 3 i d � � In creepingflow regime, the higher order velocity terms can be neglected and the drag force can be approximated as being proportional to velocity of the center of gravity of the drop. Note that the drop is translating in the negativez direction, hence the drag force is taken to be proportional to v to make it positive. Also, for a general drop material (Newtonianfluid or solid) the drag force − may depend on the viscosity ratio (m). Eq. (3.7) can be simplified as

3 ρ g ρ g f(m)µ Rv=ρ a , (3.8) o − i − 4πR3 o i d 3 ad 1 r 2 f(m)µ ov=r , (3.9) − − 4πR ρog g 3 v ad 1 r f(m)µ o =r , (3.10) − − 4πρoR√gR √gR g g(m) 1 r v∗ =ra ∗ (3.11) − − Ga d

where the superscript represents dimensionless quantities. Also, a functiong(m)(= 3f(m)/4π) ∗ has been introduced to keep the expression succinct. Rewriting Eq. (3.11) in terms of only vertical velocity and dropping the superscript from dimensionless quantities, we get

dv g(µ ) 1 ρ + r v= − r . (3.12) dt ρrGa ρr

Eq. (3.12) is aﬁrst order ordinary diﬀerential equation which can be solved using a standard method involving the integrating factor. The integrating factor can be calculated as exp g(µr ) dt ρr Ga which is equal to exp g(µr )t . Thus the solution of Eq. (3.12) can be written as �� ρr Ga � � g(µr )t 1 ρ r g(µr )t ve ρr Ga = − e ρr Ga dt+k , (3.13) ρ 1 � � r � g(µ )t Ga(1 ρ r) − r v= − +k 1e rGa . (3.14) g(µr)

Initially the drop is at rest i.e.v = 0 att = 0. This implies

Ga(1 ρ r) k1 = − . (3.15) − g(µr)

31 Therefore, the vertical velocity of the drop can be described by the following equation

g(µ )t Ga(1 ρ r) − r v= − 1 e ρr Ga . (3.16) g(µr) − � � We note, from Eq. (3.16) that the velocity of center of gravity of the drop is always negative as

ρr > 1 for a drop (by deﬁnition). Thus, we can write two separate equations for a bubble and a drop as follows g(m )t Gab(1 r b) − b v = − 1 e Gabrb , (3.17) b g(m ) − b � � and, g(m )t Gad(1 r d) − d v = − 1 e Gadrd . (3.18) d g(m ) − d � �

It is evident from Eqs (3.17) and (3.18), that for a bubble to have similar motion as that for a drop, the following conditions must be satisﬁed:

Ga r Ga r b b = d d , (3.19) g(mb) g(md)

and 1 r r 1 − b = d − . (3.20) rb rd

Additionally, the Gallilei numbers for the bubble and the drop systems must be the same for

the dynamical similarity of theflow in the respective outerfluids, i.e. Ga b = Gad. Therefore, a bubble can, in principle, be designed to behave like a drop in the creepingflow regime according to Eqs (3.19) and (3.20). An interesting result that is apparent from these conditions is that an equivalent solid drop cannot be designed for a solid bubble even in the creepingflow limit. This

is true becauseg(m b) =g(m d) = 4.5 for solid spheres translating in Stokes regime. This implies, from Eq. (3.19), that the density ratios for bubble and drop systems are identical and only a trivial solution is obtained. However, for aﬂuid bubble it is possible to design an equivalent drop which has a similar motion in the vertically downward direction as that of the bubble in the upward

direction. From the Hadamard-Rybczynski solution, the functiong(µ r) comes out to be equal to

(6 + 9µr)/(2 + 2µr). Rewriting the equivalence conditions, Eqs (3.19) and (3.20), in terms of the density and viscosity ratios r (1 +m ) r (1 +m ) b b = d d , (3.21) 2 + 3mb 2 + 3md and r r = b . (3.22) d 2r 1 b − Enforcing the condition that the drop density ratio is always positive, we arrive at a limit to the

bubble density ratio i.e.r b >0.5. On substituting the value ofr d from Eq. (3.22) in Eq. (3.21), we get 1 +m 1 1 +m b = d . (3.23) 2 + 3m 2r 1 2 + 3m b � b − � d

32 Let us deﬁneq(m b) andq(m d) such that

1 +m b q(mb) = , (3.24) 2 + 3mb 1 +m d q(md) = . (3.25) 2 + 3md

We note thatq(m b) andq(m d) lie between 1/3 and 1/2 for all positive real values ofm b andm d.

Therefore, the maximum and minimum bounds on the bubble density ratio,r b can be calculated as

1.25 and 0.83, respectively. This condition requires that the bubble density ratio,r b should always be greater than 0.834, and by definitionr b < 1. Furthermore, it is noted that not all viscosity ratios are available to choose from, forr b even slightly different from 1. Thus, it is concluded that afluid drop can be designed to behave similar to a bubble in the creepingflow limit for a bubble density ratio greater than 0.834, however not all viscosity ratios are feasible even in this regime, and one needs to satisfy Eqs (3.21) and (3.22) to choose the density and viscosity ratios.

Generally, Boussinesq approximation is applied in this limit, wherein the effect of density difference is lumped into a body force equal to the buoyant weight per unit volume of the bubble/drop. Boussinesq approximation is an approximate way of accounting for the density difference and in this regard Han & Tryggvason [6] have shown that the approximation is reasonable for drop density ratios in the range 1

Figure 3.4: Vertical location of the center of gravity as a function of time for a drop (ρr = 1.214 and µr = 76), an equivalent bubble based on Boussinesq approximation (ρr = 0.786 andµ r = 76) and an equivalent bubble based on conditions (3.21) and (3.22) (ρr = 0.85 andµ r = 0.1). The rest of 4 the parameters are Ga = 6 andBo=5 10 − . The bubble designed using the Hadamard’s solution is shown to be better than the one derived× using the often employed Boussinesq condition.

33 3.3 Bigger bubbles and drops

In the previous section, the bubbles and drops in the Hadamard regime are investigated. For the given gravitational acceleration, bigger bubbles and drops in the same kind of outerfluids have higher Ga and Bo, and therefore have inertia dominated dynamics which often involves deviation from spherical shape and even breakup/topological changes. In the inertia dominated regime, the shapes of bubbles and drops are visibly different. Typically, bigger bubbles that rise up in a straight path often develop a dimple at their rear end [1, 5] while drops for a bag like structure which thins down away from the thick, core [6, 34]. For very high inertia and low surface tension, bubbles tend to change their topology into a doughnut like shape and drops tend to break from their periphery - commonly known as bag break-up. Although there are few other breakup modes that are exhibited by bubbles and drops like shear breakup and catastrophic breakup, the important feature of bubbles is to remain in blobs (except for an exception of skirted bubbles at low Ga and high Bo) and for the drops to thin out into sheet like structures. The motion of bubbles and drops may be expressed as the continuity and momentum equations as follows: u=0, (3.26) ∇· and Du p 1 ˆn = ∇ + (µ( u+ u T)) ˆj+ ∇· δ(x x )ˆn, (3.27) Dt − ρ ρGa∇· ∇ ∇ − ρBo − s whereinρ andµ are the dimensionless density and viscosity of the entirefluid system, respectively, having a sharp change in their values at the bubble/drop interface; ˆj is the unit vector in the vertically upward direction;δ(x x ) is the Dirac-delta function being unity at the interface (which − s is defined by the position vectorx=x s) and zero elsewhere; ˆn is the unit normal to the interface, andD( ∂/∂t+u ) is the material derivative. Here the surface tension force is written as a ≡ ·∇ volume source term as proposed by Brackbill et al. [178]. Thus the surroundingfluid obeys

Du 1 o = p+ 2u ˆj, (3.28) Dt −∇ Ga∇ o − and the bubble/dropﬂuid is governed by

Du p m i = ∇ + 2u ˆj, (3.29) Dt − r rGa∇ i − with the interfacial conditions being the continuity of velocity and stress components at the interface

(x=x s). The surface tension term appears in the normal stress balance at the interface, which is p p =κ/Bo whereκ(= ˆn) is the local curvature of the interface. i|s − o|s ∇· Returning to the question of equivalence between a bubble and a drop, we observe the Eqs (3.28) and (3.29). It is noted that the viscous diffusion and surface tension terms can be matched by making the kinematic viscosity ratio (m/r), Ga andBo to be the same for the bubble and the drop systems, but the pressure gradient term cannot be made to match in both the inner and outerfluids simultaneously for the two systems. Thus we conclude that for higher Ga andBo, the equivalence is theoretically impossible for all bubbles and drops. After failing tofind any exact equivalence between a bubble and a drop at higher Gallilei and

34 Bond numbers, let us follow an order of magnitude analysis toﬁnd if a bubble can be designed to, atleast, approximately behave as a drop for this regime. The force balance in terms of the order of magnitudes may be expressed as

Δ p 2v v σ O ρ +O µ +O[gΔ ρ] +O , (3.30) R ∼ R o R2 R2 � � � � � � � � � wherein, the diﬀerent terms on the� right hand� side represent� the inertial, viscous, gravitational, and surface tension contribution to the total force per unit volume. Here, the text decoration, tilde is to represent the dimensional quantities. For pressure gradients to be the similar in both, bubble and drop, the terms involving density and viscosity. i.e.

2v 1, (3.31) gR � � or, the dimensionless velocity,v (which is also the Froude number in this case) should be several order of magnitudes less than 1. When the bubble/drop reaches a terminal velocity and shape, the drag force is comparable its buoyant weight. Therefore,

v µ gΔ ρ, (3.32) o R2 ∼ � Non-dimensionalization of this equation yields,� �

v Ga(r 1). (3.33) ∼ −

Additionally, the condition that surface tension force is very large as compared to the inertial forces, we have 2v σ ρ , (3.34) o R � R2 or, in the dimensionless form, � � 1 v . (3.35) � √Bo

From relations (3.33) and (3.35), the condition for equivalence between a bubble and a drop, in dimensionless form can be expressed as

1 Ga2(r 1) 2 . (3.36) − � Bo

According to the inequality (3.36), the equivalence can be obtained in the limit,ρ 1 even if r → Ga andBo are not small. This analysis gives the condition for equivalence of the forces in bubble and drop system. A diﬀerent route can be adopted to equate the pressure distribution on the bubble and the drop for any Ga andBo, thus providing a condition for similarity of shapes.

Shape equivalence - pressure arguement

Assuming the surface tension forces to be large compared to inertia and viscous forces, and using subscriptsb andd for the bubble and the drop and unscripted variables for the continuous phase,

35 we write the pressure diﬀerence at the tip of theﬂuid blob as

p p=σ k , b − b b

p p=σ k , d − d d rd pd =p+k dσb , rb or,

pd/rd = p/rd +k dσb/rb, (3.37)

and

pb/rb = p/rb +k bσb/rb. (3.38)

Although these equations are for the tip of the bubble, they could also be thought to be valid for the average pressure (p) and curvature (κ) over the whole surface of the bubble in an approximate sense. Subtracting equation (3.38) from (3.37), we get

1 1 σ p = b (k k ). (3.39) r − r r d − b � d b � b

Replacingr by 2 r andσ /r byσ /r , we get b − d b b d d 1 2r 1 σ p d − = d (k k ), r − r r d − b � d d � d or, 2p(1 r ) =σ (k k ). (3.40) − d d d − b For the shape of the bubble and the drop to be almost the same, i.e. mathematicallyk k �, b − d ∼ where� << 1. Eq. (3.40) suggests that

r 1+ �σ /p 1+� d ≈ d ∼

, or

rd = 1 +δ, (3.41) whereδ << 1. Hencer = (1 +δ)/(1 + 2δ) (1 +δ)(1 2δ), or b ≈ −

r 1 δ. (3.42) b ≈ −

Eqs (3.41) and (3.42) show that the equivalence in pressure at the surface of the bubble and drop is possible only when the bubble and drop densities are very close to 1. Therefore the analyses presented above suggest that the similarity between a bubble and drop is not possible for bigger and faster moving bubbles/drops.

36 (a) (b)

Figure 3.5: (a) Evolution of bubble shape with time forρ r = 0.9,µ r = 0.5, Ga = 50,Bo = 50; (b) evolution of drop shape with time forρ r = 1.125,µ r = 0.625, Ga = 50,Bo = 50. The direction of gravity has been inverted for drop to compare the respective shapes with those of the bubble. Even for high Ga andBo, the dynamics can be made similar if density ratios are close to unity.

3.4 Diﬀerences in bubble and drop dynamics

At small inertia and moderate surface tension, bubbles and drops do not remain spherical, but show similar behaviour for density ratios close to 1, as shown in Fig. 3.5. We remove the effect of gravity (which causes a pressure gradient to appear in thefluid) to observe if any differences arise between the dynamics of a bubble and drop. We replace the Gallilei number with a Reynolds number defined asRe U R/ν based on the initial drop velocityU . For a density ratio close to unity (Fig. 3.6), a ≡ 0 0 bubble and a drop behave similarly and stop after some time due to viscous dissipation. However, for density ratios far from unity (Fig. 3.7), the bubble oscillates and comes to rest, while a drop breaks up after forming a bag-like structure. This suggests that a bubble and a drop cannot behave similarly even when gravity is not present. The numerical results presented in this chapter have been obtained using the diffuse interface solver as well as gerris (see Chapter 2). A bubble is initialized as a sphere with stagnant conditions at a height of 8R from the bottom of the domain (Fig. 2.1(a)). A drop is initialized in a similar fashion, with the opposite sign of gravitational acceleration. The axis of symmetry passes through the diameter of bubble/drop and a Neumann condition on scalars (p, andc a) and vertical component of velocity, and a zero dirichlet condition on radial velocity component is imposed. Neumann boundary conditions are imposed for all variables on the remaining boundaries. Bubbles and drops of higher inertia where inequalities (3.31) and (3.36) are not followed are shown in Fig. 3.8. Drop shapes obtained numerically for a density ratio close to unity are shown for this case in Fig. 3.8(b). The Galilei and Bond numbers are the same for the bubble and the drop,

37 Figure 3.6: Dynamics in the absence of gravity: (a)evolution of bubble shape with time forρ r = 0.9, µr = 0.5, Re = 50, Bo = 50, (b) evolution of drop shape with time forρ r = 1.125,µ r = 0.625, Re = 50, Bo = 50. The initial shape of both drop and bubble was kept spherical and the initial velocity given to theﬂuid blobs isU 0 = 1 for both. The shapes of bubble and drop tend to be similar for density ratios close to unity.

38 Figure 3.7: Dynamics in the absence of gravity: (a) evolution of bubble shape with time forρ r = 0.52, µr = 0.05,Re = 50,Bo = 50, (b) evolution of drop shape with time forρ r = 13,µ r = 1.25,Re = 50, Bo = 50. The initial shape of both drop and bubble was kept spherical and the same initial velocity U0 given to bothﬂuid blobs. The bubble regains a spherical shape, whereas the drop breaks up in the bag-breakup mode.

and the densities are related by Eq. (3.22). The Reynolds number based on the terminal velocity of the bubble and the drop is about 16. The kinematic viscosity ratio m/r for the bubble is kept the same as the drop in Fig. 3.8(a) and related by Eq. (3.21) in Fig. 3.8(c). It is apparent that a drop and its equivalent bubble behave qualitatively the same. The velocities of the drop and bubble are closer to each other when the viscosity relation (3.21) is used whereas the shapes are closer together when they have the same kinematic viscosity ratio.

When the density ratio is far from unity, no equivalence is possible. Eq. (3.21) is no longer valid, nor possible to satisfy. We therefore compare drops and bubbles of the same m/r. Fig. 3.9 makes it evident that neither the shape nor the velocity of the drop and bubble are similar to each other.

Shown in color in thisfigure is the residual vorticityΩ [8], which is a good measure of the rotation in aflow. A detailed discussion of what is the best way to estimate rotation within a drop is available in [179]. In axisymmetricflow, the velocity-gradient tensor may be spilt into a symmetric part and an anti-symmetric part. The anti-symmetric part is the vorticity, of magnitudeω, oriented azimuthally. The eigenvalues of the symmetric part are given by s/2, wheres = (4u 2 +(u+w) 2)1/2. ± The vorticity in turn can be decomposed into shear part and a pure rotational part. The latter is

39 Figure 3.8: Evolution of (a) bubble shape with time forρ r = 0.9,µ r/ρr = 0.56. (b) drop shape with time forρ r = 1.125,µ r/ρr = 0.56. (c) drop shape with time forρ r = 1.125, with viscosity obtained from Eq. (3.21). The direction of gravity has been inverted for the drop in order to compare the respective shapes with those of the bubble. In all three simulations, Ga = 50, Bo = 50, and the initial shape was spherical. termed the residual vorticity, deﬁned [8] as

W=s 2 +s 2 ω 2, (3.44) n s − whereins ( ∂ u ∂ w) ands ( ∂ w ∂ u) are the normal and the shear components of the n ≡ x − z s ≡ x − z strain rate tensor respectively, is another measure of rotation in theflow. Both measures give similar images of the vortex cores in our simulations, but since the residual vorticity takes care to remove the shear part of the vorticity, we present results using this quantity. At later times in Fig. 3.9, it is evident that residual vorticity is concentrated within the bubble but outside the drop. This is the primary difference between a bubble and a drop. The region of low pressure and high vorticity tends to lie in the lighterfluid. In the case of the bubble, this causes an azimuthally oriented circulation in the lower reaches, which then leads to a fatter base and aids in a pinch-off at the top of the bubble. In the case of a drop, the vorticity being outside means that the lower portion of the drop is stretched into a thin cylindrical sheet, and an overall bag-like structure is more likely. Also a pinch off in this sheet region is indicated rather than a central pinch-off. We present in Figs 3.10 and 3.11 streamlines at various stages of evolution in this simulation. Closed

40 (a) (b)

Figure 3.9: Evolution of (a) bubble and (b) drop shapes with time, when densities of outer and innerfluid are significantly different. As before, for the drop (b), the direction of gravity has been inverted. In both simulations Ga = 50 andBo = 10. The other parameters for the bubble system areρ r = 0.5263 andµ r = 0.01, while for the drop systemρ r = 10 andµ r = 0.19. Note the shear breakup of the drop at a later time. Shown in color is the residual vorticity [8].

41 Figure 3.10: Streamlines in the vicinity of a bubble fort = 1, 2, 3 and 4 for parameter values Ga = 50,Bo = 10,ρ r = 0.5263 andµ r = 0.01. The bubble is shown in grey and a red outline. The circulation can be seen lying inside the bubble, which does not allow the bubble to thin out at its base.

Figure 3.11: Streamlines in the vicinity of a drop fort = 1, 2, 3 and 4 for parameter values Ga = 50, Bo = 10,ρ r = 10 andµ r = 0.19. The bubble is shown in grey and a red outline. The direction of gravity has been inverted to compare the shapes with those in Fig. 3.10. The circulation is seen to move out of the drop, making the drop to thin out at its trailing end. streamlines are visible in the region of lower density, indicative of regions of maximum vorticity being located in the lighterfluid. The fact that regions of low pressure and high vorticity would concentrate in the less densefluid follows directly from stability arguments. A region of vorticity involves a centrifugal force oriented radially outwards, i.e., pressure increases as one moves radially outwards from a vortex. There is a direct analogy between density stratification in the vicinity of a vortex and in a standard Rayleigh Bénardflow [180]. In the latter, we have a stable stratification when density increases downwards. In the former, we have a stable stratification when density increases radially outwards, i.e., when the vortex is located in the less dense region. Fig. 3.12 is a demonstration that for the same outerfluid even if the viscosity of the bubble and the drop were kept the same, and only the densities of the two were different, the behavior discussed above is still displayed. Our results thus indicate that density is the dominant factor rather than viscosity in determining the shapes of inertial drops and bubbles. In particular, the vorticity maximum tends to migrate to the region of lower density, and this has a determining role in the shape of the structure. Since large density differences bring about this difference, these are effectively non-Boussinesq effects. We note that given the large number of parameters in the problem, including initial conditions, which we have keptfixed, the location of maximum vorticity in the less dense region may not be universally observed in all bubble and drop dynamics. For example, the Widnall instability [181] in drops resembles the central break-up we have discussed. In the usual set-up of the Widnall instability, the densities of the inner and outerfluid are close to each other, so we may expect the

42 Figure 3.12: Evolution of (a) bubble and (b) drop (gravity reversed) shapes with time. Parameters for both bubble and drop systems are: Ga = 100,Bo = 50 andµ r = 10. The density ratio for the bubble and drop areρ r = 0.52 andρ r = 13 respectively, based on Eq. (3.22). Theﬁgure shows that the density, rather than viscosity, decides the location of vortical structures, which results in altogether diﬀerent deformation in bubbles and drops.

43 drop to behave similar to a bubble, and the initial conditions are not stationary. Our arguments above on vorticity migrating to the lighterﬂuid do not depend on gravity being present. We also conﬁrm this in simulations which obtain the motion of a bubble and a drop started with a particular initial velocity in a zero-gravity environment. These are shown in Fig. 3.7.

3.5 Before breakup

Break-up of drops and bubbles is typically a three-dimensional phenomenon on which much has been said, see e.g. the experimental studies of Elzinga & Banchero [182], Blanchard [183] and Hsiang & Faeth [184], the theoretical work of Kitscha & Kocamustafaogullari [185] and Cohen [186], and numerical studies of Jing & Xu [187] and Jalaal & Mehravaran [34]. The transient behavior of liquid drops has been discussed extensively [188–191], especially in the context of internal combustion engines, emulsification, froth-formation and rain drops. However, the transient behavior of bubbles has not been commented upon as much, and we make a few observations, regarding large-scale oscillations, that are not available in the literature to our knowledge. We note that since our simulations are restricted to axisymmetric break-up they may not always capture the correct breakup location or shape. Various parameter ranges are covered in numerous papers in the past 100 years, and it is known that drops and bubbles break up at higher Bond numbers. The Bond number below which a bubble of very low density and viscosity ratio does not break, but forms a stable spherical cap, is about 8 [192] which is similar to that found in our simulations. We begin by associating a time scale ratio with the Bond number. It may be said that surface tension would act to keep the blob together whereas gravity, imparting an inertia to the blob, would act to set it asunder. The respective 3 time-scales over which each would act may be written asT s = ρR /σ for surface tension and Tg = R/g for gravity. The ratio � Tg 1 � = , (3.45) Ts √Bo is a measure of the relative dominance. At Bo >> 1, surface tension is ineffective in preventing break-up, and we may expect a break-up at a time ofO(1), since we use gravitational scales. For Bo 1, it is reasonable to imagine a tug-of-war to be played out between surface tension and inertia ∼ in terms of shape oscillations, with a frequency ofO(1). Since a bubble usually breaks up at the centre by creating a dimple, the vertical distanceD d of the top of the dimple from the top of the bubble is a useful measure to observe oscillations. Fig. 3.13 shows the dimple distance as a function of time for various Bond numbers, and our expectations are borne out. Figs 3.14 and 3.15 are typical streamline patterns in the vicinity of bubbles in the break-up and recovery cases respectively. Instantaneous streamlines are plotted by taking the velocity of the foremost point of the bubble as the reference, but the picture is qualitatively unchanged when the velocity of the centre of gravity of the drop is chosen instead. Both cases are characterized by a large azimuthal vortex developing within the bubble initially. In the break-up case (Fig. 3.14), this vortex is sufficient to cause the bubble surface to rupture and obtain a topological change, from a spherical-like bubble into a toroidal one. In all the cases of bubble recovery we have simulated, of which a typical one is shown (Fig. 3.15), there develop at later times several overlaying regions of closed streamlines, which act to counter the effect of rupture by the primary vortex, and to bring

44 Figure 3.13: Variation of dimple distance versus time for diﬀerent Bond numbers for Ga = 50, 4 6 ρr = 7.4734 10 − ,µ r = 8.5136 10 − . The tendency of a bubble to break from the center is evident. However,× a bubble may form× a skirt for intermediate Bond numbers (Bo = 15), which may lead to breakup or shape oscillations in certain cases.

Figure 3.14: Streamlines in and around the bubble at time,t = 1, 1.5, 2.0 and 2.5 respectively, for 4 6 Ga = 50,Bo = 29,ρ r = 7.4734 10 − andµ r = 8.5136 10 − . The shape of the bubble is plotted in red. The toroidal vortex inside× the bubble maintains× the thickness of its base as the liquid jet penetrates the remaining airﬁlm at the top.

Figure 3.15: Streamlines in and around the bubble at time,t=2.5, 5, 7, 9 and 11 respectively, for 4 6 Ga = 50,Bo = 15,ρ r = 7.4734 10 − andµ r = 8.5136 10 − . Three toroidal vortices form inside and outside the bubble which× compete with the surface× tension force to make the bubble shape oscillate.

45 back the drop to a shape that is thicker at the centre. During each oscillation, we see the upper and lower vortices form and disappear cyclically.

3.5.1 Eﬀects of viscosity

In this section we study the effects of viscosity on the tendency to break-up. That decreasing external fluid viscosity (increasing Gallilei number) will increase the tendency to break-up is demonstrated in Figs 3.16 and 3.17 for two Bond numbers. Also oscillations become more prominent at higher Ga. If the viscosity ratio is low enough, the outerfluid is able to shear-break the drop. It is already known [186] that a higher Weber number is required to break a drop when the surroundingfluid is more viscous. This indicates that the more the viscous drag, i.e. the less the inertia of the blob, the less willing it is to break. If the surroundingfluid is more viscous, we would need to increase gravity or reduce surface tension to break a blob. Thus, in effect, a higher Bond number is needed to break a blob. Fig. 3.17 shows that when all the physical properties are kept the same while the kinematic viscosities are reduced in the same proportions for inner and outerfluids, the bubble tends to break up.

Figure 3.16: Variation of dimple distance versus time for diﬀerent Gallilei numbers for Bo = 8, 4 6 ρr = 7.4734 10 − ,µ r = 8.5136 10 − . The bubble shapes are shown at corresponding times for Ga = 5 (top)× and 125 (bottom).× The shape oscillations ensue after a threshold in outerﬂuid’s viscosity i.e. Ga.

3.5.2 Drop breakup

A typical breaking drop, with its associated streamlines is shown in Fig. 3.18. As discussed above, the breakup is very diﬀerent from that of a bubble, since the primary vortical action is outside, and causes a thinly stretched cylindrical, rather than toroidal shape. The vortex in the wake of the drop tends to stretch the interface (and surface tension is not high enough to resist the stretching) which leads to an almost uniformly elongated backward bag. New eddies are formed due toﬂow separation at the edge to the “bag” and a toroidal rim is detached

46 4 Figure 3.17: Variation of dimple distanceD d versus time for Bo = 29,ρ r = 7.4734 10 − ,µ r = 6 × 8.5136 10 − . Bubble shapes are shown for non-oscillating (top, black), oscillating (blue) and breaking× (bottom, black) bubbles.

Figure 3.18: Streamlines in and around the drop at time,t=4.5, 6 and 7.5, respectively (from left to right), for Ga = 50,Bo = 5,ρ r = 10 andµ r = 10. The circulation zones form outside the drop, as observed in Fig. 3.9. after some time from this bag. A drop too responds to Bond number, but the response is shown in terms of an early break-up at high Bond numbers, as seen in Fig. 3.19. The shape at break-up too evolves with the Bond number, as shown.

3.6 Summary

A bubble and a drop, starting from rest and moving under gravity in a surroundingfluid, cannot in general be designed to behave as one another’s mirror images (one rising where the other falls). We have shown that the underlying mechanism which differentiates the dynamics is that the vorticity tends to concentrate in the lighterfluid, and this affects the entire dynamics, causing in general a thicker bubble and a thinner drop. However, if inertia is small, surface tension is large, and a drop is only slightly heavier than its surroundingfluid, a suitably chosen bubble can display dynamics similar to it. In this limit, the Hadamard solution can be exploited to design a bubble with its

47 Figure 3.19: Variation of break-up time with Bond number for Ga = 50,ρ r = 10 andµ r = 10. A typical bag breakup mode is shown in thisfigure. Shapes of the drop just before breakup are shown for various Bond numbers. density and viscosity suitably chosen to yield the same acceleration at any time as a given drop. We are left with an interesting result: while a solid ‘bubble’ can never display aflow history which is the same as a solid ‘drop’, a Hadamard bubble can. Also, although density differences are small, the Boussinesq approximation cannot lead us to the closest bubble for a given drop. Wefind numerically that a similarity in bubble and drop dynamics and shape is displayed up to moderate values of surface tension and inertia, so long as the density ratio is close to unity. In axisymmetricflow, the vorticity concentrates near the base of the bubble, which results in a pinch-off at the centre whereas the cup-like shape displayed by a drop, and the subsequent distortions of this shape due to the vorticity in the surroundingfluid, encourage a pinch-off at the sides. Bubbles of Ga higher than a critical value for a givenBo will break up at an inertial time between 2 and 3. For Ga orBo just below the critical value, oscillations in shape of the same time scale occur before the steady state is achieved.

48 Chapter 4

Three dimensional bubble and drop motion

4.1 Introduction

In the preceding chapter, we have investigated the motion of rising bubbles and falling drops under the assumption that the dynamics is axisymmetric. This assumption breaks down for large Gallilei and Bond numbers, as observed in several experimental as well as numerical studies [1, 11, 32, 83]. Interestingly, the fact that this dynamics is three-dimensional wasfirst documented by Leonardo Da Vinci in the 1500s, in his book Codex leicester which was recently discovered by Prosperetti [18]. Leonardo Da Vinci found that the bubbles rise in zigzagging and spiralling trajectory even if released axisymmetrically under a coloumn of water. This is currently known as path instability and it has been the subject of a host of experimental [9, 42, 78, 79], numerical [10, 80] and analytical [81, 82] studies. Most of the workers embarking on this studyfind it satisfactory to investigate the effect of initial bubble diameter on the rising dynamics, and no experimental investigations are available to our knowledge which study the effects of just the surface tension or viscosity of liquid on the bubble rise. However, a recent study discusses the effect of viscosity ratio on the drop dynamics and breakup for immiscible liquid-liquid systems [193]. A vast majority of the earlier experimental and theoretical studies have had one of the following goals (i) to obtain the rise velocity (ii) to evaluate the path instability (iii) to understand bubbly flows, (iv) to make quantitative estimates for particular industrial applications, and (v) to derive models for estimating different bubble parameters. Most of these restrict themselves to only a few Ga or Bo. Our study, in contrast, is focussed on the dynamics of a single bubble/drop. Starting from the initial condition of a spherical stationary blob, we are interested in delineating the physics that can happen. We cover a range of several decades in the relevant parameters. In thefirst part of this chapter, a study of bubbles rising due to buoyancy in a far denser and 3 more viscousfluid is presented. Therefore in this part of the work,ρ r andµ r arefixed at 10 − 2 and 10− , respectively. We show that as the size of the bubble is increased, the dynamics goes through three abrupt transitions from one known class of shapes to another. A small bubble will attain an axially symmetric equilibrium shape dictated by gravity and surface tension, and travel vertically upwards at a terminal velocity thereafter. A bubble larger than afirst critical size loses

49 axial symmetry. We show that this can happen in two ways. Beyond the next critical size, it breaks up into a spherical cap and many satellite bubbles, and remarkably, the cap regains axial symmetry. Finally, a large bubble will prefer not to break up initially, but will change topologically to become an annular doughnut-like structure, which is perfectly axisymmetric. In the latter part of this chapter, we present a three-dimensional study of the effect of viscosity and density ratios on drop dynamics. It is shown that the effect of density ratio is to increase the inertia of the drop and thus the drop tends to breakup with an increase in the density ratio. The effect of viscosity ratio is shown to delay the breakup. Also, it is confirmed by three-dimensional simulations that a drop tends to break up from periphery rather than its center, whereas a bubble often breaks up at its center, at high inertia and low surface tension forces. A large portion of this chapter is contained in one of our published works [19].

4.2 Bubbles

The results of more than 130 three-dimensional simulations of single bubbles rising due to buoyancy are presented in this section. The open-source volume-of-ﬂuid solver, gerris has been used due to its dynamic adaptive grid reﬁnement feature and one of the best algorithms for inclusion of surface tension force in Navie-Stokes equation [175]. The simulation domain is shown in Fig. 2.1(b). The boundary conditions on all sides of the domain is symmetry i.e. Neumann condition for scalars

(p andc a) and velocity components tangential to the given boundary, and zero dirichlet condition for velocity components normal to the given boundary. The bubble is initialized as a sphere of unit radius in the dimensionless terms. The dimensionless governing equations and the constitutive relations (Eqs 2.15-2.27) can be simpliﬁed to:

u=0, (4.1) ∇· ∂u 1 δ ρ +u u = p+ µ( u+ u T ) ρ�e + κ�n, (4.2) ∂t ·∇ −∇ Ga∇· ∇ ∇ − z Bo � � � � ∂c a +u c = 0, (4.3) ∂t ·∇ a along with, ρ = (1 c )ρ +c , (4.4) − a r a µ = (1 c )µ +c , (4.5) − a r a The results obtained from the simulations are discussed next.

4.2.1 Regimes of diﬀerent behaviours

Fig. 4.1 represents a summary of what happens to an initially spherical bubble rising under gravity in a liquid. A range of ratios of gravitational, viscous and surface tension forces have been simulated (in about 130 simulations). Several features emerge from this phase plot, which is divided intoﬁve regions. Region I, at low Bond and Galilei numbers, is shown in pink. In this region, surface tension is high and gravity is low, so it is understandable that the bubble retains its integrity. It attains a constant ellipsoidal shape, of which a typical example is shown in theﬁgure in that region, and takes

50 Figure 4.1: Different regimes of bubble shape and behaviour. The different regions are: axisymmetric (circle), asymmetric (solid triangle) and breakup (square). The axisymmetric regime is called region I. The two colors within the asymmetric regime represent non-oscillatory region II (shown in green), and oscillatory region III (blue) dynamics. The two colors within the breakup regime represent the peripheral breakup region IV (light yellow), and the central breakup region V (darker yellow). 3 The red dash-dotted line is theMo = 10 − line, above which oscillatory motion is not observed in experiments [1,9]. Typical bubble shapes in each region are shown. In this and similarfigures below, the bubble shapes have been made translucent to enable the reader to get a view of the internal shape. on a terminal velocity going straight upwards. The bubble is axisymmetric in this region. Region II, at high Bond numbers and low Galilei numbers, is demarcated in green color. The bubble here has two distinct features, an axisymmetric cap with a thin skirt trailing the main body of bubble. The skirt displays small departures from axisymmetry in the form of waves, e.g., a wavenumber 4 mode is barely discernible in the typical shape shown. Bubbles in this region travel upwards in a vertical line as well, and practically attain a terminal velocity after the initial transients and display shape changes only in the skirt region. The extreme thinness, in parts, of the skirt presents a great challenge for numerical analysis, and a detailed study of this region is left for the future. Region III, depicted in blue colour, occupies lower Bond and higher Galilei numbers. Here surface tension and inertial forces are both significant, and of the same order. Bubbles display strong deviations from axisymmetry in this region, at relatively early times, and rise in a zigzag or a spiral manner. Bubbles remain integral but their shapes change with time. Region IV is shown in light yellow colour, and region V is in dark yellow. The bubble, faced with higher gravity and relatively weak surface tension, breaks up or undergoes a change of topology in these regions. Remarkably, the dynamics may be described well as axisymmetric up almost to the break-up. Region IV is a narrow region which may be described roughly as having a moderate value of the product GaBo. At low Ga and highBo (i.e high Morton number) the bubble in this regime breaks into a large axisymmetric spherical cap and several small satellite bubbles in the cap’s wake. We term this a peripheral break-up, since it involves

51 a pinch-off of a skirt region of the kind seen in region II. For high Ga and low Bo (i.e lowerMo) a new breakup dynamics is observed, not hitherto described to our knowledge, which is discussed below with Fig. 4.9. Significant among the results is the fact that in region IV, after break-up axisymmetry is regained and thefinal spherical cap bubble attains a constant shape and terminal velocity. Finally, the bubbles shown is region V are under the action of high inertial force and low surface tension force. A qualitatively different kind of dynamics is seen here. A dimple formation in the bottom centre leads to a change of topology: to a doughnut-like or toroidal shape as seen in thefigure. Close to the boundary of region IV, the change of topology may be accompanied by an ejection of small satellite bubbles. As Ga and Bo are increased further in this region, a perfectly axisymmetric change of topology of the whole bubble is observed. Unlike in the other regions, this new shape is not permanent. It eventually loses symmetry, and evolves into multiple bubble fragments. The boundaries between thefive regions are easy to distinguish because the time evolution is qualitatively different on either side. Details of how a bubble is assigned to a particular region are provided in the below. Moreover the sum of the kinetic and surface energies usually goes to a maximum at the transition between two regions, and falls on either side. This is discussed in more detail below. We had mentioned the Morton number above, defined asMo=Bo 3/Ga4. This

Figure 4.2: Dynamics expected for bubbles in different liquids. Constant Morton number lines, each corresponding to a different liquid, are overlayed on the phase-plot to demonstrate that our transitions can be easily encountered and tested in commonly found liquids. The initial radius of the air bubble increases from left to right on a given line. Circles, triangles and squares represent air bubbles of 1 mm, 5 mm and 20 mm radii, respectively. combination deserves a separate name because it depends only on thefluid properties and not on the bubble size. Air bubbles in a particularfluid at a particular temperature will lie on constant Morton number lines, which are straight lines in the log-log phase plot. The red dashed line in Fig. 3 4.1 corresponds to a Morton number of 10− , which is the Morton number mentioned in numerous experiments, see e.g. [1, 9] below which spiralling and zigzagging trajectories are seen. Note that

52 the boundary between regions II and III, i.e. between straight and zigzagging trajectories, in our simulations lies very close to this. The lines of constant Morton number corresponding to some common liquids at different temperatures are shown in Fig. 4.2. Since we have used very small viscosity and density ratios, our results apply to various air-liquid systems. In the examples given, the liquid densities are not far from water, and we know from [52,194] that the dynamics is insensitive to viscosity ratio for smallµ r. Moving upwards and to the right on a given line, the bubble size increases, and typical bubble sizes are indicated in thefigure. We see that our results apply to a range of liquids in which an estimate of bubble motion may be desired, for instance crude oil, water at different temperatures and cooking oils. A 1 mm radius bubble in water at room temperature will execute spiral or zigzag motion whereas a 20 mm bubble in honey will develop a skirt but move upwards in a straight line. We had recently shown [22] that a bubble is more likely to lose its original topology to attain a doughnut shape at high inertia and low surface tension, whereas a drop under the same Bond and Galilee numbers would tend to break into several drops. We predicted that non-Boussinesq effects are qualitatively different in drops and bubbles, since highly vortical regions are stable when situated within the lighterfluid. The present three dimensional simulations are a confirmation of this physics.

4.2.2 Path instability and shape asymmetry

Figure 4.3: Agreement and contrast between present and previous results for differentflow regimes. Comparison between the onset of asymmetric bubble motion obtained in the numerical stability analysis of Cano-Lozano et al. [10] (solid black line), and the present boundary between regions I and II. Also given in thisfigure arefive different conditions (diamond symbols) studied by Baltussen et al. [11]. The dynamics they obtain are as follows: A - Spherical, B - Ellipsoidal, C - Boundary between skirted and ellipsoidal, D - Wobbling and E - Peripheral breakup. The correspondence between present results and [11] is excellent. Grace et al. [12] obtained spherical bubbles below the solid blue line shown.

53 Two portions of our phase space have received particular attention earlier. Thefirst, which we have spoken of earlier, is the onset of zigzagging motion, famously referred to as the path instability. Ryskin & Leal [195] and many other studies believed the path instability to occur due to vortex shedding from the bubble. Indeed in the motion of solid objects throughfluid this is the only way in which one can get a path which is not unidirectional. Magnaudet & Mougin [80] assumed the bubble to be ellipsoidal in shape, and studied a constant velocityflow past such a bubble to obtain the instability of its wake. The bubble shape and position were heldfixed during the simulation. An asymmetric wake was taken to be indicative of the onset of zigzagging motion. Cano-Lozano et al. [10] repeated a similar analysis, but on a realistic bubble shape, which they obtained from axisymmetric numerical simulations. The bubble was held in a constant velocity inletflow equal to the terminal velocity obtained in their simulations for the axisymmetric shape. Wake instabilities were the investigated from a three-dimensional simulation of thisfixed bubble. Wefind that this simplified method yields a good qualitative estimate of the onset of zig-zagging motion at low inertia. A comparison with our more exact three-dimensional simulations is shown in Fig. 4.3, where quantitative discrepancies are noticed, especially for large inertia, i.e., Ga > 50. Also given in thisfigure is a comparison with the very recent results of Baltussen et al. [11]. Forfive different pairs of Ga and Bo, the dynamics predicted by these authors may be seen to be confirmed by present results. While we did not distinguish our shapes into spherical and ellipsoidal, we note that the boundary provided by Grace et al. [12], also shown in thisfigure, between spherical and non-spherical shapes, is consistent with ourfindings. The line falls well within our regions I and II where we have ellipsoidal drops, and in region II lies close to the minimumBo of our computations.

(a) (b) (c)

Figure 4.4: Dynamics and shapes of region III bubbles: trajectory of the bubble centroid for (a) Ga = 70.7,Bo = 10, and (b) Ga = 100,Bo = 4, and (c) shape evolution of bubble corresponding to the latter case. In panel (c), the radial distance of the center of gravity (rs) of the bubble measured in the horizontal plane from the original location is shown below the shapes at each time.

A point to note is that unlike solid spheres, departures from vertical motion in a bubble can be caused either by shape asymmetries, or unsteady vortex shedding, or both. The stability analyses discussed above take account only of the latter, whereas experiments, e.g. those of De Vries et al. [196] in clean water found a regime of path instability where no vortex shedding was expected.

54 (a) (b) (c)

Figure 4.5: Diﬀerences between two dimensional and three dimensional bubble shapes: (a) A region III bubble att = 20 for Ga = 100 andBo=0.5, (b) att = 30 for Ga = 100 andBo = 4, again in reign III, and (c) a region IV bubble att = 5 for Ga = 70.71 andBo = 20. The second row shows the side view of the three-dimensional shapes of bubbles rotated by 90 degrees about thex = 0 axis with respect to the top row.

(a) (b)

Figure 4.6: Characteristics of a region III bubble of Ga = 100 andBo=0.5. (a) Oscillating upward velocity, with diﬀerent behaviour at early and late times, (b) trajectory of the bubble centroid. The two regions corresponding to two diﬀerent behaviours in the rise velocity correspond to the inline oscillations and zig-zagging motion.

In fact a recent analytical study [82] attempts to explain that vortex shedding is the effect, rather than a cause of the path instability in rising bubbles. Without a statement as to cause and effect, we expect an intimate connection between loss of symmetry and loss of a straight trajectory. Any asymmetry in the plane perpendicular to gravity should result in an imbalance of planar forces. Similarly any asymmetry in the planar forces, due to vortex shedding or otherwise, should result in shape asymmetry. In accordance with these expectations, wefind that path instability and shape asymmetry go hand in hand, so the onset of path instability is just the boundary between regions I and III. Not just the onset, but the entire region III, where the bubble shape is strongly non- axisymmetric, coincides with the regime where path instability is displayed. Fig. 4.4 shows the trajectory, and the shape of a typical bubble in this region at different times. A helical-like motion is executed in the cases shown, while the shape is continuously changing. The bubble does not adopt a standard geometry. Incidentally, in several of the simulations, the centre of the helix does not coincide with the original location in the horizontal plane. Nor are the windings of the helix

55 (a) (b)

Figure 4.7: Region III bubble corresponding to that shown in Fig. 4.6 (Ga = 100 andBo=0.5). (a) Iso-surfaces of the vorticity component in thez direction at timet = 15 (ω z = 0.0007) and 26 (ω = 0.006), (b) The evolution of the shape of the bubble. The radial distance± of the center z ± of gravity (rs) of the bubble measured in the horizontal plane from the original location is shown below the shapes at each time.

periodic or regular. Most trajectories in this regime are indicative of chaotic dynamics. We also obtain trajectories resembling widening spirals, or those which execute a zig-zag motion with the centroid lying close to some vertical plane and there seems to be no particular region in the Ga Bo − plane where one or the other dominates. Zig-zag and helical motion is accompanied by oscillations in the vertical velocity as well, so the bubble alternately speeds up and slows down on its way. An example is seen in the vertical velocity plot of Fig. 4.6a. Two kinds of oscillatory behaviour in the velocity are clearly visible in thefigure, one with increasing oscillations at early times, and one with a different character at later times. At later times the dynamics is more erratic, but amplitudes of variation are lower. In thefirst part vorticity is generated in the wake but remains vertically aligned and attached to the bubble. At timet> 14 the drop begins to display zig-zag motion (see Fig. 4.6b). The wake now consists of a pair of counter-rotating two-threaded vortices, often considered to be afirst sign of path instability [80]. This is soon followed by shedding of the vortices, which begins att> 20. Wefind that the onset of the second type of unsteadiness may be attributed to the start of the vortex shedding off the bubble surface. The vertical component of vorticity in this regime is shown in Fig. 4.7a. In some cases wefind resemblences to the hairpin vortices of Gaudlitz & Adams [83]. The manner in which the shape of the bubble evolves during this process is shown in Fig. 4.7b. The correspondence between asymmetry in shape and the path instability is obvious. A few animations are available in http://www.iith.ac.in/ ksahu/bubble.html. ∼ We bring out the importance of three-dimensional simulations in Fig. 4.5 in regions III and IV. We saw that the path instability is deeply connected to shape asymmetries, so region III dynamics are inherently three-dimensional. In region IV the break-up is not axisymmetric. We note that region I can be well obtained from axisymmetric simulations.

56 4.2.3 Breakup regimes

We now examine the dynamics of bubbles destined for break-up, of regions IV and V. The contrast in bubble behaviour between these two regions is evident in Fig. 4.8. At early times both bubbles are axisymmetric. The region IV bubble develops a skirt, in this case similar to the one seen in region II, with the difference that this skirt then breaks off in the form of satellite bubbles, leaving an axisymmetric spherical cap. The region V bubble was seen tofirst undergo a change in topology into a doughnut or toroid shape. Beyond timet = 5, the toroid is subject to further instability, and breaks into a number of droplets. Pedley [197] had predicted that a perfectly toroidal bubble of circular cross section will undergo instability beyond a timet c. In our scales, the instability time of Pedley may be written ast GaBo1/2f 3/2, wheref is the ratio of the inner radius of the toroid c ∼ to the initial radiusR. Given that our toroidal bubble has a cross section very far from circular, we expect instability to set in much sooner, andfind break-up at times an order of magnitude lower thant c. In addition the history of theflow, including the vortex patterns, contribute to hastening instability. (a) (b)

Figure 4.8: Time evolution of bubbles exhibiting a peripheral and a central breakup. Three- dimensional and cross-sectional views of the bubble at various times (from bottom to top the dimensionless time is 1, 2, 4 and 5). (a) region IV, a bubble breaking into a spherical cap and several small satellite bubbles, Ga = 70.7 and Bo = 20, and (b) region V, a bubble changing in topology from dimpled ellipsoidal to toroidal, Ga = 70.7 andBo = 200.

Figure 4.9: A new breakup mode in region IV for Ga = 500 andBo = 1. Bubble shapes are shown at dimensionless times (from left to right)t = 2, 4, 6, 7, 8, 9 and 9.1).

Region IV bubbles show diﬀerent breakup dynamics at higher inertia and surface tension (low

57 Mo). For largeMo, a wide skirt was seen to form which then broke oﬀ into small bubbles, whereas for lowerMo values small bubbles are ejected from the rim of the bubble while it recovers from an initially elongated shape to the spherical cap shape. Bubbles of even lowerMo values, i.e., at high inertia and surface tension, are subjected to strong vertical stretching giving rise to a far narrower skirt, which results in an ellipsoidal rather than a cap-like bubble, and a small tail of satellite bubbles, as seen in Fig. 4.9. This type of break-up has not been reported before, to our knowledge.

Figure 4.10: Comparison of our 3D results with those of Bonometti & Magnaudet [13] for bubble breakup. The light yellow and dark yellow colours represent the regions for peripheral and central breakup. The corresponding data points from the present numerical simulation are shown as blue and black squares, respectively.

Before break-up, departures from symmetry are small in region IV bubbles. Similarly region V bubbles are symmetric up to toroid formation. We may thus ask whether cap or toroid formation requires three-dimensionality. The transition from a spherical cap to a toroidal shape, as obtained by Bonometti & Magnaudet [13] by means of axisymmetric computations are compared in Fig. 4.10 to our region IV – region V boundary, showing that the two trends agree qualitatively. Theﬁrst diﬀerence between the axisymmetric and 3D simulations was seen in region IV in Fig. 4.5. While the 2D simulations can only obtain break-up in the form of a ring that detaches from the spherical cap, our simulations enable the ejection of satellite bubbles. Another feature which the axisymmetric simulations will miss is the fact that the centre of gravity moves in the horizontal plane. Thirdly, just below the lowest point given by Bonometti & Magnaudet [13], we obtain a protrusion of region V (seen in deep yellow in Fig. 4.10) pointing to the left and downwards in the Ga Bo plane. − The dynamics in this protrusion region is asymmetric, and seems to have been missed by other axisymmetric simulations. We have now seen that a bubble which is initially spherical with a Ga andBo corresponding to regions IV and V will break up eventually. Does this mean that no single bubble can display a Ga andBo corresponding to this region? The answer is a no. Large single bubbles have been created

58 experimentally by many [33, 198]. It has been found in all of these studies that the stable shape for large-sized bubbles is a spherical cap. The initial conditions are extremely important for large bubbles, and experimenters take great care to generate an initial bubble which itself is in the form of a spherical cap. This is done by specially designed dumping cups. In fact Landel et al. [33] noted that only with a cup whose shape was very close to thefinal spherical cap bubble shape could they generate a stable bubble. Not just the curvature but particular care had to be taken to match the angle subtended by the cup shape at the centre of curvature to that of thefinal bubble shape, and to minimize external perturbations. In summary it was very difficult to create a single large spherical cap bubble since if these conditions were not enforced, the bubble would break up and satellite bubbles were inevitably present in the wake. Additionally, Wegener & Parlange [28] observes that in general spherical cap bubbles undergo tilting and wrinkling of their bottom, which results in the occasional peel off of satellite bubbles.

The largest spherical cap bubbles that have been thus observed, to our knowledge, have Ga 10 4 ∼ andBo 10 2 [33], which are well beyond the regime we have investigated. Batchelor [199] conducted ∼ a stability analysis of a steady rising spherical cap bubble to obtain an estimate of the largest stable bubble. This size is far smaller, and lies in regime V of our phase plot. These studies, and the computations of Ohta et al. [200], underline the importance of initial conditions in this problem. In addition to spherical cap bubbles, toroidal bubbles too of much larger size have been experimentally observed by Landel et al. [33] for diﬀerent initial conditions and parameters. Our results show that a bubble which starts from a spherical shape has a vastly diﬀerent fate, and can stay integral only when much smaller.

4.2.4 Upward motion

The vertical velocities of bubbles in the diﬀerent regions is characterised in Fig. 4.11. In region I, the vertical velocity monotonically increases and saturates at a terminal value. In region II, some minor oscillations are displayed initially owing to the skirt formation, but again a terminal velocity is reached. Region III displays oscillations of amplitude 25% of the average velocity, but these were ∼ seen to quieten down somewhat once vortex shedding begins. Regions IV and V display irregular but large oscillations in the velocity. In both regions the oscillations are small at later times, but while in region IV, theﬁnal velocity is close to its maximum, in region V the upward movement of the centre of gravity of the dispersed phase has slowed down to about half its original velocity. This is because the bubble has disintegrated considerably in the latter case.

The variation of dimensionless terminal velocity,w T versusBo for diﬀerent values of Ga is plotted in Fig. 4.12. It can be seen that decreasing the value ofBo results in an increase in the terminal velocity for all values of Ga; however, as expected the rate of increase of the terminal velocity is higher for higher values of Ga. The bubbles which exhibit peripheral breakup (i.e. bubbles lying in region IV in our phase-plot, Fig. 4.1) tend to have an increase in their average rise velocity because of the presence of satellite bubbles [33].

59 (a) (b) (c) 2.5 2.5 2.5 2 2 2 1.5 1.5 w w w1.5 1 1 1 0.5 0.5 0.5

00 2 4 6 8 00 2 4 6 8 10 00 10 20 30 40 t t t (d) (e) 2.5 2.5

2 2

1.5 1.5 w w 1 1

0.5 0.5

00 2 4 6 8 10 00 5 10 15 t t

Figure 4.11: Rise velocity for bubbles having markedly diﬀerent dynamics. (a) region I: axisymmetric (Ga = 10,Bo = 1) (b) region II: skirted (Ga = 10,Bo = 200), (c) region III: zigzagging (Ga = 70.7, Bo = 1), (d) region IV: oﬀset breaking up (Ga = 70.7,Bo = 20) and (e) region V: centrally breaking up bubble (Ga = 70.7,Bo = 200). In addition to the upward velocity, the in-plane components are unsteady too in regions III to V.

4.3 Determination of the behaviour type

4.3.1 Shape analysis

Assignment of a given bubble dynamics to a region is straightforward given that behaviour is so different on either side of each boundary. Bubbles which break up and those which do not are clearly evident in visual examination of the time evolution of the shape. Similarly the difference between the two kinds of break-up (region IV-V) is very evident. The boundary between regions II and III is again evident by visual examination, since (a) the shapes are very different on either side of the boundary (b) the path in region II is oscillatory whereas region III bubbles move up in a straight line. The green and blue colour in the phase plot (Fig. 4.1) combine to give the region in the Ga-Bo plane where the bubble assumes an asymmetric shape. The asymmetry is computed as follows. The bubble is cut with 8 vertical planes in order to get 8 cross sections, each successive plane separated by an angle ofπ/8 radians. The area of a vertical face of each cross-section of the bubble is calculated and the percentage difference in the area with respect to a reference cross-section (lying in they z − plane) is obtained. The root-mean-squared value of this data at each time step represents the degree

of asymmetry,δ a. Because of theO(Δx 2) scheme used inﬁnite volume discretization, the error in calculation of area of cross-section(A) may be estimated as

ΔA ΔL ΔL v + h , (4.6) A ≈ Lv Lh

60 Figure 4.12: Variation of dimensionless terminal velocity with Bo for diﬀerent Ga. The terminal velocity tends to decrease with decreasing surface tension because of the increased drag on the bubble.

whereL v andL h are the bubble dimensions in the vertical and horizontal directions, in the cross- sectional plane. The errors in the bubble dimensions,ΔL v andΔL h are of the order of the square of the smallest grid size, i.e. 0.0292 for a simulation with the coarsest mesh used in our study i.e. Δx=0.029. Thus we obtainΔA/A 2 0.029 2, or 0.0017 which is about 0.2%. The root-mean- ≈ × ≈ squared error percentage is calculated for all the cross-sections, which is denoted byδ a in this text.

To be conservative, any variation within 0.5% inδ a is considered to represent a symmetric bubble whereas, the bubble is considered to be asymmetric whenδ a exceeds 0.5%. As described in the main manuscript, shape asymmetry can be seen with or without an ac- companying asymmetrical motion in the horizontal plane. The motion of the bubble is obtained by tracking the center of gravity (centroid of the bubble) of the bubble with time. Our measure of deviations from azimuthal symmetry,δ a is for both kinds of asymmetry i.e. oscillatory as well as non-oscillatory, whereas the centroid motion gives information about the deviation from vertical motion, i.e. the path instability.

4.3.2 Energy analysis

We found that the sum of the kinetic and surface energies usually goes to a maximum at the transition between any two of theﬁve regions identiﬁed in the phase plot (Fig. 4.1), showing minima on either side (Fig. 4.13). The kinetic and surface energies have been computed at the steady state/quasi- steady state for bubbles lying in all the regions of the phase plot except for the region V.

4.4 Drops

Three-dimensional study of drops falling under gravity has been presented in this section. This is an ongoing work, therefore only some of the preliminary work has been discussed here. Eﬀect of inertia on the drop dynamics is shown in Fig. 4.14. For low density ratios the drop remains almost spherical

61 (a) (b) 250

60 200

40 150 TE TE 100 20 50

0 0 10 100 0.1 1 10 100 Ga Bo

Figure 4.13: Variation of the sum of kinetic and surface energies (TE) for (a) Bo = 20, and (b) Ga = 100. The peak in energy corresponds to the boundaries of the regions shown in Fig. 4.1.

and deforms into a dimpled ellipsoidal shape for slightly higher values of the parameter. The drop tends to take an upward opening cup-like structure for higher values of density ratio, which is also

observed in axisymmetric simulations [6,22]. For the values ofρ r approximately greater than 20, the surrounding medium tends to shear oﬀ a thin portion of the drop leading to a thin skirt-like structure emanating from the periphery of the drop. This shearing may occur at multiple locations at drop surface, resembling a Kelvin-Helmholtz like instability which is more pronounced for larger values of th th ρr (for instance see the 6 and 7 row of Fig. 4.14). For density ratios of the order of 100 or more, a violent breakup may occur, leading to multiple fragments of the drop. These regimes may change

depending on the other parameters i.e. Ga, Bo andµ r. A computational study of fragmentation of falling drops has been conducted by Jalaal and Mehravaran [34]. For higher density ratios, the dynamics may be more chaotic and would need further attention. Drop fragmentation results due to Villermaux & Bossa [191] show a reverse bag breakup mode which are not observed for density ratios upto 100 in the present results. Although this regime(very low Bond number, high Gallilei number and high density ratio) is very diﬃcult to simulate and the computational cost is very high, the dynamics need to be understood as it is an essential part in understanding rain. One such result

is shown forρ r = 1000 in Fig. 4.15. The drop has a very high inertia, which causes the Kelvin- Helmholtz instabilities to grow (Fig. 4.15(a)) on the surface of the drop and shear away fragments of it(Fig. 4.15(b)) violently.

The eﬀect of viscosity ratio on drop dynamics is depicted in Fig. 4.16. It is observed that the breakup is delayed as the viscosity ratio is increased. The breakup occurs from the periphery of the drop after it forms a skirt-like structure. The instability grows on the skirt which ultimately separates ring-like structures from the drop, in its wake. It can be noticed that the phenomenon is largely axisymmetric before breakup, thus can be understood with axisymmetric simulations in these cases. However for low Bond and Gallilei numbers the drop starts to deviate from axisymmetry and may execute zigzagging or spiralling motion, which is a part of the ongoing work.

62 Figure 4.14: Time evolution of drops for diﬀerent values of density ratios (ρr) for parameter values: Ga = 40,Bo = 5 andµ r = 10.

4.5 Conclusions

In thefirst part of this work, we studied the rise under gravity of an initially static and spherical bubble whose density and viscosity arefixed to be much smaller than that of the surroundingfluid. The parameters that govern the dynamics are the Galilee and the Bond numbers. Our extensive fully three-dimensional study, with Ga andBo ranging from 7 to 500 and 0.1 to 200, respectively, brings to light a number of features. Wefindfive distinct regions in the phase plot, with sharply defined boundaries. The bubble is axisymmetric in region I, non-axisymmetric in regions II and III, and breaks in regions IV and V. Region II, where the bubble consists of an axisymmetric 3 spherical cap and a skirt with minor asymmetries, is distinguished by theMo 10 − line from the ∼ dramatically asymmetrical bubbles of region III. This Morton number has been found in experiments to be the highest at which path instabilities are seen. Region II bubbles are non-oscillatory whereas all bubbles of region III display path instabilities, in the form of spirals or zig-zags. This shows an intimate connection between shape and path asymmetries. In regions IV and V the bubble motion is unsteady and shows two different kinds of topology change: peripheral break-up and toroid formation respectively, the latter is followed by break-up. Moving along lines of constant Morton number on

63 (a) (b)

Figure 4.15: A large liquid drop violently breaking up while falling in the air at timest = 4 and 5 (from left to right) for parameter values: Ga = 40,Bo = 5,ρ r = 1000 andm = 10 . this plot, i.e., for bubbles of increasing radius placed in a given surrounding liquid, there are thus up to three transitions which take place. Some older experiments [1] have given crude boundaries between different shapes of bubbles in regions I to III, with very good agreement with present simulations in the transition from axisymmetric to wobbly. At low Morton number in region IV, we show a new kind of bubble break-up, into a bulb-shaped bubble and a few satellite drops. Each transition is clearly distinguishable in terms of the completely different behaviour on either side. A maximum in kinetic plus surface energy occurs on the transition boundaries as shown in Fig. 4.13. The importance of studying this problem in three-dimensions is brought out at many places in this chapter. Other three-dimensional studies have obtained the path instability, but not the transition to other regimes. We hope that this work will motivate experiments on initially spherical bubbles to check our phase plot. In the latter part of this work, we studied the dynamics of falling drops. As opposed to bubbles, falling water drops in air are challenging to study due to high inertia and high surface tension forces. To our knowledge no numerical work exists on extensive simulations of high density ratio drops falling in air, whereas a vast literature exists on simulation of bubbles. Most of the numerical work on fragmentation and atomization of sprays has been done for low density ratios [34, 174]. A preliminary study of effect of density and viscosity ratio has been carried out in this work. A thorough study of the effect of Ga andBo is being carried out currently.

64 Figure 4.16: Time evolution of drops for diﬀerent values of viscosity ratios (µr) for parameter values: Ga = 40,Bo = 5 andρ r = 10.

65 Chapter 5

Bubble rise in a Bingham plastic

5.1 Introduction

The motion of droplets influids that exhibit yield stress is important in many engineering applications, including food processing, oil extraction, waste processing and biochemical reactors. Yield stressfluids or viscoplastic materialsflow like liquids when subjected to stress beyond some critical value, the so-called yield stress, but behave as a solid below this critical level of stress (see Chapter 1 for a brief review). As a result the gravity-driven bubble rise in a viscoplastic material is not always possible as in the case of Newtonianfluids but occurs only if buoyancy is sufficient to overcome the material’s yield stress [86, 87]; the situation is also similar for the case of a settling drop or solid particle [88]. Thefirst constitutive law proposed to describe this material behavior is the Bingham model [89] which was later extended by Herschel & Bulkley [90] to take into account the effects of shear-thinning (or thickening). According to this model the material can be in two possible states; it can be either yielded or unyielded, depending on the level of stress it experiences. As the common boundary of the two distinct regions the so-called yield surface is approached, the exact Bingham model becomes singular. In simpleflows this singularity does not generate a problem, but, in more complexflows the discontinuous behaviour of the Bingham model may pose significant difficulties due to the fact that in most cases the yield surface is not known a priori but must be determined as part of the solution. Nevertheless, there are examples of successful analysis of two-dimensionalflows using this model at the expense of relatively complicated numerical algorithms [88, 91–93]. A simpler way to overcome these difficulties is to modify the Bingham constitutive equation in order to produce a non-singular constitutive law, by introducing a ‘regularization’ parameter [94]. This method has been used with success by several researchers in the past [87, 95–98] and when used with caution can give significant insight in the behaviour of viscoplastic materials. As mentioned in the literature review, several authors have investigated the creepingflow [110,111] and steady state dynamics [87] of bubbles in viscoplastic media. In the numerical simulations of [87], even in the cases where a steady solution could be obtained it is not certain that this solution is stable. Therefore a question that arises is whether for some parameter values it is possible to get a time-dependent solution and what would be the dynamics of the bubbleflow in this case. This is the question that our study attempts to address.

66 In this part of the work, we assume axial symmetry and study the buoyancy-driven rise of a bubble inside an inﬁnite viscoplastic medium. To account for the viscoplacity we consider the regularized Herschel-Bulkley model. We employ the Volume-of-Fluid (see Chapter 2 for more details) method to follow the deforming bubble along the domain. Most of the work presented here is contained in one of our papers in press [24].

5.2 Formulation

We consider the rise of a bubble (Newtonianfluid ‘B’) in a viscoplastic material (fluid ‘A’) under the action of buoyancy within a cylindrical domain of diameterH and heightL, as shown in Fig. 5.1. We use an axisymmetric, cylindrical coordinate system, (x, z), to model theflow dynamics, in whichx andz denote the radial and axial coordinates, respectively, the latter being aligned in the

opposite direction to gravity. The bubble is initially present at a distancez i above the bottom of the domain located atz = 0. The governing equations of the problem correspond to those of mass and momentum conservation (Eq. (2.15)-(2.16)) as described in Chapter 2. After dropping oﬀ the energy and vapour advection-diﬀusion equations, the dimensionless governing equations relevant to this problem are,

u=0, (5.1) ∇· ∂u 1 δ ρ +u u = p+ µ( u+ u T ) ρ�e + κ�n, (5.2) ∂t ·∇ −∇ Ga∇· ∇ ∇ − z Bo � � � � ∂c a +u c = 0, (5.3) ∂t ·∇ a along with the following dependence of density on the volume fraction of the outerﬂuid:

ρ = (1 c )ρ +c . (5.4) − a r a

The outerﬂuid viscosity (dimensional),µ o is given by the regularized Herschel-Bulkley model

τ0 n 1 µ = +µ (Π+�) − , (5.5) o Π+� 0

whereτ 0 andn are the yield stress andflow index, respectively,� is a small regularization param- 1/2 eter, andµ 0 is thefluid consistency;Π=(E ij Eij ) is the second invariant of the strain rate tensor, whereinE 1 (∂u /∂x +∂u /∂x ). The effect of the regularization parameter� has been ij ≡ 2 i j j i presented in Figs 2.13 and 2.14 of Chapter 2. Finally, we setn = 1 henceforth so that our non- Newtonianfluid corresponds to a Bingham plastic and the effect of a shear-dependent viscosity will be ignored for the purposes of the present study. An important ingredient of every study that con- cerns theflow of a viscoplastic material is the determination of the position of the yield surface and when using a regularized model this can be achieved a posteriori by using the following criterion:

yielded material:T>τ 0, (5.6)

unyielded material:T τ , (5.7) ≤ 0

67 Figure 5.1: Schematic diagram of a bubble ofﬂuid ‘B’ rising inside a Binghamﬂuid ‘A’ under the action of buoyancy. The bubble is placed atz=z i; the value ofH,L andz i are taken to be 20R, 48R, and 10.5R, respectively. Initially the aspect ratio of the bubble, h/w is 1, whereinh andw are the maximum height and width of the bubble.

whereT denotes the second invariant of the stress tensor in material ‘A’,

1 1/2 T= τ τ , (5.8) 2 ij ji � �

andτ ij is given by

τij =µ oEij . (5.9)

In addition to the non-dimensionalization procedure presented in Chapter 2, the viscosity,µ, is non-dimensionalized as:

Bn n 1 µ= +m(Π+�) − c + (1 c)µ , (5.10) Π+� − r � �

where the definitions ofBn,m andµ r are given in Chapter 2. The position of the yield surface is determined by evaluating the dimensionless second invariant of stress tensor,T , insidefluid ’A’ and finding the locus of points for whichT= Bn. Rest of the dimensionless parameters are same as those discussed in Chapter 2.

5.3 Results

We numerically solve the governing equations in afinite-volume framework using gerris as well as a bespoke diffuse-interface solver. The results from the open-source solver, gerris have been cross-checked with our diffuse-interface code results for accuracy. Note that in the framework of the diffuse-interface method, the advection equation of the colour function is modified to contain

68 a diffusion term with a very low dimensionless diffusion coefficient, of the order of grid size (see Chapter 2 for details).

We assume that theﬂow is symmetrical about the axisx = 0. Stress free boundary conditions are imposed at the rest of the boundaries. The domain width is chosen such that the yielded region is well within the boundaries. A dimensionless domain ofH = 32 andL = 48 has been found to be a reasonable choice for the set of parameter values considered in the present study. We compare the shape of the bubble along with the unyielded region obtained using the simple regularized viscosity model (Eq. (5.10)) with the Papanastasiou’s model [109] (Eq. (5.11)) in Fig. 2.15 of Chapter 2. It can be seen that the results agree qualitatively. Thus in this part of the work, the rest of the results are generated using the simple regularized viscosity model (Eq. (5.10)).

NΠ 1 e − n 1 µ =Bn − +m(Π+�) − . (5.11) o Π � � In Fig. 5.2, we present an illustration of the convergence of the numerical solutions upon mesh

reﬁnement. The parameters chosen for this case areRe = 70.71,Bn = 14.213,µ r = 0.01,ρ r = 0.001, m = 1 andBo = 30. Other validations for this code can be found in Chapter 2.

(a) (b)

Figure 5.2: The shape of the bubble along with the mesh att=1.5 are shown for (a)finer and (b) coarser grids. Adaptive grid refinement has been used in the interfacial and yielded regions. The smallest mesh size in thefiner and coarser grids are 0.015 and 0.0625, respectively. Note that the finer grid has been used to generate the results presented in the subsequentfigures. The parameter values are Ga = 70.71,Bn = 14.213,µ r = 0.01,ρ r = 0.001,m = 1 andBo = 30. The aspect ratios of the bubble obtained using thefiner and courser grids are 1.002 and 1.003, respectively.

69 5.3.1 Discussion

We begin the discussion of our results by examining the dependence of the results on the regularisation parameter,�, used in the viscosity model forﬂuid ‘o’, given by Eq. (5.10), for Ga = 70.71,

Bn = 14.213,µ r = 0.01,ρ r = 0.001,m = 1 and Bo = 30. In Fig. 2.13, we show that the bubble rise is accompanied by its deformation and the development of a yielded region which surrounds the bubble att = 10, in which the stress generated from the bubble motion is sufficient to exceed the yield stress influid ‘o’; this region is itself surrounded by unyieldedfluid. Also shown in Fig. 2.13 is the formation of three small unyielded regions: two at the bubble equator, and one near the dimple located at the bubble base; similar predictions have been presented by Tsamopoulos et al. [87] using the Papanastasiou model [109]. It is seen that the dependence of the shapes of the bubble and the yielded region surrounding it, as well as the extent of the unyielded regions immediately adjacent to the bubble becomes progressively weaker with increasing�. At this point, it should be noted that decreasing the value of� the system of equations becomes stiffer and more difficult to handle numerically. This may also result in the appearance of numerical noise and therefore very small values of� should actually be avoided. A more accurate evaluation of the yield surface position is possible, as was shown recently by Dimakopoulos et al. [93] using the augmented Langrangian method at the expense of a significantly more complex numerical algorithm. Nevertheless, for the purposes of this study, the calculated yield surfaces are considered to be reasonably accurate. We have also found that the time evolution of the bubble aspect ratio (h/w), which is defined as the ratio of instantaneous maximum height of the bubble to its maximum width, and its centre of gravity,z CG, exhibit a similar dependence on� and become virtually indistinguishable with decreasing�, as shown in Fig. 2.14. Thus, the rest of the results discussed in this chapter have been generated using�=0.001. Next, we study the bubble rise dynamics by examining the temporal evolution of the bubble aspect ratio and centre of gravity for varying Bingham number, Bn, with Ga=7.07, Bo = 10,

µr = 0.01,ρ r = 0.001, andm = 1. It is seen in Fig. 5.3 that for lowBn values, which reflect the presence of a weak yield stress, the bubble undergoes severe deformation at relatively early times before assuming a constant aspect ratio. More specifically, forBn=0.071 the aspect ratio is found to be approximately equal to 0.48 in good agreement with the predictions given by Tsamopoulos et al. [87]. We also found that, as expected, the rise velocity of the bubble decreases withBn due to the increased resistance associated with the larger yield stresses (see Fig. 5.3). In the lowBn range, the bubble achieves a constant rise speed rapidly, as shown by the linear dependence ofz CG on time. In particular for Bn=0.071 the calculated terminal velocity is approximately equal to 0.765 in agreement with the predicted value of 0.75 given in Tsamopoulos et al. [87]. The extent of bubble deformation and rise speed decrease with increasingBn forBn less than unity for the parameters used to generate the results shown in Fig. 5.3; the same trend was also found in Tsamopoulos et al. [87]. For higher Bn values, e.g. Bn=0.99, we notice that the bubble aspect ratio (1.05) and terminal velocity (0.226) differ significantly from the predictions of Tsamopoulos et al. [87], i.e. 1.25 and 0.07, respectively. The difference cannot be attributed to the

finite viscosity of thefluid since, as shown in Fig. 5.4, increasing the viscosity ratio,µ r, leads to the decrease of the rise velocity. We notice though that even at late times the deformation of the bubble has not reached a steady state (see Fig. 5.4b) and continues to change. As shown in Fig. 5.3b, the latter effect is more prominent for even higher values of theBn number where we see clearly that

70 (a) (b)

1.2 16

Bn 0 1 Bn 0.071 0 14 0.354 0.071 zCG 0.99 h/w 0.354 1.343 0.8 0.99 1.343 12 0.6

10 0.4 0 10 20 30 0 10 20 30 t t

Figure 5.3: (a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for diﬀerent values of Bn. The parameter values are Ga=7.071,µ r = 0.01,ρ r = 0.001,m = 1 and Bo = 10.

(a) (b) 3

1.06

2 1.04 µ µ zCG r h/w 0.001 0.001 1.02 1 0.005 0.005 0.01 0.01 0.05 0.05 1

0 0 3 6 9 12 15 0 3 6 9 12 15 t t

Figure 5.4: (a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for different values ofµ r. The parameter values are Ga=7.071, Bn=0.99,ρ r = 0.001,m = 1 and Bo = 10. theflow does not reach a steady state and that the bubble aspect ratio exhibitsfinite amplitude oscillations. These oscillations in the bubble deformation may lead to yielding of the surrounding material and thereby could be responsible for the enhancement of the bubble motion. Fig. 5.5 depicts the spatio-temporal evolution of the shape of the bubble and its surrounding unyielded region as a function ofBn for the same parameters used to generate Fig. 5.3. Inspection of thisfigure shows that the extent of the unyielded region increases withBn, as expected, and for Bn < 1, the bubble widens as it rises, which is consistent with the results shown in Fig. 5.3(b) for the same range of parameter values and in accordance with thefindings of Tsamopoulos et al. [87]. These shapes become steady with increasing time. For Bn=1.34, it is evident that the bubble aspect ratio exceeds unity, which is also consistent with Fig. 5.3(b), likely brought about by the confinement due to the smaller yielded region associated with this value of Bn; it is also evident that the shapes of the bubble and unyielded regions do not achieve a steady-state in this case. In Fig. 5.6 we show contour plots of the radial and axial components of the velocityfield for Bn=0,0.35,0.99,1.34, and the rest of the parameters remain unchanged from those of Fig. 5.3. The case with Bn = 0 corresponds to the Newtonian case. It is clearly seen that the radial and

71 axial velocity components exhibit stagnation contours that separate regions of outward and inward, and upwards and downwards motion, respectively; at regions where the radial motion of theﬂuid is negligible, unyielded zones are likely to occur. The stagnation contour associated with the vertical component moves progressively closer to the interface with increasingBn; for the largestBn studied, it is evident that the regions nearest the top and bottom of the bubble move upwards, while the remaining regions move downwards leading to bubble elongation. The stagnation contour associated with the radial component emanates from rightmost bubble edge at a negative angle to the horizontal in the Newtonian case. This contour becomes essentially horizontal and the bubble, whose bottom is dimpled in the Newtonian case, becomes well-rounded with inceasingBn as the bubble becomes ﬂatter at the equatorial plane.

It is also important to study the effect of bubble deformability on its dynamics; this is done by varying the Bond number, Bo, which reflects the relative significance of surface tension to gravti- ational forces. In Fig. 5.7, it is seen clearly that for low Bo, for which surface tension forces are dominant, bubble deformation is small and its rise speed is constant, increasing withBo. For larger Bo, however, the bubble dynamics gain in complexity. The bubble appears to undergo sudden acceleration between periods of constant rise speed; these periods become shorter and the magnitude of the acceleration increases with Bn, as shown in Fig. 5.7a. Thisz CG dynamics is associated with large bubble deformation as can be ascertained upon inspection of Fig. 5.7b: the aspect ratio undergoes nonlinear oscillations about unity as the bubble ‘swims’ upwards, whose wavelength and amplitude increase withBo.

In order to rationalise the results presented in Fig. 5.7 and further highlight the role of bubble deformation in the ‘swimming’ motion discussed, we show in Fig. 5.8 the spatio-temporal evolution of the shape of the bubble and the unyielded regions for Bo = 1 and Bo = 30 while the rest of the parameters remain unchanged from those of Fig. 5.7. It is seen that forBo = 30, at relatively early times unyielded regions are situated in the equatorial region of the bubble, and the bubble aspect ratio is close to unity. With increasing time, the extent of the unyielded regions decreases due to the shear stress associated with the bubble acceleration and the bubble elongates as it rises through a yielded region of increasing size. The bubble then decelerates to a constant rise speed, its aspect ratio decreases, and the decrease in shear stress in the vicinity of the interface leads to the development of unyielded zones in the equatorial and south pole regions; the former become more pronounced with increasing time, and the bubble aspect ratio decreases below unity as the bubble decelerates. The process is then repeated. In contrast, no such process is evident in the case of Bo = 1 for which the bubble appears to suﬀer negligible deformation and the size of the unyielded regions remains largely unaltered.

In Fig. 5.9, we show the effect of Bo on the contour plots of the radial and axial velocity components forBo = 1 andBo = 30; these plots are shown fort = 6 andt=8.5 that correspond to the times at which the bubble achieves its maximal and minimal aspect ratio for Bo = 30, respectively. As can be seen from thisfigure, the magnitude of both components remains essentially unchanged for theBo = 1 case, while, for the same times in theBo = 30 case, the axial and radial velocity components dominate att = 6 andt=8.5, resulting in bubble elongation, andflattening and dimpling, respectively.

72 Figure 5.5: The evolution of the shape of the bubble (shown by red lines) and the unyielded region in the non-Newtonianﬂuid (shown in black) for diﬀerent values of Bingham number. The results of the Newtonian case are shown for the comparison purpose. The rest of the parameter values are the same as those used to generate Fig. 5.3.

73 (a) (b)

Figure 5.6: Contour plots for the radial (right) and axial (left) velocity components for (a)Bn=0 att = 6 (Newtonian case), (b)Bn=0.354 att = 6, (c)Bn=0.99 att = 20 and (d)Bn=1.34 at t = 20. In each panel the shape of the bubble is shown by red line. The rest of the parameter values are the same as those used to generate Fig. 5.3.

5.4 Conclusions

In this chapter, we have examined the axisymmetric dynamics of bubble rise in Binghamfluids. We have used an open-sourcefinite-volumeflow solver, gerris based on volume-of-fluid methodology to study theflow, which involves the numerical solution of the equations of mass and momentum conservation, and an equation of the volume fraction of the Binghamfluid. The momentum equation accounts for surface tension and gravitational effects, while the density and viscosity are volume fraction-weighted with respect to the corresponding quantities of the twofluids. For the Bingham fluid, the formula for the viscosity contains a regularisation parameter; convergence of our results was achieved upon mesh-refinement and reduction of the magnitude of this parameter to sufficiently small values. Our numerical results indicate that in the presence of weak yield stress the bubble achieves a constant rise speed relatively rapidly, whilst its aspect ratio, defined as the ratio of its height to its width asymptotes to a value less than unity; unyielded zones are confined to regions that surround but are not immediately adjacent to the bubble. With increasing yield stress, the bubble

74 (a) (b) 15 Bo Bo 0.1 1.4 0.1 14 1 1 5 5 30 30 13 1.2 z CG h/w 12 1

11 0.8

10 0 5 10 15 20 0 5 10 15 20 t t

Figure 5.7: (a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for different values ofBo. The rest of the parameter values are Re = 70.71, Bn = 14.213,µ r = 0.01, ρr = 0.001, andm = 1. rise is unsteady, and the bubble aspect ratio exhibits oscillations above a value that exceeds unity. Unyielded zones near the equatorial and south pole regions of the bubble have also been observed to form for sufficiently large yield stress in agreement with earlier studies in the literature [87, 93]. We have also shown that bubble deformation has a profound impact on the dynamics. In the case of strong surface tension, the rise is steady and the bubble suffers negligible deformation. For weak surface tension, however, the rise is unsteady, periods of approximately constant rise speed are separated by rapid acceleration stages that coincide with oscillations in the bubble aspect ratio about unity whose amplitude increases with decreasing surface tension. These oscillations also coincide with the formation and destruction of unyielded zones in the equatorial regions. The motion executed by the bubble for this range of parameters resembles ‘swimming’ as the bubble appears to grab hold of the unyielded zones to propel itself upwards.

75 Figure 5.8: The evolution of the shape of the bubble (shown by red lines) and the unyielded regions in the Binghamﬂuid (shown in black) for diﬀerent values ofBo. The rest of the parameter values are the same as those used to generate Fig. 5.7.

76 (a) (b)

Figure 5.9: Contour plots for the radial (right) and axial (left) velocity components for (a)Bo=1 att = 6, (b)Bo = 1 att=8.5, (c)Bo = 30 att = 6 and (d)Bo = 30 att=8.5. In each panel the shape of the bubble is shown by red line. The rest of the parameter values are the same as those used to generate Fig. 5.7.

77 Chapter 6

Non-isothermal bubble rise

6.1 Eﬀect of temperature gradients

Interfacialflows with temperature gradients in the surrounding medium invariably create interfacial tension gradient along the interface separatingfluid pairs. A gradient in interfacial tension cause the fluid toflow towards the regions of high surface tension so as to minimize the surface energy of the system. A typical problem where Marangoni stresses play a significant role is the thermocapillary migration of bubbles and drops. Much of the work in thisfield has been reviewed by [112] and [113]. A brief literature review is presented in Chapter 1. In this chapter, we present the buoyancy-driven rise of a bubble inside a tube imposing a constant temperature gradient along the wall. To account for the non-monotonicity of surface tension we consider a quadratic dependence on temperature. We examine the Stokesflow limitfirst and derive conditions under which the motion of a spherical bubble can be arrested in self-rewettingfluids even for positive temperature gradients. We then employ a diffuse-interface method [169] to follow the deforming bubble along the domain in the presence of inertial contributions. Our results indicate that for self-rewettingfluids, the bubble motion departs considerably from the behaviour of ordinaryfluids and the dynamics may become complex as the bubble crosses the position of minimum surface tension. As will be shown below, under certain conditions, the motion of the bubble can be reversed, and then arrested, or the bubble can become elongated significantly. A large portion of this chapter appears in one of our published works [23].

6.2 Formulation

Apart from the continuity, Navier-Stokes and Cahn-Hilliard equation/advection equation within the diffuse interface/volume offluid framework, we solve the temperature equation to allow for the conduction and convection within thefluid. The set of equations to be solved are the same as discussed in Chapter 2. An axisymmetric domain is considered to solve the governing equations. It is to be noted that only in this work, a bounded domain has been considered. The geometry is shown in Fig. 6.1 for clarity, where a no-slip boundary condition is imposed on the wall of the cylinder. Other boundary conditions are same as those employed in Chapter 3. The viscosity (dimensional) is assumed to depend on the temperature and the volume fraction

78 Figure 6.1: Schematic diagram of a bubble moving inside a Newtonianﬂuid under the action of buoyancy. The initial location of the bubble is atz=z i; unless speciﬁed, the value ofH,L and zi are 6R, 48R, and 10.5R, respectively. The acceleration due to gravity,g, acts in the negativez direction. as follows (from Eqs 2.12 and 2.13):

3/2 T T c T T c ( T − T ) µ= cµ e− m− c + (1 c)µ 1 + − , (6.1) o − i T T � � m − c � � whereT c andT m are the temperature at the bottom of the tube (z = 0) and temperature atz=z m, whereinz m corresponds to the vertical location where surface tension reaches its minimum;µ A and

µB are the viscosity of the liquid and gas at temperatureT c, respectively. This viscosity dependence on temperature for the liquid and gaseous phases are taken from [201]. The density (ρ) and thermal conductivity (λ) are calculated as linear functions of the volume-fraction of the outerﬂuid [128]. In order to model the behaviour of a self-rewettingﬂuid, we use the following relationship for the functional dependence of the surface tension on temperature (from Eq. 2.7):

σ=σ β (T T ) +β (T T )2, (6.2) 0 − 1 − c 2 − c

dσ 1 d2σ whereβ 1 andβ 2 2 . A linear temperature variation is imposed in the vertical ≡ − dT |Tc ≡ 2 dT |Tc direction with a constant gradientγ, and Fig. 6.2 shows the corresponding variation inσ withz for diﬀerent values ofβ 1 andβ 2. As can be seen from thisﬁgure, the parabolic dependence ofσ onz becomes more pronounced, with a deeper minimum, located atz=z m, for increasingβ 1 and

β2; this is expected to alter the type of Marangoniflow observed in casefluids that exhibit a simple linear variation ofσ withT which we will refer to in this chapter as ‘linear’fluids. Below, we will explore the dynamics of the bubble as it rises starting fromz i, which may be either below or above z=z m, for both linear and self-rewettingfluids. The dependence of this dynamics onβ 1 andβ 2, which parameterise the behaviour of various self-rewettingfluids, will also be studied. The governing equations are non-dimensionalized with the characteristic scales as described in

79 1.2

M1=0.4, M2=0.2 1.12 M1=0.2, M2=0.1 M =0.1, M =0.05 1.04 1 2 σ 0.96

0.88

0.8

0 5 10 15 20 25 z

Figure 6.2: Variation of the liquid-gas surface tension along the wall of the tube forΓ=0.1 and various values ofM 1 andM 2.

Chapter 2, however the following variables are non-dimensionalized diﬀerently:

σ σ (T T ) T= T(T T ) +T ,β = 0 M ,β = 0 M ,γ= m − c Γ, (6.3) m − c c 1 T T 1 2 (T T )2 2 R m − c m − c � � � where the velocity scale isV= √gR,σ 0 is the surface tension atT c, and the tildes designate dimensionless quantities. After dropping tildes from all non-dimensional terms, the governing dimensionless equations are given by u=0, (6.4) ∇· ∂u 1 +u u= p+ µ( u+ u T ) ρ�e +F , (6.5) ∂t ·∇ −∇ Ga∇· ∇ ∇ − z s ∂(ρc T) � 1 � p + (uρc T)= (λ T), (6.6) ∂t ∇· p GaPr∇· ∇ ∂c +u c=0, (6.7) ∂t ·∇ where Ga ρ V R/µ denotes the Gallilei number; Pr c µ (T )/λ is the Prandtl number, wherein ≡ o o ≡ p o c o cp is the speciﬁc heat capacity at constant pressure, andλ o is the thermal conductivity of the liquid.

The dimensionless viscosity,µ, has the following dependence onT andc a:

T 3/2 µ=c e− + (1 c )µ 1 +T . (6.8) a − a r � � The dimensionless density and thermal diﬀusivity are the same as those described in Chapter 2. In

Eq. (6.5), the surface tension forceF s is given by (continuum surface force formulation [178])

κδ F = 1 M T+M T 2 �n. (6.9) s Bo − 1 2 � � In this relation, the dependence ofσ onT , using Eq. (6.2) has been included. As the bubble is

80 assumed to reach a terminal location for some values ofM 1 andM 2, the Stokesflow assumption is valid when the bubble slows down and tends to stop theflow. Therefore, we can solve the equations of motion in creepingflow regime andfind the terminal location of the bubble without fully solving the Navier-Stokes equation.

6.3 Analytical results: Stokesﬂow limit

In this section, we provide a discussion of our analytical results. We derive expressions for the terminal velocity of a spherical bubble rising vertically through a quiescent liquid in the Stokesflow limit in which thermocapillary stresses arise due to a temperature gradient imposed on the liquid. For the purpose of this calculation we will ignore the presence of walls and consider the case of unconfinedflow. We show how the dependence of the surface tension on temperature, represented by Eq. (6.2), affects the terminal bubble speed, and the magnitude and sign of the temperature gradient required to arrest bubble motion; this is also contrasted with the case of a linearfluid. We adopt a spherically-symmetric coordinate system, (r,θ), with the polar angle,θ, measured from the bottom of the bubble (θ = 0) to the top of the bubble (θ=π);u=u rir +u θiθ is the velocityfield in whichu r andu θ represent its radial and azimuthal components, andi r andi θ denote the unit vectors in ther andθ directions, respectively. Thez-axis originates at the bubble centre and is oriented vertically upwards so that it coincides with the axis of symmetry of the bubble. The unit vector in thez-direction is expressed byi = ηi + (1 η 2)1/2i in whichη cosθ. Note that z − r − θ ≡ we have adopted a different coordinate system from that in Section?? temporarily for the purpose of this calculation and redefined the origin of thez-axis. At the end of the present subsection, we shall revert to the use of cylindrical coordinates. We also adopt a frame of reference that moves with the centre of the bubble, which is scaled on the steady translational speed of the bubble,U; this speed will be determined as part of the solution. Note that all quantities presented in this subsection are in dimensional terms. We assume that heat transfer is dominated by conduction so that the temperaturefield in the liquid,T , is governed by 2T=0. (6.10) ∇ We impose a linear temperature distribution in the liquid, so that at large distances from the bubble we have

T (z)=T (0) +γz �. (6.11) ∞ ∞

Here,T (0) denotes the temperature atz � = 0, the position of the centre of the bubble. At the ∞ bubble surface,r=R, we demand continuity of the thermalﬂux:

∂T λ ∂T = B g , (6.12) ∂r λ ∂r � A � whereλ denotes the thermal conductivity of the gas. We assume thatλ λ so that Eq. (6.12) B A � B reduces to ∂T = 0. (6.13) ∂r

81 The general solution of Eq. (6.10) is given by

∞ r n r (n+1) T= A +B − P (η), (6.14) n R n R n n=0 � � � � � � �

whereP n(η) are Legendre polynomials of theﬁrst kind of degreen. We apply the no-ﬂux condition given by Eq. (6.13) atr=R: n B = A . (6.15) n n+1 n � � Substitution of Eq. (6.15) into Eq. (6.14) gives

∞ r n n r (n+1) T= A + − P (η). (6.16) n R n+1 R n n=0 � ��

To match to the farﬁeld condition, we setz � =r cos(π θ)= r cosθ= rη in Eq. (6.11) so that − − − T (z)=T (0) γrη. Matching this equation to Eq. (6.16) yields ∞ ∞ −

A0 =T (0),A1 = γR,A n = 0 forn 2. (6.17) ∞ − ≥

Substitution of these values into Eq. (6.16) gives

1 r 3 T=T (0) γr 1 + − η. (6.18) ∞ − 2 R � � � �

The solution for the velocityﬁeld in theﬂuid,u=(u r, uθ), is subject to the following boundary conditions: u i as r , (6.19) →− z | |→∞

ur = 0 atr=R, (6.20) ∂σ τ + = 0 atr=R, (6.21) rθ ∂θ

whereτ rθ is the tangential stress. As noted above, theﬂow is axisymmetric about thez-axis, hence the solution to the Stokesﬂow problem can be expressed in terms of the streamfunction,ψ [202]:

r 2 ∞ r 2 n r n ψ=UR 2 Q (η)+ C − +D − Q (η) . (6.22) − R 1 n R n R n � n=1 � � � � � � � � � � This is the general solution forﬂow past an axisymmetric body of arbitrary shape in the Stokes

ﬂow limit. Here,Q n are integrals ofP n(η), and closely related to the Gegenbauer polynomials. The polynomials relevant to the present work are

1 η Q (η)= (η2 1),Q (η)= (η2 1). (6.23) 1 2 − 2 2 −

The solution expressed by Eq. (6.22) is chosen to satisfy the following equation

E4ψ=0, (6.24)

82 whereE 2 is given by ∂2 (1 η 2) ∂2 E2 + − , (6.25) ≡ ∂r2 r2 ∂η2

as well as the farﬁeld condition given by Eq. (6.19). The streamfunctionψ is related tou r andu θ by 1 ∂ψ 1 ∂ψ u = , u = . (6.26) r − r2 ∂η θ − r(1 η 2)1/2 ∂r − It can be shown that forﬂow problems that haveψ expressed as in Eq. (6.22), the component

of the dimensional force along the axis of symmetry,F z, exerted by the surroundingﬂuid on an axisymmetric body of arbitrary shape with its centre of mass at x = 0 is given by the following | | general formula

Fz = 4πµAURC1. (6.27)

At steady-state, this drag force balances the buoyancy force:

4 4 4πµ URC = π(ρ ρ )R3g πρ R3g, (6.28) A 1 3 A − B ≈ 3 A

where we have assumed thatρ ρ . Equation (6.28) suggests thatC is the only coeﬃcient that A � B 1 we need to compute in order to determine the terminal velocity of the bubble.

The no-penetration condition given by Eq. (6.20) can be re-expressed as∂ψ/∂η = 0 atr=R. It follows from this condition thatψ is constant atr=R. Sinceψ = 0 atθ = 0 andθ=π, corresponding toη= 1, for allr because of symmetry, then Eq. (6.20) can be re-written as ±

ψ = 0 atr=R. (6.29)

Application of this condition yields

∞ 0 = Q (η)+ (C +D )Q (η). (6.30) − 1 n n n n=1 �

The tangential stress balance given by Eq. (6.21) can be re-expressed as

∂ 1 ∂ψ ∂σ µ r + = 0 atr=R. (6.31) − A ∂r r2(1 η 2)1/2 ∂r ∂θ � − � Using Eq. (6.2), the surface tension gradient,∂σ/∂θ is then given by

∂σ ∂σ ∂T = ∂θ ∂T ∂θ 3 1 r − = β1 + 2β2 T (0) T c γr 1 + η − ∞ − − 2 R � � � � � � �� 1 r 3 γr 1 + − (1 η 2)1/2 , (6.32) × 2 R − � � � � � � 83 whereT c is a reference temperature. Substitution of Eqs. (6.22) and (6.32) into Eq. (6.31) yields

∞ 2Q (η) + 2C Q (η) 4D Q (η)+ ((2 n)(1 +n)C n(n + 3)D )Q (η) − 1 1 1 − 1 1 − n − n n n=2 � 3γR 3 2 = − β1 + 2β2 T (0) T c γRη (1 η ) 2µ U − ∞ − − 2 − A � � �� 2 2 3γRβ1 2β2 9γ R β2 = − 1 (T (0) T c) Q1(η) Q2(η). (6.33) µ U − β ∞ − − µ U A � 1 � A From Eq. (6.30) we get

∞ Q + (C +D )Q + (C +D )Q = 0, (6.34) − 1 1 1 1 n n n n=2 � whence C +D = 1, andC = D forn 2. (6.35) 1 1 n − n ≥

Forβ 2 = 0, i.e. for a linearﬂuid, then from Eq. (6.33) we get

3γRβ n(n + 3) 2Q + 2C Q 4D Q = 1 Q , andC = D , forn 2. (6.36) − 1 1 1 − 1 1 − µ U 1 n (n + 1)(2 n) n ≥ A − Thus, we deduce that γRβ1 γRβ1 C1 = 1 ,D1 = . (6.37) − 2µAU 2µAU Forn 2, ≥ n(n + 3) n(n + 3) C = D = D + 1 D = 0. (6.38) n (n + 1)(2 n) n − n ⇒ (n + 1)(2 n) n − � − � However, the coeﬃcient ofD is not zero, soD =C = 0 forn 2. From Eq. (6.28), we arrive at n n n ≥ the following expression forU, after making use ofC 1 from Eq. (6.37):

ρ R2g 3 γβ U= A 1 + 1 . (6.39) 3µ 2 ρ Rg A � A � Thus, ifU = 0 the bubble rise is arrested provided

2ρARg γ=γ c = . (6.40) − 3β1

This equation implies that for a linearfluid, the temperature gradient must be negative in order for the bubble to come to rest [202]. For a self-rewettingfluid withβ = 0, and following a similar procedure to that discussed above, 2 � the relevant coefficients are

84 γRβ1 2β2 C1 = 1 1 (T (0) T c) , (6.41) − 2µ U − β ∞ − A � 1 � 2 2 9γ R β2 C2 = , (6.42) − 10µAU γRβ1 2β2 D1 = 1 (T (0) T c) , (6.43) 2µ U − β ∞ − A � 1 � 2 2 9γ R β2 D2 = , (6.44) 10µAU

andC =D = 0 forn 3. Substitution ofC into Eq. (6.28) yields the following expression for n n ≥ 1 the terminal velocityU:

2 ρAR g 3 γβ1 2β2 U= 1 + 1 (T (0) T c) . (6.45) 3µ 2 ρ Rg − β ∞ − A � A � 1 ��

The expression forγ c that leads to bubble arrest andU = 0 is given by:

2 ρARg γc = . (6.46) − 3 β2 β1 1 2 (T (0) T c) − β1 ∞ − � � Forβ 2 = 0, this equation reduces to Eq. (6.40). Equation (6.46) suggests thatγ c is positive

(negative) if 2β2 (T (0) T c)/β 1 > 1 (2β2 (T (0) T c)/β 1 < 1) in the case of a self-rewetting ∞ − ∞ − ﬂuid, in contrast to the case of a linearﬂuid in whichγ c < 0.

We note that if we had assumed that the temperature distribution given by Eq. (6.11) applied everywhere, including at the bubble surface,r=R, then the formula for the terminal velocity would have been expressed by

2 ρAR g γβ1 2β2 U= 1 + 1 (T (0) T c) , (6.47) 3µ ρ Rg − β ∞ − A � A � 1 ��

and the expression forγ c by

ρARg γc = . (6.48) − β2 β1 1 2 (T (0) T c) − β1 ∞ − � �

Reverting back to the coordinate system of the previous section the position of the bubble centre is given by T (0) T c z= ∞ − (6.49) γ and using Eq. (6.46) it is possible to derive an expression for the terminal vertical position of the

bubble,z c: β γ ρ Rg 1 z = 1 c + A . (6.50) c 2 3 β γ2 � � 2 c In the following section, we compare the predictions of Eq. (6.50) with those obtained from the numerical simulations. We turn our attention now to the numerical results.

85 15 Isothermal Linear 14 Self-rewetting

13 zCG 12

10 0 2 4 6 8 10 t

Figure 6.3: Temporal variation of the center of gravity of the bubble for the parameter values 2 3 2 Ga = 10,Bo = 10 − ,ρ r = 10− ,µ r = 10− ,Γ=0.1 andα r = 0.04. The plots for the isothermal (M1 = 0 andM 2 = 0), linear (M1 = 0.4 andM 2 = 0) and self-rewetting (M1 = 0.4 andM 2 = 0.2) cases are shown in theﬁgure. The horizontal dotted line indicates the prediction of Eq. (6.51) for the self-rewetting case.

6.4 Numerical results

In this section, we present a discussion of our numerical results starting with a presentation of the numerical procedures used to carry out the computations. To account for the effects of inertia and confinement we solve the governing equations numerically. For this part of the study we have made use of the cylindrical coordinates. We use a bespokefinite-volumeflow solver as well as gerris (more details are provided in Chapter 2) to simulate the bubble rise in a non-isothermal medium. 2 Below, we present a discussion of our results for the following set of ‘base’ parameters: Bo = 10− , 2 3 Ga = 10,H=6, µ r = 10− ,M 1 = 0.4,ρ r = 10− ,z i = 10.5, andΓ=0.1, which are consistent with the case of a small air bubble rising in water due to buoyancy, in the presence of strong mean surface tension and Marangoni effects, and appreciable inertial contributions. We will contrast the difference in behaviour between bubble motion in linear and self-rewettingfluids by studying the effect of parameterM 2 on the dynamics. We begin the discussion of our results by showing in Fig. 6.3 the temporal variation of the centre of gravity,z CG, of a rising bubble for three different cases: the isothermal case, and the cases of a simple linearfluid, and a self-rewetting one rising in a tube whose walls are heated with a linear temperature profile of constant gradientΓ> 0. It can be seen from thisfigure that following an initial, relatively short, acceleration period, the bubble reaches a constant, terminal speed for both the isothermal case, and the linearfluid in the non-isothermal case. The terminal velocity is higher for the non-isothermal case due to the presence of Marangoni stresses driving liquid towards the cold region of the tube and thereby enhancing the upward motion of the bubble. For the self-rewetting

ﬂuid in the non-isothermal case,z CG also reaches a constant speed for a certain time duration; this is, however, followed by a drop inz CG before a terminalz CG value is reached. Thus, the motion of a bubble rising initially in a self-rewettingﬂuid, whose temperature is essentially increasing linearly,

86 isfirst reversed and then arrested. The fact that the bubble motion comes to a halt in the self- rewetting and not the linearfluid in a positive temperature gradient was also suggested by Eq. (6.45) in the Stokesflow limit. This property can be used to manipulate bubbles by simply shifting the temperature gradient along the wall appropriately. This might be of interest to researchers working in microfluidics and multiphase microreactors. The predictions of the dimensionless version of Eq. (6.50), given by ΓM Bo 1 z = 1 + , (6.51) c 2 3 M Γ2 � � 2 are also shown in Fig. 6.3. For the parameters used to generate the results presented in thisfigure, z 11.67, which is in good agreement with the numerical predictions, despite the fact that strong c ∼ inertial contributions are present in theflow as represented by Ga = 10. We will examine the mechanism underlying this behaviour below.

In Fig. 6.4a, we examine the dependence of the terminal velocityV t of a bubble rising in a linear

fluid on the parameterM 1; the latter governs the strength of the linear variation of the surface tension with temperature. As explained above, for positive values ofΓ the induced Marangoni stresses increase the rise velocity of the bubble. The terminal velocity, though, appears to reach a plateau withM 1 indicating that the strength of Marangoni stresses saturates at largeM 1, and the dynamics are dominated by the remaining parameters. In the case of self-rewettingfluids, one parameter that we need to take into account is the position of minimum surface tension with respect to the bubble because it will affect the action of induced Marangoni stresses; this effect will be studied in detail below. For the time being, we have positioned the center of gravity of the bubble above the position of minimum surface tension (zi = 10.5, zm = 10) att = 0. In this case, as the bubble rises, it comes into contact with liquid of increasingly lower surface tension. The induced Marangoni stresses drive liquid upwards, towards the hot region of the tube, and inhibit the upward motion of the bubble. This is evident at early times in Fig. 6.3 where it is shown that the rise velocity for the self-rewettingfluid is lower than for the isothermal case. In order to study the effect of Marangoni stress in more detail, we examine in Fig. 6.4b the terminal distance reached by bubbles moving in a self-rewettingfluid as a function ofM 1 with

M2 =M 1/2; the latter restriction is imposed in order to keep the position where the minimum surface tension arises constant. As shown in Fig. 6.4b, this distance increases with decreasingM 2, which indicates that an increase in the self-rewetting character of theﬂuid leads to a larger degree of bubble retardation: in the limitM 0, a steady, terminal speed is reached forΓ> 0. The 2 → numerical predictions for the terminal distance shown in Fig. 6.4b are also in good agreement with those obtained from Eq. (6.51): for the parameters used here,z (16.67, 13.33, 12.22, 11.67, 11.33) c ∼ 1 forM = (0.1,0.2,0.3,0.4,0.5). Also, forBo 0, andM =M /2, Eq. (6.51) reduces toz Γ − , 1 ∼ 2 1 c ∼ which for the parameters in Fig. 6.4b leads toz 10; this appears to be the value to which the c ∼ terminal distance limits with increasingM 1.

Interestingly, the onset time for motion reversal,t reversal, has a non-monotonic dependence on

M1: starting from a global maximum at smallM 1,t reversal exhibits a shallow minimum, followed by a local maximum, before undergoing a sharp decrease with increasingM 1. This is probably due to the eﬀect of inertia and the interplay of buoyancy and Marangoni stresses which act in opposite directions. For lowM 1 values, the Marangoni stresses are initially relatively weak and take a long time before they grow to change the direction of motion of the bubble. AsM 1 increases, Marangoni

87 (a) 5 1.18 1.12 1.07 4 1.03

Vt 3

Ar = 1.00 1 0 0.1 0.2 0.3 0.4 M1 (b) 25 M1 = 0.1

M1 = 0.2

M1 = 0.3 20 M1 = 0.4

M1 = 0.5 z z = CG c 16.67

13.33 12.22 11.67 11.33 10 0 2 4 6 8 10 t (c) 7

5 reversal t

3 0.1 0.2 0.3 0.4 0.5 M1

Figure 6.4: (a) The terminal velocity of the center of gravity of the bubble along with the aspect ratio for different values ofM 1 forM 2 = 0; (b) temporal variation of the center of gravity of the bubble forM 2 =M 1/2; (c) variation of the time at whichz CG reaches its maximum for different 2 3 2 values ofM 1. The rest of the parameter values are Ga = 10, Bo = 10− ,ρ r = 10− ,µ r = 10− , Γ=0.1 andα r = 0.04. The numerical predictions of Eq. (6.51) are shown by thefilled square symbols on the right vertical axis.

88 25 Ga 1 5 20 10 20 zCG

10 0 2 4 6 8 10 t

2 Figure 6.5: Effect of Ga on the temporal evolution of the bubble centre of gravity forBo = 10 − , 3 2 ρr = 10− ,µ r = 10− ,M 1 = 0.2,M 2 = 0.1,Γ=0.1 andα r = 0.04. The prediction of Eq. (6.51) is shown by the dotted line. stresses become stronger initially, so the bubble decelerates faster and less time is needed for the motion reversal. However, as Marangoni stresses gain in relative significance, the initial acceleration of the bubble becomes considerably smaller and this results in small rise velocities initially and therefore the bubble now has to move for longer times before it reaches the position of motion reversal. Finally for even higher values ofM 1, the Marangoni stresses are so strong that they outweigh buoyancy, and the bubble very soon starts moving in the opposite direction. The effect of inertia on the bubble motion in a self-rewettingfluid, parameterised by the Galileo number, Ga, is also of interest, and is shown in Fig. 6.5. As can be seen from thisfigure, at low values of Ga, the bubble centre of gravity,z CG, increases monotonically with time before reaching a terminal value. With increasing Ga, however, the bubble exhibitsflow reversal; the maximalz CG values reached increase progressively with Ga prior toflow reversal, which then culminates in the bubble motion being arrested. The onset offlow reversal also appears to be an increasing function of Ga. For the parameters used in Fig. 6.5, wefind using Eq. (6.51) thatz 13.33. As it is shown c ∼ in Fig. 6.5 this value is quite close to our calculations for the terminal position of the bubble even for high values of Ga and in the presence wall confinement despite the fact that Eq. (6.51) was derived an unconfined bubble moving in the Stokesflow limit. This is explained by the fact that at the latter stages of theflow, for all values of Ga, the migration velocity of the bubble decreases significantly, entering into the creepingflow regime. Next we examine the effect of mean surface tension, characterised by the Bond number,Bo, on the bubble dynamics in a self-rewettingfluid; this is shown in Fig. 6.6a,b for Ga = 5 and Ga = 10, respectively. It is seen clearly in Fig. 6.6a,b that there exists a critical value ofBo above whichflow reversal is no longer possible andz CG undergoes a monotonic rise with time whose rate decreases, and eventually saturates, with increasingBo. These results highlight the role of bubble deformation in the dynamics: minimising deformation, which is promoted by small values ofBo, acceleratesflow reversal, leading to lower terminalz CG values. For Ga = 5, measures of interfacial deformation are provided by the bubble length,l B and aspect ratio of the bubble,A r whose temporal variation are

89 (a) (b) 30 25 Bo 0.01 25 0.02 0.03 20 0.05 z 1 z CG 20 5 CG 10 15 15

10 10 0 2 4 6 8 10 0 2 4 6 8 10 t t (c) (d) 2.1 2 Bo 1.8 1 5 1.6 2 10 1.4 lB lB 1.2 1.9 Bo 0.01 1 0.02 0.03 0.8 0.05 1.8 0.6 0 2 4 6 8 10 0 2 4 6 8 10 t t (e) (f) 4 Bo 1.08 0.01 0.02 0.03 0.05 3 1.04 A r Ar 1 2 Bo 1 5 0.96 10 1 0 2 4 6 8 10 0 2 4 6 8 10 t t

Figure 6.6: Eﬀect ofBo on bubble motion for (a) Ga = 10 and (b) Ga = 5; eﬀect ofBo on the (c,d) length of the bubble,l B, (e,f) aspect ratio of the bubble,A r for Ga = 5. The rest of the parameters 3 2 valuesρ r = 10− ,µ r = 10− ,M 1 = 0.4,M 2 = 0.2,Γ=0.1 andα r = 0.04.

90 shown in Fig. 6.6c,d and Fig. 6.6(e), (f), respectively for small and largeBo values. Inspection of these panels reveals that the extent of deformation increases withBo, as expected. In order to elucidate the reasons underlying the behaviour depicted in Fig. 6.6, we show in Fig. 6.7 the evolution of the bubble shape and that of the temperature distribution in thefluid surrounding the bubble for two values ofBo; the rest of the parameters remainfixed at their ’base’ values. Also shown in Fig. 6.7 are streamlines which represent the structure of theflow within the bubble, in the surroundingfluidflowing past it, as well as in its wake region. It is seen that the bubble in the Bo = 10 case (shown in Fig. 6.7(a)), which rises starting fromz i = 10 that coincides with the surface tension minimum in Fig. 6.2, undergoes significant deformation; this begins at relatively early times, and culminates in the formation of a cap bubble. This deformation is accompanied by the formation of a pair of counter-rotating vortices within the bubble, and another pair in the wake region; the lateral and vertical extent of the latter increase with time, as the bubble rises towards the warmer regions of the tube. No evidence offlow reversal is observed, which is consistent with the results presented in Fig. 6.6. 2 For theBo = 10 − case, it is seen from Fig. 6.7b that the bubble suffers negligible deformation, and its rise is accompanied by the formation of a pair of counter-rotating vortices form inside the bubble at early times, as was also observed in the Bo = 10 case (shown in Fig. 6.7(a)). This flow structure persists untilt = 5 at which the bubble is seen to develop a wake region, and two more vortices are formed within the bubble; this coincides with the onset offlow reversal, as can be ascertained upon inspection of Fig. 6.6(a). At later times, the direction of theflow is reversed as 2 indicated by the direction of the streamlines associated with thet = 10 panel forBo = 10 − in Fig. 6.7(b), which points upwards since the liquidflows past a descending bubble. This is brought about by the fact that the vertical temperature gradient across the bubble is positive which gives rise to a positive surface tension gradient sincez>z m (viz. Fig. 6.2). This, then, sets up a Marangoni stress, which acts in the opposite direction to theflow past the rising bubble, retarding its motion. This stress becomes increasingly dominant, counterbalances, and then exceeds the magnitude of the buoyancy force, leading to the reversal of the bubble motion and its eventual arrest. We have also studied the effect of varying the initial location of the bubble on the dynamics of the centre of gravity of bubble,z CG. In Fig. 6.8, we show the temporal evolution ofz CG as a parametric function ofz i with the rest of the parametersfixed at their ‘base’ values. For situations in which the initial location of the bubble is lower than that associated with the surface tension minimum in Fig. 6.2,z=z m, the surface tension gradient across the bubble re-inforces the buoyancy-driven bubble rise. This results in an increase inz CG with time until the bubble reaches elevations such thatz>z m for which the sign of the surface tension gradient across the bubble is reversed, which drives Marangoniflow that acts to retard and eventually reverse the direction of bubble motion. The time associated with the onset offlow reversal decreases with increasingz i. For sufficiently large values ofz i, the bubble moves in the negativez-direction under the action of Marangoni stresses whose magnitude exceeds that of the buoyancy force. The terminal value ofz CG appears to be weakly-dependent onz for largez values. Also,z 13.33 from Eq. (6.51), which is in good i i c ∼ agreement with the numerical predictions. As was mentioned in the introduction, self-rewettingfluids have been used in heat pipes associated with substantially higher heatfluxes than normal liquids. In these applications, the bubbles are very confined, usually forming slugs. It therefore seems appropriate to investigate the effect of

91 (a) (b)

Figure 6.7: Evolution of bubble shape (blue line), streamlines (lines with arrows), and temperature 2 contours (shown in color) with time for (a)Bo = 10 and (b)Bo = 10 − . The initial location of the bubble,z i = 10. The inset at the bottom represents the colormap for the temperature contours. 3 2 The rest of the parameter values are Ga = 10,ρ r = 10− ,µ r = 10− ,M 1 = 0.4,M 2 = 0.2,Γ=0.1 andα r = 0.04.

92 24 zi 5 20 8 10.5 12 16 15 zCG 12

4 0 2 4 6 8 10 t

Figure 6.8: The eﬀect of initial location of the bubble on the temporal evolution of the center of 2 3 2 gravity,z CG. The rest of the parameter values are Ga = 10, Bo = 10− ,ρ r = 10− ,µ r = 10− , M1 = 0.2,M 2 = 0.1,Γ=0.1 andα r = 0.04. The prediction of Eq. (6.51) is shown by the dotted line.

(a) (b) 2.8 Bo 3.2 zi 10 5 2.6 8 100 10.5 2.8 12 2.4 15 lB lB 2.4 2.2

2 2

0 2 4 6 8 10 12 0 2 4 6 8 10 t t

Figure 6.9: (a) Evolution of the length of the bubble,l B for two values ofBo when he initial location of the bubblez i = 8. (b) The eﬀects of initial location of the bubble on elongation of the bubble for 3 Bo = 100. The radius of the tube,H=2.5. The rest of the parameters are Ga = 10,ρ r = 10− , 2 µr = 10− ,M 1 = 0.4,M 2 = 0.2,Γ=0.1 andα r = 0.04.

93 (a) (b)

Figure 6.10: Evolution of bubble shape (blue line), streamlines (lines with arrows), and temperature contours (shown in color) with time for (a) Bo = 10 and (b) Bo = 100, andH=2.5. The initial location of the bubblez i = 8. The inset at the bottom represents the colormap for the temperature 3 2 contours. The rest of the parameters are Ga = 10,ρ r = 10− ,µ r = 10− ,M 1 = 0.4,M 2 = 0.2, Γ=0.1 andα r = 0.04.

94 (a) (b) 4 11 M1=1.8, M2=0.9 M =1.0, M =0.5 3.5 1 2 M =0.4, M =0.2 1 2 10 Isothermal l 3 B zCG 9 2.5

2 8 0 2 4 6 8 10 0 2 4 6 8 10 t t

Figure 6.11: Evolution of (a) the length of the bubble,l B, (b) the location of center of gravity, in a tube havingH=2.1. The initial location of the bubblez i = 8. The rest of the parameters are 3 2 Ga = 5,ρ r = 10− ,µ r = 10− . The non-isothermal curve is plotted forΓ=0.1 andα r = 0.04.

conﬁnement. We have done this by varying the value of the dimensionless radius of the tube,H and

plotted in Fig. 6.9(a) the temporal evolution of the bubble length,l B, forBo = 10 andBo = 100, andH=2.5. For this set of simulations we place the bubble below the position of minimum surface

tension (zi = 8, zm = 10). As seen in thisﬁgure, the bubble undergoes a contraction at early times, which is followed by rapid expansion for both values of Bo. This is then followed by a sustained

increase (decrease) inl B with time for Bo = 100 (Bo = 10). The eﬀect of the initial location of

the bubble,z i on the elongation of the bubble is investigated in Fig. 6.9(b). It can be seen that

the length of the bubble,l B increases as we move the initial location of the bubble in the positivez direction. The evolution of the bubble shape, temperature distribution, andflow structure for the dynamics associated with Bo = 10 and Bo = 100 for the parameter values the same as those used in Fig. 6.9(a) are shown in Fig. 6.10. Inspection of thisfigure reveals that the bubble remains essentially bullet-shaped for Bo = 10, which is in contrast to the cap-like shape adopted by the bubble for the sameBo and largerH value. ForBo = 100, the bubble developsfilaments in its wake region, driven by the formation of a pair of counter-rotating vortices in this region, which leads to bubble elongation. This elongation is sustained by the action of the vortices whose size grows with time and they cause the stretching of thefilaments from the main body of the bubble towards the wake region. Next, we study the effect of self-rewetting character of the liquid surrounding the bubble on its elongation in the presence of confinement effects. The results are shown in Fig. 6.11 forBo = 10, Ga = 5 while the tube radius has been reduced toH=2.1 to intensify the effect of confinement; the rest of the parameters remainfixed at their ‘base’ values. It is seen clearly from Fig. 6.11(a) that

the bubble elongation rate increases withM 2(=M 1/2); the maximall B is reached at an earlier time

with increasingM 2. For the largestM 2 values studied, the bubble undergoes a weak contraction

to an essentially terminall B value. The bubble rise speed also increases withM 2, as shown in Fig.

6.11(b) which depicts the temporal evolution of the bubble centre of gravity,z CG.

We contrast in Fig. 6.12 theﬂow dynamics associated with the isothermal and (M 1 = 1.8,M2 = 0.9) cases shown in Fig. 6.11. It is seen that in contrast to the isothermal case in which the bubble

95 (a) (b)

Figure 6.12: Evolution of bubble shape with time for (a) isothermal case, and (b)M 1 = 1.8,M2 = 0.9 (temperature contours shown in color). The inset at the bottom represents the colormap for the temperature contours. The rest of the parameters are the same as those used to generate Fig. 6.11.

96 shape remains approximately spherical, the bubble rising in a non-isothermal, self-rewettingfluid deforms significantly in the presence of confinement, and assumes the shape of a Taylor bubble. It is difficult to compare this with the behaviour of a bubble in a linearfluid since there is no obvious basis for such a comparison.

6.5 Concluding remarks

We have carried out an analytical and numerical investigation of a gas bubble rising in a non- isothermal liquid in a cylindrical tube whose walls have a linearly-increasing temperature. Two types of liquids were considered: a ‘linear’ liquid whose surface tension decreases linearly with temperature; and a so-called ‘self-rewetting’ liquid which exhibits a parabolic dependence of the surface tension on temperature, with a well-defined minimum. Attention was focused on how the latter can affect the development of thermocapillary Marangoni stresses and, in turn, the bubble dynamics. We have shown that in the Stokesflow limit, the motion of a spherical bubble can be arrested in a self-rewetting liquid, and derived a formula for the terminal distance in this case, even if the temperature gradient in this liquid, which surrounds the bubble, is positive. This is in contrast to the case of a linear liquid in which a negative gradient is necessary to bring the bubble motion to a halt. We have also studied the bubble motion numerically to account for the presence of thermocapillarity, buoyancy, inertia, interfacial deformation, and confinement effects. Our results have demonstrated that the motion of the bubble in a self-rewettingfluid can be reversed and then arrested in the limit of weak bubble deformation. In this limit, good agreement between the numerical and analytical predictions for the terminal distance was found, even for appreciable inertial contributions; this is due to the fact that during the latter stages of theflow, the bubble enters the creepingflow regime prior to reaching its terminal location. Theflow reversal becomes accentuated for strongly self-rewetting liquids in the presence of significant inertia. These phenomena are absent in the case of linear liquids and are attributed to the thermocapillary Marangoni stresses which oppose the direction of the buoyancy-driven bubble rise when the bubble crosses the vertical location associated with the surface tension minimum. These stresses gain in significance during the course of theflow and eventually become dominant leading to reversal and arrest of the bubble motion. We have also shown that a bubble in a self-rewettingfluid undergoes considerable elongation for significant confinement, forming a Taylor bubble; this is absent in the case of isothermalflows in which the bubble remains essentially spherical. Non isothermal bubbles may appear in a variety of situations and this study is but a glimpse of what may happen when a bubble rises in afluid with temperature gradients. At higher temperatures,the liquid may boil and increase the size of the bubble, or generate vapour bubbles on walls of the container. But even at lower temperatures, liquid surfaces undergo phase change in an open atmosphere under certain conditions. Evaporation can occur at room temperature also and cause lakes and creeks to dry. A preliminary study was carried out to understand evaporation of liquid drops falling under gravity.

97 Chapter 7

Evaporating falling drop

7.1 Introduction

Phase-change is commonly observed in unconfined air-liquidflows, for instance gasoline droplets evaporating in an internal combustion engine, solidification of alloys, evaporation of ocean water during wave-breaking and condensation of snow on falling snow crystals. Many industrial processes require melting, evaporation and solidification of certain materials to either manufacture a desired product or as coolants and other supporting components. Therefore understanding phase-change is very important for industries as well as for natural phenomena. Phase-change is difficult to measure experimentally and hence numerical techniques would be a great contribution to the research in this area. As discussed in Chapter 1, there have been a significant amount of work on modelling the boiling phenomenon. However the evaporation of drops below their boiling point is difficult to simulate because of the dependence of evaporation rate on the local pressure at the drop surface and the change in temperature as the evaporation proceeds. To model the process of evaporation accurately, pressure at the liquid surface should be calculated with good accuracy. Due to the presence of high density ratio and surface tension for a liquid drop in air, most of the numerical techniques generate spurious currents, which are errors in the velocityfield in the vicinity of the interface. Therefore, a good interface capturing scheme is necessary to simulate evaporation below boiling point of the liquid. Below, we discus the numerical scheme and incorporation of a phase- change model in a state-of-the-artflow solver - gerris, created by Stephane Popinet [170].

7.2 Formulation

We conducted three-dimensional numerical simulation of a drop (at temperatureT c) of radiusR falling under the action of gravity,g inside anotherﬂuid initially kept at a higher temperature

(temperatureT h). The schematic showing the initial conﬁguration of the drop is presented in Fig. 2.1 (b). The inner and outerﬂuids are designated by ‘i’ and ‘o’, respectively. At time,t=0a spherical drop is placed at a height,H = 42R from the bottom of the computational domain of size 30R 30R 60R. The governing equations are the same as those described in Chapter 2 (Eqs × × (2.15)-(2.18)). The vapour volume fractionc v is initialized as zero at timet = 0. The boundary conditions on all sides of the domain is imposed as follows: Neumann condition for scalars (p,c a,T,

98 andc v) and the velocity components tangential to the given boundary, and zero dirichlet condition for velocity components normal to the given boundary. The evaporation model is discussed below.

7.2.1 Evaporation model

The interfacial mass source per unit volume,˙mv is given by

ρ ˙m = Dav g ω ˆn, (7.1) v A 1 ω ∇ · � − � where is the area of the interface per unit volume, andω is the mass fraction of vapour inside A the gas phase, given by c ρ ω= v v . (7.2) caρg The gas-liquid interface is assumed to be at the saturation condition, such that the gradient of vapour mass fraction across the interface can be estimated as,

ω ω ˆn ω n − sat , (7.3) ·∇ ≈ x x n − sat

wherex n andx sat are the locations of a neighbouring point lying in the gas phase and the location of the interface, respectively. Correspondingly,ω n andω sat are the vapour mass fractions atx n andx sat, respectively. The saturation vapour mass fraction is calculated, using Dalton’s law of partial pressure, as

p ω =M sat , (7.4) sat gv p whereM gv is the molar mass ratio of vapour to gas-mixture,p is the pressurefield, andp sat is the saturation vapour pressure depending on the local temperature as follows (employing the Wagner equation) p a τ+a τ 1.5 +a τ 3 +a τ 6 ln sat = 1 2 3 4 , (7.5) p 1 τ � cr � − wherep is the critical pressure, andτ=1 T/T , whereinT is the critical temperature. For cr − cr cr water, the coefficientsa (i = 1 to 4) in the above equation are,a = 7.76451,a = 1.45838, i 1 − 2 a = 2.77580 anda = 1.23303. The critical pressure (p ) and temperature (T ) for water 3 − 4 − cr cr are 220.584 kPa and 647 K, respectively. The values ofT h andT c arefixed at 343 K and 293 K, respectively.

The dimensionless interfacial source term for mass transfer (˙mv) is given by

ρ ˙m = rg ω ˆn, (7.6) v A P e(1 ω) ∇ · � − �

cv cv √gRR whereρ rg = ρrv + 1 ,Pe( ) is the peclet number for the diﬀusion of the vapour in ca − ca ≡ Dav dry air, and is the area� of the� interface per unit volume. In Eq. (6.5), surface tension of the liquid A gas interface is given by the following constitutive equation

σ=1 T, (7.7) −M T

99 Figure 7.1: Vapour mass source calculated only in the interfacial cells. Normal to the interface (yellow, dashed line) and its components (yellow, solid lines) are shown.

whereM T =γ T T1/σ0.

7.2.2 Model implementation in gerris

The model implementation is similar to that of Hardt & Wondra [203], in that the mass source is smeared about the interface. A positive and a negative source term is added to the vapour and liquid side of the interface without having to add any source term right at the interface. This approach is simple to implement as it does not require one to modify the advection scheme of the

VOF volume-fraction variable,c a. The pressurefield thus generated, automatically drives thefluid from the interface towards the vapour phase. As shown in Fig. 7.1, the mass source is computed in the cells containing the interface. The geometric reconstruction of a sharp interface is exploited to accurately calculate the vapour mass fraction gradient in a cell given by Eq. (7.3). The geometric location of the interface is calculated asx sat, which is not possible for diffuse interface or other methods without a sharp interface reconstruction or tracking. The mass source is thus calculated using Eq. (7.6), where is derived from the geometry of the interface. The mass source thus A calculated is smeared in 3-4 cells about the interface and the mass source in the interfacial cells is made zero. The mass source thus obtained of either side of the interface is weighted such that the total massflux per unit time remains same as that for the interfacial mass source. Next, the mass source is made positive and negative on the gas and liquid side of the interface, respectively. This is a simple model which doesn’t need any extensive modification of the existing code and is easy to implement. This method is faster than that of Hardt & Wondra [203] in that a diffusion equation is not solved to smear the mass source about the interface, instead a corner averaging is performed. Some of the preliminary results are presented in the next section.

100 7.3 Results: Evaporating falling drops

We present the three-dimensional simulations of evaporating drops of three diﬀerent volatilities. The properties of the three cases considered are as follows:

(a) (b) (c) (d)

Figure 7.2: Drop shape and vapour volume fraction contours with minimum and maximum levels 3 as 0 and 10− , for a water drop falling in air at time,t = 1, 3, 4 and 5 (from left to right). The other parameters are: Ga = 500,Bo=0.025,ρ rb = 1000,ρ rv = 0.9,µ rb = 55,µ rv = 0.7,Pe = 200, λ = 26,λ = 1.0,c = 4,c = 2, = 0.2,T = 293K, andT = 343K. rb rv p,rb p,rv M T c h

Water drop in air

As mentioned earlier, the coefficients for Wagner equation for water drop in dry air are,a 1 = 7.76451,a = 1.45838,a = 2.77580 anda = 1.23303. The critical pressure (p ) and − 2 3 − 4 − cr temperature (Tcr) for water are 22.06 MPa and 647 K, respectively. The values ofT h andT c are fixed at 343 K and 293 K, respectively. The drop shape at different times is shown in Fig. 7.2 along with the contours of vapour volume fraction. Other parameters are mentioned in thefigure caption. Due to evaporation, a spherical envelope of water vapour is formed at an initial time (Fig. 7.2(a)), which is then convected to the wake of the drop. The amount of vapour keeps on increasing as the time progresses. A lower pressure in the wake region promotes evaporation at the rear side of the drop which reaches a saturation as the wake becomes saturated with the vapour. Also, a drop in temperature in the wake region causes the evaporation to slow down after some time. Deformation and breakup increase the rate of evaporation which is evident from Fig. 7.2(d).

Chloroform drop in dry air

The coefficients for Wagner equation for chloroform drop in dry air are,a = 6.50419,a = 1 − 2 0.010117,a = 0.37359 anda = 2.2322. The critical pressure (p ) and temperature (T ) for 3 − 4 − cr cr water are 5.33 MPa and 537 K, respectively. The values ofT h andT c arefixed at 343 K and 293 K, respectively. A chloroform drop falling in dry air is shown at different times in Fig. 7.3 for the parameter values mentioned in the caption. It is noted that the amount of vapour generated is more as compared to a falling water drop. The spherical envelope of vapour at an initial time,t = 1 (Fig. 7.3(a)) convects to the wake of the drop as the drop accelerates downwards. A vapour trail is seen

101 (a) (b) (c) (d)

Figure 7.3: Drop shape and vapour volume fraction contours with minimum and maximum levels 3 as 0 and 3 10 − , for a chloroform drop falling in air at time,t = 1, 3, 5 and 7 (from left to right). × The other parameters are: Ga = 100, Bo=0.1,ρ rb = 1480,ρ rv = 0.9,µ rb = 281.2,µ rv = 0.7, P e = 230,λ = 6,λ = 1.0,c = 1.05,c = 2, = 0.2,T = 293K, andT = 343K. rb rv p,rb p,rv M T c h behind the drop as it falls. The drop elongates and breaks up into several fragments (Fig. 7.3(d)) which promotes the evaporation further.

(a) (b) (c) (d)

Figure 7.4: Drop shape and vapour volume fraction contours with minimum and maximum levels 3 as 0 and 3 10 − , for a chloroform drop falling in air at time,t = 1, 3, 5 and 7 (from left to right). × The other parameters are: Ga = 100, Bo=0.1,ρ rb = 1480,ρ rv = 0.9,µ rb = 281.2,µ rv = 0.7, P e = 230,λ = 6,λ = 1.0,c = 1.05,c = 2, = 0.2,T = 293K, andT = 343K. rb rv p,rb p,rv M T c h

Octane drop in dry air

The coefficients for Wagner equation for an octane drop in dry air are,a = 8.1622,a = 2.1052, 1 − 2 a = 5.4164 anda = 0.1583. The critical pressure (p ) and temperature (T ) for water are 24.9 3 − 4 − cr cr MPa and 568.5 K, respectively. The values ofT h andT c arefixed at 343 K and 293 K, respectively. An octane drop falling in dry air is shown at different times in Fig. 7.4 for the parameter values

102 mentioned in its caption. The octane drop behaves similar to a chloroform drop, with a difference in the breakup dynamics (Fig. 7.4) , which alters the vapour generation. This would lead to an altogether different vapour concentration for the octane drop even when the dynamics is slightly different from that of the chloroform drop.

7.4 Future work

We have recently started the study of evaporation and it would be our future task to study this parametrically vast problem and identify the variables which play crucial role to govern this phenomenon. The dynamics of the drop can change the evaporation rate and vapour distribution in air. It would be important to study the effect of inertia and volatility on evaporation. It would also be interesting to study the effect of breakup on the vapourfield generated. This would require a more accurate calculation of mass source on the drop, which is an ongoing work and has not been included in this thesis. A better numerical method is needed to develop where the source term does not have to be smeared and can be treated sharply at the interface.

103 Chapter 8

Conclusions

We started off to understand canonical bubble and drop motion under gravity. Some time ago, Rama madam saw a big bubble at Visvesvaraya industrial and technological museum, Bangalaore (India) and was fascinated by it. When she came to hyderabad, she asked something to the effect of - how big can a bubble get? Soon after we started investigating bubble dynamics, I had the opportunity to present my preliminary work at the Fluids Days 2013 meeting organized on the birthday of Prof. Roddam Narasimha. As mentioned in the introduction (Chapter 1), Prof. Garry Brown asked us the question - ”why should a bubble and drop behave differently?” - which led us to ponder over it and come up with a vorticity argument as mentioned in Chapter 3. We analysed bubble and drop motion under axisymmetric assumption, and later on showed regions of axisymmetry and asymmetry by performing extensive three-dimensional simulations. The fully three-dimensional nature of rising bubbles and falling drops was observed and was attempted to quantify. Bubbles and drops may not always be in an isothermal or a Newtonianfluid. A few complexities were added to the system and the assumptions of constant temperature and Newtonian nature of thefluid were relaxed. Hence, in the next part of the work, we considered the problem of a rising bubble inside a viscoplastic material. This problem had been studied either assuming theflow to be steady or in Stokesflow regime, previously. All of the previous studies reported steady shapes of bubble or a at least a terminal shape. By computer simulations, we showed that for a particular range of parameters, the bubble motion may become unsteady and presented a mechanism for the “crawling” motion of the bubble. In another study, inspired from an experiment done by Khellil Sefiane, we tried to investigate the bubble rise dynamics in a confined tube with a “self-rewetting”fluid as the surrounding medium (see Chapter 6 in detail). It was observed that a positive temperature gradient can also act to reverse theflow when a “self-rewetting”fluid is used instead of afluid for which surface-tension depends linearly on temperature. A theoretical expression was derived to estimate the terminal location of the bubble which agreed very well with the computational result. In real-life drops like rain drops, fuel droplets in an internal combustion engine, paint droplets through a nozzle, evaporation and condensation have to be accounted for, to capture the essential physics. Thefinal part of this work was to develop a nuemrical technique to simulate evaporation of falling drops under the action of gravity. This was done by modifying the source-code of an already existingfinite volume intefacialflow solver - gerris [170] to include phase change. A preliminary

104 study of falling drops of diﬀerent volatility were studied to obtain the vapourﬁeld and drop shapes. In conclusion, this work has developed from a basic study of bubbles and drops and went on to add complexities such as non-Newtonian behaviour, Marangoni forces, and evaporation, keeping the focus on their motion and deformation.

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