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.