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Proquest Dissertations Two Cases of Symmetry Breaking of Free Surface Flows Hamid AIT ABDERRAHMANE A thesis In The department of Mechanical and Industrial Engineering Presented in Partial Fulfilment of the Requirements For the Degree of Doctor Philosophy (Mechanical Engineering) at Concordia University Montreal, Quebec, Canada November 2008 © Hamid Ait Abderrahmane , 2008 Library and Archives Bibliothgque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l'6dition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre r6f6rence ISBN: 978-0-494-63388-5 Our file Notre r6f6rence ISBN: 978-0-494-63388-5 NOTICE: AVIS: The author has granted a non- L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. 14-1 Canada ABSTRACT Two Cases of Symmetry Breaking of Free Surface Flows Hamid AIT ABDERRAHMANE, Ph.D. Concordia University, 2008 The present thesis consists of two parts; both are devoted to two celebrated old problems in fluid dynamics. The first deals with symmetry breaking in a liquid layer flowing down an inclined plane. The second problem concerns the equilibrium and symmetry breaking of interfacial polygonal patterns generated by a system of vortices arranged on a circular ring. The first problem dates back to Nusselt (1916) who obtained the solution for the basic flow. Since then, thin layers of liquid falling down inclined plane continues to be the subject of extensive studies for both their practical applications and theoretical value. In this thesis, the problem is approached analytically. Three new mathematical models are proposed. The first two involve three and four equations respectively. These produce linear stability results that agree fairly with past experimental outcomes and results obtained with similar models. For a deeper and qualitative analysis a lower dimension model that retains the physics is needed. Hence, a two-equation model (involving only two fundamental flow parameters namely the film thickness and flow rate) is derived. The new model taking account of the iii shear stress at the free surface is shown to be superior to the existing two- equation model of Usha and Uma in Phys Fluid (2004). The influence of electrical and magnetic fields on the stability of falling film of an electrically conductor fluid is also investigated. In comparison with the model of Korsunsky (Eur.J.F.M.1999) for higher Reynolds numbers. The proposed model takes account of the inertia terms, which are of second order with respect to a small parameter namely the long wave parameter. As shown through the chapter four of the part one, the proposed two-equation model improves significantly Korsunsky's model. The second problem dates back to Kelvin (1867) who hypothesized atoms to be point vortices arranged in circular ring forming symmetrical polygonal patterns. Although, the atomic vortex model is long abandoned, the problem of system of point vortices has become of great interest in superfluidity and by analogy in plasma physics. Moreover, polygonal patterns, which are the signature of the presence of vortices, equally distributed in rings were also observed in several engineering problems and geophysical flows in nature. In fluid dynamics, polygonal patterns become clearly visible in swirling flows where the vortex core is hollow. The empty core can eventually support polygonal shapes (up to hexagonal). The first experimental report on the phenomenon was by Vatistas in 1990. In this thesis the phenomenon is revisited using image-processing technique that allows a deeper and more precise investigation. The dynamics of the patterns iv is investigated and for the first time the transition from one pattern to another is explored in detail. The stability condition for a system of point vortices on circular ring derived first by J.J Thomson (1897) and generalized later by Havelock (1931) for N point vortices including the influence of circular boundaries surrounding the equilibrium is confirmed. Frequency locking between the pattern and the disk frequencies which are suspected in the previous experiments is established and quantified. Moreover, the transition from the elliptical to the hexagonal pattern is found that it follows a "devil's staircase" scenario. Due to the similarity between the problem under the scope and other fields of physics, the present experimental results are anticipated to go beyond the field of fluid mechanics. v ACKNOWLEDGEMENTS I would like to thank my supervisors, Dr. G.H Vatistas and K. Siddiqui for their help and support during the preparation of my PhD. In particular Dr G.H Vatistas who is more than a supervisor; he was for me a mentor. I would also like to thank my examiners for managing to read the present manuscript. Much respect to my officemates, and hopefully still friends, Mohamed, Jomeir, and Saadi, for putting up with me for almost three years for all that serious discussion (!) and all those lunches. vi TABLE OF CONTENTS List of the figures Xi General Introduction 1 Chapter 1 Introduction 10 Chapter 2 Four-equation and Three-equation models 1. Introduction 16 2. Formulation of the problem 18 3. Models 22 3.1 Four-equation model 22 3.1.1 Linear stability 24 3.1.2 Conclusion 41 3.2 Three-equation model 41 3.2.1 Linear stability 44 3.2.2 Conclusion 56 Chapter 3 Two equation model 1. Model 57 2. Dynamic system formulation of permanent waves 62 3. Linear stability 63 4. Fixed points 68 vii 4.1 Stability and bifurcations of the stationary solutions 69 4.1.1 Transcritical bifurcation 74 4.1.2 Hopf bifurcation 75 5. Numerical Simulations 85 5.1 Illustration Bifurcation scenarios 85 5.2 Numerical implication of the correction 93 5.2.1 Transcritical bifurcation 93 5.2.2 Hopf bifurcation 94 6. Conclusion 104 Chapter 4 M.H.D falling film stability 1. Introduction 105 2. Formulation of the problem 108 3. Linear stability 112 4. Stability and bifurcation of the stationary solution 126 4.1. Transcritical bifurcation 129 4.2. Hopf bifurcation 132 5. Conclusion 138 viii Chapter 5 Introduction 1. Historical background 137 2. Patterns in several fields 138 3. Analogies and Pattern's stability 148 4. The fundamental nature of the phenomenon 149 5. Outline 156 Chapter 6 Confirmation of Kelvin's equilibria 1. Introduction 158 1.1 Outline 158 1.2 Survey on Swirling Flows 160 2. Experimental setup and measurement technique 163 2. 1 Experimental setup 163 2.2 Measurement technique 165 3. Results and discussion 168 4. Conclusions 186 Chapter 7 Transition between two Kelvin equilibria 1. Introduction 187 1.1 Outline 187 1.2 Bifurcations in swirling flows 187 ix 2.1 Experimental setup and measurement technique 189 2. 1 Experimental setup 189 2. 2 Measurement technique 190 3. Results and discussion 192 3.1 Transition in laboratory frame of reference 192 3.2 Transition in frame of reference moving with the patterns 208 3.3 Transition in term of pattern's area 213 4. Concluding Remarks 215 General conclusion 216 References 221 Appendixes 232 X List of the Figures Figure 2-1: Schematic of the problem 18 Figure2-2: Critical wave number in the case of vertical plane 28 Figure 2-3: Cutoff frequency 0 = 5.6 28 Figure2-4: Cutoff frequency <9 = 4.6 29 c Figure 2-5: Phase velocity — as function of wave number^ 29 kr Figure 2-6: Temporal growth rate -c, function of the wave number kr 30 Figure 2-7: Angular frequency cr function of the frequency kr 30 Figure 2-8: Image of different branches when the growth rate is c, = 0.02 32 Figure 2-9: Image of different branches when the growth rate is c, =0 33 Figure 2-10: Pinching process in the complex wave number plane(ki,kr) 34 Figure 2-11: Pinching process in the complex wave number plane 35 Figure 2-12: Pinching process in the complex wave number plane,kr) 36 Figure 2-13: Pinching process in the complex wave number plane 37 Figure 2-14: Spatial growth ratefunction of the frequency cr 39 Figure 2-15: Wave number k r function of the frequency 39 Figure 2-16: Spatial growth rate -fc. as function of wave number kr 40 Figure 2-17:
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