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Rates of Mixing in Models of Fluid Devices with Discontinuities

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Rates of Mixing in Models of Fluid Devices with Discontinuities Rates of mixing in models of fluid devices with discontinuities Hannah Elizabeth Kreczak Submitted in accordance with the requirements for the degree of Doctor of Philosophy University of Leeds Centre for Doctoral Training in Fluid Dynamics September 2019 I confirm that the work submitted is my own, except where work which has formed part of jointly authored publications has been included. No material is included which has been accepted for the award of any other degree or diploma at any university or equivalent institution. The contribution of the candidate and the other authors to this work has been explicitly indicated below. The candidate confirms that appropriate credit has been given within the thesis where reference has been made to the work of others. The work in Chapter 5 of the thesis has appeared in publication as follows: Deceleration of one-dimensional mixing by discontinuous mappings, H. Kreczak, R. Sturman, and M. C. T. Wilson, Phys. Rev. E 96, 053112 (2017). Conception of the model that is investigated, computational simulations, data analysis of the results and writing of the paper is attributable to myself. Supervision of the research with suggestions on lines of enquiry, manuscript re- vision and suggested edits of the paper were carried out by my supervisor team Dr Rob Sturman and Dr Mark Wilson. 2 \That continued mixing results in uniformity, and that uniformity is only to be obtained by mixing, will be generally acknowledged, but how deeply and universally this enters into all the arts can but rarely have been apprehended." Osborne Reynolds, 1894 This thesis is dedicated to my parents, Stephen and Lynn. Acknowledgements First of all I would like to express my immense gratitude to my su- pervisors, Rob Sturman and Mark Wilson. I am thankful for the guidance they have both shown me throughout this supervisory pe- riod, not only relating to my academic studies, but also in my per- sonal wellbeing. I am appreciative of Mark's engineering eye, keeping the research grounded in reality, and patience during the discussions which spiralled into abstract dynamical theory. I extend additional gratitude to Rob, for the mentorship during my attendance at Snow- bird, and most importantly the initial encouragement to purse a PhD in the first place. I would like to thank Jean-Luc Thiffeualt, and the extended network of academics at the University of Madison-Wisconsin, who warmly received me during my academic placement. I would also like to thank the teams at Northwestern University, and the Universities of Exeter and Manchester, for inviting me to speak about my work and engaging me in insightful discussions, especially those over a few pints. I would like to thank EPSRC for providing the funding allowing me to undertake this research as part of the Leeds Centre of Doctoral training in Fluid Dynamics. I have benefited massively from the fi- nancial support, providing me with the opportunities to visit national and international conferences and academic institutions. I am par- ticularly grateful to Claire Savy and the CDT management team, for arranging extra professional development to accompany my academic studies, preparing me for my future career. On a personal note, I am extremely indebted to my friends and col- leagues for the necessary distractions over the last five years. Special mentions to Ollie, Inna, Georgie, Jacob, Jenny, and Andy, for the lunch time chats and cryptic crosswords, laughs over after work wines, and encouragements during climbs. I would especially like to thank my house mate Caitlin, for being both a fun and supportive friend over the years. Last but not least, I would like to thank my parents and brothers for the unending support I have received during my education to date. Although the financial support my parents have given me has helped dispel worries that would have made it difficult to complete this thesis, it is their unwavering love for which I am most grateful, and which has allowed me to pursue my passions and achieve my many successes. Abstract In the simplest sense, mixing acts on an initially heterogeneous sys- tem, transforming it to a homogeneous state through the actions of stirring and diffusion. The theory of dynamical systems has been suc- cessful in improving understanding of underlying features in fluid mix- ing, and how smooth stirring fields, coherent structures and bound- aries affect mixing rates. The main stirring mechanism in fluids at low Reynolds number is the stretching and folding of fluid elements, although this is not the only mechanism to achieve complicated dy- namics. Mixing by cutting and shuffling occurs in many situations, for example in micro–fluidic split and recombine flows, through the closing and re- orientation of values in sink{source flows, and within the bulk flow of granular material. The dynamics of this mixing mechanism are subtle and not well understood. Here, mixing rates arising from fundamental models capturing the essence of discontinuous, chaotic stirring with diffusion are investigated. In purely cutting and shuffling flows it is found that the number of cuts introduced iteratively is the most important mechanism driv- ing the approach to uniformity. A balance between cutting, shuffling and diffusion achieves a long-time exponential mixing rate, but sim- ilar mechanisms dominate the finite time mixing observed through the interaction of many slowly decaying eigenfunctions. The time to achieve a mixed condition varies polynomially with diffusivity rate κ, obeying t / κ−η. For the transformations meeting good stirring criteria, η < 1. Considering the time to achieve a mixed condition to be governed by a balance between cutting, shuffling, and diffusion derives η ∼ 1=2, which shows good agreement with numerical results. In stirring fields which are predominantly chaotic and exponentially mixing, it is observed that the addition of discontinuous transforma- tions contaminates mixing when the stretching rates are uniform, or close to uniform. The contamination comes from an increase in scales of the concentration field by the reassembly of striations when cut and shuffled. Mixing stemming from this process is unpredictable, and the discontinuities destroy the possibility to approximate early mixing rates from stretching histories. A speed up in mixing rate can be achieved if the discontinuity aids particle transport into islands of the original transformation, or chops and rearranges large striations generated from highly non-uniform stretching. The long-time mix- ing rates and time to achieve a mixed condition are shown to behave counter-intuitively when varying the diffusivity rate. A deceleration of mixing with increasing diffusion coefficient is observed, sometimes overshooting analytically derived bounds. 8 Contents 1 Introduction1 2 Literature Review7 2.1 Mixing and Applications . .8 2.1.1 Advection-diffusion equation . .8 2.1.2 Scaling numbers . .9 2.2 Fluid transport as a dynamical system . 11 2.2.1 Chaotic advection . 11 2.2.2 Ergodic hierarchy . 14 2.3 Mixing via chaotic advection . 16 2.3.1 Iterative advection-diffusion equation . 17 2.3.2 Properties of Transfer Operators . 17 2.3.3 Numerical approximations of transfer operators . 21 2.3.4 Numerical approach using Discrete Fourier expansion . 23 2.4 Measures of mixing . 26 2.4.1 Functional norms and the mix norm . 27 2.4.2 Measures of mixing in Fourier space . 29 2.4.3 Additional measures of mixing . 30 2.5 Mixing rates in chaotic advection . 31 2.5.1 Non-uniformity and strange-eigenmodes . 31 2.5.2 Boundaries and walls . 35 2.5.3 Barriers to mixing . 36 2.6 Discontinuous mixing . 37 2.6.1 Dynamics of Piecewise Isometries . 37 2.6.2 Examples of discontinuous mixing . 39 2.6.3 Dynamics of mixed SF and CS systems . 40 2.6.4 Dynamics of mixed CS and D systems . 41 i CONTENTS 3 Fundamental models of mixing by chaotic advection 45 3.1 Discrete maps as models for mixing . 45 3.1.1 Toral Automorphisms . 45 3.1.2 Poincar´esections . 50 3.1.3 Lyapunov exponents . 51 3.2 Discrete chaotic maps with diffusion . 55 3.2.1 Analytic solutions of idealised chaotic mixing . 55 3.2.2 Numerical solutions of non-uniform chaotic mixing . 58 3.2.3 Time to a mixed condition . 64 3.3 Conclusions . 65 4 Mixing by cutting and shuffling 67 4.1 Interval exchange transformations . 69 4.1.1 Formal definitions . 69 4.1.2 Model parameters for investigation . 71 4.2 Iterative Advection-Diffusion Transformation . 74 4.2.1 Transfer operators for permutations of equal sized cells . 74 4.2.2 Initial conditions of interest . 75 4.2.3 Resolution of Fourier Modes . 77 4.2.4 An initial example of mixing by Interval Exchange trans- formations with diffusion . 78 4.3 Stages of mixing in Diffusive Interval Exchange Transformations . 82 4.3.1 Asymptotic mixing rates . 83 4.3.2 Finite time mixing: interaction of slowly decaying eigen- functions . 85 4.3.3 Effect of initial condition . 91 4.4 Dependence on parameters . 93 4.4.1 Effect of permutation rearrangement . 93 4.4.2 Effect of sub-interval ratio r ................. 99 4.4.3 Effect of the number of sub-intervals N ........... 104 4.5 Polynomial dependence on the time to achieve a mixed condition with diffusivity rate . 107 4.5.1 Numerical results for t95 ................... 108 4.5.2 Comparison to leading eigenvalue . 110 4.5.3 Mechanism for polynomial scaling relations . 113 4.6 Conclusions . 117 ii CONTENTS 5 Mixing by piecewise, uniformly expanding maps 121 5.1 Formulation of the problem . 122 5.1.1 Baker's transformation with permutations . 122 5.2 Dynamics in the absence of diffusion . 124 5.2.1 Space-time plots of segregation . 124 5.2.2 Rate of decay of correlations .
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