Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical Transactions paper: part 2

In 1923, the Philosophical Transactions published G. I. Taylor’s seminal paper on the stability of what we now call Taylor–Couette flow. In the century since the paper was published, Taylor’s ground-breaking linear stability analysis of fluid flow between two rotating cylinders has had an enormous impact on the field of fluid mechanics. The paper’s influence has extended to general rotating flows, geophysical flows and astrophysical flows, not to mention its significance in firmly establishing several foundational concepts in fluid mechanics that are now broadly accepted. This two-part issue includes review articles and research articles spanning a broad range of contemporary research areas, all rooted in Taylor’s landmark paper. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical Transactions paper (part 2)’

Cutting and Shuffling a Line Segment: Mixing by Interval Exchange Transformations

We present a computational study of finite-time mixing of a line segment by cutting and shuffling. A family of one-dimensional interval exchange transformations is constructed as a model system in which to study these types of mixing processes. Illustrative examples of the mixing behaviors, including pathological cases that violate the assumptions of the known governing theorems and lead to poor mixing, are shown. Since the mathematical theory applies as the number of iterations of the map goes to infinity, we introduce practical measures of mixing (the percent unmixed and the number of intermaterial interfaces) that can be computed over given (finite) numbers of iterations. We find that good mixing can be achieved after a finite number of iterations of a one-dimensional cutting and shuffling map, even though such a map cannot be considered chaotic in the usual sense and/or it may not fulfill the conditions of the ergodic theorems for interval exchange transformations. Specifically, good shuffling can occur with only six or seven intervals of roughly the same length, as long as the rearrangement order is an irreducible permutation. This study has implications for a number of mixing processes in which discontinuities arise either by construction or due to the underlying physics.Comment: 21 pages, 10 figures, ws-ijbc class; accepted for publication in International Journal of Bifurcation and Chao

Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua

We compare and contrast two types of deformations inspired by mixing applications -- one from the mixing of fluids (stretching and folding), the other from the mixing of granular matter (cutting and shuffling). The connection between mechanics and dynamical systems is discussed in the context of the kinematics of deformation, emphasizing the equivalence between stretches and Lyapunov exponents. The stretching and folding motion exemplified by the baker's map is shown to give rise to a dynamical system with a positive Lyapunov exponent, the hallmark of chaotic mixing. On the other hand, cutting and shuffling does not stretch. When an interval exchange transformation is used as the basis for cutting and shuffling, we establish that all of the map's Lyapunov exponents are zero. Mixing, as quantified by the interfacial area per unit volume, is shown to be exponentially fast when there is stretching and folding, but linear when there is only cutting and shuffling. We also discuss how a simple computational approach can discern stretching in discrete data.Comment: REVTeX 4.1, 9 pages, 3 figures; v2 corrects some misprints. The following article appeared in the American Journal of Physics and may be found at http://ajp.aapt.org/resource/1/ajpias/v79/i4/p359_s1 . Copyright 2011 American Association of Physics Teachers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the AAP

Influence of boundaries on pattern selection in through-flow

The problem of pattern selection in absolutely unstable open flow systems is investigated by considering the example of Rayleigh-B\'{e}nard convection. The spatiotemporal structure of convection rolls propagating downstream in an externally imposed flow is determined for six different inlet/outlet boundary conditions. Results are obtained by numerical simulations of the Navier-Stokes equations and by comparison with the corresponding Ginzburg-Landau amplitude equation. A unique selection process is observed being a function of the control parameters and the boundary conditions but independent of the history and the system length. The problem can be formulated in terms of a nonlinear eigen/boundary value problem where the frequency of the propagating pattern is the eigenvalue. PACS: 47.54.+r, 47.20.Bp, 47.27.Te, 47.20.KyComment: 8 pages, 5 Postscript figures, Physica D 97, 253-263 (1996

Pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem: Taylor vortices in axial flow

A unique pattern selection in the absolutely unstable regime of a driven, nonlinear, open-flow system is analyzed: The spatiotemporal structures of rotationally symmetric vortices that propagate downstream in the annulus of the rotating Taylor-Couette system due to an externally imposed axial through-flow are investigated for two different axial boundary conditions at the in- and outlet. Unlike the stationary patterns in systems without through-flow the spatiotemporal structures of propagating vortices are independent of parameter history, initial conditions, and system's length. They do, however, depend on the axial boundary conditions, the driving rate of the inner cylinder and the through-flow rate. Our analysis of the amplitude equation shows that the pattern selection can be described by a nonlinear eigenvalue problem with the frequency being the eigenvalue. Approaching the border between absolute and convective instability the eigenvalue problem becomes effectively linear and the selection mechanism approaches that one of linear front propagation. PACS:47.54.+r,47.20.Ky,47.32.-y,47.20.FtComment: 15 pages (LateX-file), 8 figures (Postscript

Reconstruction of one-dimensional chaotic maps from sequences of probability density functions

In many practical situations, it is impossible to measure the individual trajectories generated by an unknown chaotic system, but we can observe the evolution of probability density functions generated by such a system. The paper proposes for the first time a matrix-based approach to solve the generalized inverse Frobenius–Perron problem, that is, to reconstruct an unknown one-dimensional chaotic transformation, based on a temporal sequence of probability density functions generated by the transformation. Numerical examples are used to demonstrate the applicability of the proposed approach and evaluate its robustness with respect to constantly applied stochastic perturbations

Sources of potential bias when combining routine data linkage and a national survey of secondary school-aged children: a record linkage study

Background Linking survey data to administrative records requires informed participant consent. When linkage includes child data, this includes parental and child consent. Little is known of the potential impacts of introducing consent to data linkage on response rates and biases in school-based surveys. This paper assessed: i) the impact on overall parental consent rates and sample representativeness when consent for linkage was introduced and ii) the quality of identifiable data provided to facilitate linkage. Methods Including an option for data linkage was piloted in a sub-sample of schools participating in the Student Health and Wellbeing survey, a national survey of adolescents in Wales, UK. Schools agreeing to participate were randomized 2:1 to receive versus not receive the data linkage question. Survey responses from consenting students were anonymised and linked to routine datasets (e.g. general practice, inpatient, and outpatient records). Parental withdrawal rates were calculated for linkage and non-linkage samples. Multilevel logistic regression models were used to compare characteristics between: i) consenters and non-consenters; ii) successfully and unsuccessfully linked students; and iii) the linked cohort and peers within the general population, with additional comparisons of mental health diagnoses and health service contacts. Results The sub-sample comprised 64 eligible schools (out of 193), with data linkage piloted in 39. Parental consent was comparable across linkage and non-linkage schools. 48.7% (n = 9232) of students consented to data linkage. Modelling showed these students were more likely to be younger, more affluent, have higher positive mental wellbeing, and report fewer risk-related behaviours compared to non-consenters. Overall, 69.8% of consenting students were successfully linked, with higher rates of success among younger students. The linked cohort had lower rates of mental health diagnoses (5.8% vs. 8.8%) and specialist contacts (5.2% vs. 7.7%) than general population peers. Conclusions Introducing data linkage within a national survey of adolescents had no impact on study completion rates. However, students consenting to data linkage, and those successfully linked, differed from non-consenting students on several key characteristics, raising questions concerning the representativeness of linked cohorts. Further research is needed to better understand decision-making processes around providing consent to data linkage in adolescent populations

Mechanisms for secondary instabilities in Couette-Taylor flow with superimposed axial and radial throughflows

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The transition to wavy vortices

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