Pattern formation and bistability in flow between counterrotating cylinders

Golubitsky, Martin; Langford, William F.

doi:10.1016/0167-2789(88)90063-2

Cited by 72 publications

(57 citation statements)

References 22 publications

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“…The focus of our study differs from that of Golubitsky and Langford [6], Langford et al [12] and Chossat and Iooss [2]. These authors examine the low order dynamics of flows in the Taylor apparatus, in particular how counter-rotation of the outer cylinder effects the sequence of time-dependent flow regimes which develop on increasing the angular velocity of the inner cylinder.…”

Section: W Introductionmentioning

confidence: 47%

DegenerateO(2)-Equivariant bifurcation equations and their applications to the taylor problem

Okamoto

Tavener

1991

Japan J. Indust. Appl. Math.

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A coupled pair of O(2)-equivariant bifurcation equations ate developed to model steady axisymmetric Navier-Stokes flow between two concentric cylinders. The flow is driven by the rotation of the inner cylinder and axially periodic boundary conditions are applied at the two end surfaces. We examine the stabilities of steady axisymmetric flows and determine how these stabilities change upon varying the geometric parameters in the model. Our abstract bifurcation equations axe of degree ¡ and are dependent upon three physical paxameters. By considering unfoldings about degenerate singularities we reproduce the qualitative behaviour generated numericaUy by Tavener aad Cliffe [18]. New bifurcation diagrams are produced by varying a parameter held constant by these authors.Key words: Taylor vortex, bifurcation from double eigenvalue, degenerate singularity w IntroductionIn this paper we perform a mathematical analysis of steady, axisymmetric Navier-Stokes flows between a pair of concentric cylinders in which the flow is driven by the rotation of the inner cyUnder alone. In the physical experiment, both the outer cylinder and top and bottom surfaces are stationary and for smaU Reynolds numbers, (the dimensionless angular velocity of the inner cylinder), there is a unique steady flow in which the particle paths ate essentiaUy circular except near the top and bottom surfaces. This flow loses stability with respect to axisymmetric disturbances above a critical Reynolds number. With a gradual increase in the Reynolds number an axisymmetric steady flow develops, comprising a discrete number of toroidal shaped Taylor cells in which the particle paths ate spirals around the in_ ner cylinder. The cellular flow which develops on gradual increase of the Reynolds number is caUed the 'primary' flow. Taylor cells prefer to have ah approximately square cross-section, hence for cylinders with length to gap width ratios near two, the stable primary flow is a two-cell flow. For cylinders with length to gap width ratios near four, the stable primary flow is a four-cell flow. Except in very short cylinders, the primary flow is always observed to have even number of ceUs. We wish to understand how the change from a two-cell primary flow to a four-cell primary flow occurs with increasing length to gap width ratio. We relax the physical boundary conditions and apply axially periodic boundary conditions (as opposed to nonslip conditions) at the two ends. With these boundary conditions, a purely axThe ¡ author is partially supported by Ohbayashi Corporation.

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Section: W Introductionmentioning

confidence: 47%

DegenerateO(2)-Equivariant bifurcation equations and their applications to the taylor problem

Okamoto

Tavener

1991

Japan J. Indust. Appl. Math.

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“…This resonance was studied in [4], [5], [13], [14], [19] for m = 0, n = 1 and [4], [5] for m = 1, 2, n = m + 1, and in [16] for arbitrary m, n. The codimension-2 points, corresponding to resonance Res 1, form in the four-dimensional space Π of parameters (α, η ,R 1 , R 2 ) certain surfaces (two-parametric families). Projections of the sections of such surfaces for η = 1.2 into the plane (R 1 , R 2 ) are shown in Fig.…”

Section: Introductionmentioning

confidence: 99%

Resonances in the Codimension-2 Bifurcations in the Couette–Taylor Problem

Yudovich

Ovchinnikova

2009

J. Math. Fluid Mech.

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The investigation of codimension-2 bifurcations, in particular in systems with cylindric symmetry, enables us to deduce new types of secondary regimes branching-off from the symmetric regimes. This investigation also allows us the unique possibility of a rigorous treatment of chaotic solutions to Navier-Stokes and other nonlinear PDE's. The central manifold approach combined with the reduction to the normal form lead to the so-called amplitude systems. These ODE systems describe the nonlinear interaction between the neutral modes, and always include several nonlinear terms due to so-called intrinsic resonances. However, sometimes additional resonances appear. In this paper we present the complete list of all possible resonances in dynamic systems with cylindric symmetry and the corresponding forms of the amplitude equations. Further, we present the results of extensive numerical investigation of the resonant codimension-2 bifurcations in the Couette-Taylor problem, thus creating an intriguing subject for further investigation. (2000). 76B47 (34D20, 70K30). Mathematics Subject Classification

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“…Motivation for the present work is provided also by the classical Taylor-Couette experiment, in which the flow of a fluid (usually water) between two differentially rotating long coaxial cylinders is studied, see for example [1,6,8,16] …”

Section: Relationship To Previous Workmentioning

confidence: 99%

Hysteresis in a Rotating Differentially Heated Spherical Shell of Boussinesq Fluid

Lewis

Langford

2008

SIAM J. Appl. Dyn. Syst.

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A mathematical model of convection of a Boussinesq fluid in a rotating spherical shell is analyzed using numerical computations guided by bifurcation theory. The fluid is differentially heated on its inner spherical surface, with the temperature increasing from both poles to a maximum at the equator. The model is assumed to be both rotationally symmetric about the polar axis and reflectionally symmetric across the equator. This work is an extension to spherical geometry of previous work on the differentially heated rotating annulus. The spherical geometry is motivated by applications to planetary atmospheres. As the temperature gradient increases from zero, large Hadley cells extending from equator to poles form immediately. For larger temperature differences, two or three convection cells appear in each hemisphere. An organizing centre is shown to exist, at which two saddle-node bifurcations come together in a codimension-2 hysteresis bifurcation (or cusp) point, providing a mechanism for hysteretic transitions between different cell patterns as the temperature gradient is varied.

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Pattern formation and bistability in flow between counterrotating cylinders

Cited by 72 publications

References 22 publications

DegenerateO(2)-Equivariant bifurcation equations and their applications to the taylor problem

DegenerateO(2)-Equivariant bifurcation equations and their applications to the taylor problem

Resonances in the Codimension-2 Bifurcations in the Couette–Taylor Problem

Hysteresis in a Rotating Differentially Heated Spherical Shell of Boussinesq Fluid

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