Contributions to the nonlinear theory of stability of viscous flow in pipes and between rotating cylinders

Joseph, Daniel D.; Hung, W. L.

doi:10.1007/bf00250825

Cited by 42 publications

(43 citation statements)

References 16 publications

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“…This feature appears to reflect the experimental observations of the full three-dimensional Taylor-Couette problem, where results of linear stability analysis closely predict the appearance of the first bifurcation which develops into a set of toroidal vortices, as exemplified by the early work of Taylor (1923). More formally, such results has been proven by Joseph and Hung (1971) for the case of no-slip boundary conditions. For the present model with double-periodic boundary conditions, Huang et al (2015) performed nonlinear stability analysis using a novel methodology based on sum-of-squares-of-polynomials optimisation techniques, extended from the work of Goulart and Chernyshenko (2012), and demonstrated that the flow is in fact stable to finite amplitude perturbations if it is linearly stable, in the range Ω = [0.2529, 0.7471].…”

Section: Linear Stability Analysis Of the First Bifurcationsupporting

confidence: 53%

Flow regimes in a simplified Taylor–Couette-type flow model

Lasagna

Tutty

Chernyshenko

2016

European Journal of Mechanics - B/Fluids

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In this paper we introduce a simplified variant of the well-known Taylor-Couette flow.The aim is to develop and investigate a model problem which is as simple as possible while admitting a wide range of behaviour, and which can be used for further study into stability, transition and ultimately control of flow. As opposed to models based on ordinary differential equations, this model is fully specified by a set of partial differential equations that describe the evolution of the three velocity components over two spatial dimensions, in one meridian plane between the two counter-rotating coaxial cylinders.We assume axisymmetric perturbations of the flow in a narrow gap limit of the governing equations and, considering the evolution of the flow in a narrow strip of fluid between the two cylinders, we assume periodic boundary conditions along the radial and axial directions, with special additional symmetry constraints. In the paper, we present linear stability analysis of the first bifurcation, leading to the well known Taylor vortices, and of the secondary bifurcation, which, depending on the type of symmetries imposed on the solution, can lead to wave-like solutions travelling along the axial direction. In addition, we show results of numerical simulations to highlight the wide range of flow structures that emerge, from simple uni-directional flow to chaotic motion, even with the restriction placed on the flow.

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Section: Linear Stability Analysis Of the First Bifurcationsupporting

confidence: 53%

Flow regimes in a simplified Taylor–Couette-type flow model

Lasagna

Tutty

Chernyshenko

2016

European Journal of Mechanics - B/Fluids

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“…Joseph and Hung (1971) ity boundary, which we briefly discuss in the Appendix. In summary, we have used the energy method to determine regions of absolute stability of plane Couette flow.…”

Section: Resultsmentioning

confidence: 99%

Convective instabilities in concurrent two‐phase flow: Part II. Global stability

Gumerman

Homsy

1974

AIChE Journal

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The method of energy is used to determine global stability limits for thermally stratified two-phase plane Couette flow. Instabilities due to surface tension variations, buoyancy effects, and shear are allowed for, but surface waves are specifically excluded from consideration. Under these assumptions, the Marangoni, Rayleigh, and Reynolds numbers completely describe the system. Stability plots, valid for disturbances of any magnitude, are presented for both streamwise oriented roll vortices and two-dimensional transversely oriented disturbances. It is shown that rolls are the most dangerous disturbances in the sense that they cannot be shown to be stable relative to transverse disturbances at any nonzero Reynolds numbers. Comparisons are made with existing linear limits, and these are seen to be close only for moderate Reynolds numbers.This study is the second in a series which focuses on convective instabilities due to surface tension, density, or velocity gradients in two phase flow. The need for a unified stability theory for these phenomena is clear: in the cases in which they occur, they significantly affect the rates of heat, mass, or momentum transport between the phases. These increases have yet to be exploited in any systematic manner in applications for a variety of reasons. The foremost of these pertains to the relative uncertainty in being able to predict when instabilities will and will not occur. As we have noted (Gumerman and Homsy, 1974), most theoretical analyses have been for stagnant plane layers, while most practical applications involve the presence of shear, for example, liquid-liquid extraction and film evaporation. In that work, we employed the techniques of linear theory to discuss the various modes of instability and gave quantitative stability SCOPE CONCLUSIONS AND SIGNIFICANCEThe model system employed here is that of thermally stratified plane Couette flow. The model is specifically restricted to fluid pairs with large density differences and large interfacial tension. These restrictions exclude the existence of interfacial waves discussed extensively in Part I (Gumerman and Homsy, 1974). Thus, instabilities due to surface tension, density, and velocity gradients are allowed for.It is shown that of the possible disturbance modes, streamwise oriented roll vortices are the most dangerous, that is, they yield the lowest global stability boundaries. These results are valid for disturbances of arbitrary magnitude. For the longitudinal roll mode, the linear limits are

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“…It has already been established that G(co) has positive coefficients in powers of (-(To), when -a2 < Co < ||B|lo ^ as i11 (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16). By similar reasoning, it is observed that G(ai) has the same properties as G(ao) since, for example, the second coefficient G((Tq)]\IG(<7o) in (4.35) behaves like a typical factor in (4.21) when expanded.…”

Section: The Problem Of Pes and The Methods Of Positivementioning

confidence: 99%

“…However, his proof does not lead to any conclusions about the behavior of the imaginary part of a. The Synge result has been extended to finite amplitude disturbances by Joseph & Hung [12]. As for instability results, rigorous proof was first provided by Velte [27], and by Yudovich [30], [31].…”

Section: Basic Formulationmentioning

confidence: 99%

The principle of exchange of stabilities for Couette flow

Herron¹,

Ali²

2003

Quart. Appl. Math.

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Abstract.The eigenvalue problem for the linear stability of Couette flow between two rotating concentric cylinders to axisymmetric disturbances is considered. It is proved that the principle of exchange of stabilities holds when the cylinders rotate in the same direction and the circulation decreases outwards. The proof is based on the notion of a positive operator which is analogous to a positive matrix. Such operators have a spectral property which implies the principle of exchange of stabilities.

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Contributions to the nonlinear theory of stability of viscous flow in pipes and between rotating cylinders

Cited by 42 publications

References 16 publications

Flow regimes in a simplified Taylor–Couette-type flow model

Flow regimes in a simplified Taylor–Couette-type flow model

Convective instabilities in concurrent two‐phase flow: Part II. Global stability

The principle of exchange of stabilities for Couette flow

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