On the behaviour of small disturbances in plane Couette flow

Gallagher, A. P.; Mercer, A. McD.

doi:10.1017/s0022112064000258

Cited by 23 publications

(9 citation statements)

References 5 publications

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“…For example, when Re = 1,000 it is necessary to take N 1 = N 2 = 45. A first check of the accuracy of the numerical method was performed by comparing the present results with available results for the plane Couette flow of a homogeneous fluid (e.g., [11]), and confirming excellent agreement. A further check was performed by comparing the growth rates of the two-layer channel flow with those obtained by an alternative shooting method based on fourth-order Runge-Kutta integration (e.g., [12, p. 452]).…”

Section: Methodssupporting

confidence: 63%

Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

Blyth

Pozrikidis

2004

J Eng Math

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The effect of inertia on the Yih-Marangoni instability of the interface between two liquid layers in the presence of an insoluble surfactant is assessed for shear-driven channel flow by a normal-mode linear stability analysis. The Orr-Sommerfeld equation describing the growth of small perturbations is solved numerically subject to interfacial conditions that allow for the Marangoni traction. For general Reynolds numbers and arbitrary wave numbers, the surfactant is found to either provoke instability or significantly lower the rate of decay of infinitesimal perturbations, while inertial effects act to widen the range of unstable wave numbers. The nonlinear evolution of growing interfacial waves consisting of a special pair of normal modes yielding an initially flat interface is analysed numerically by a finite-difference method. The results of the simulations are consistent with the predictions of the linear theory and reveal that the interfacial waves steepen and eventually overturn under the influence of the shear flow.

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Section: Methodssupporting

confidence: 63%

Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

Blyth

Pozrikidis

2004

J Eng Math

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“…structure for all values of IY.R. They also clearly exhibit the phenomenon of "mode crossing" concerning which there has been some controversy recently, but they do not bear directly on the disagreement found by Gallagher and Mercer (1964) between their results for IY. = 2 which did not show any mode crossing and the results of Grohne (1954) which did.…”

Section: Discussionmentioning

confidence: 67%

Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation. Part 1. Plane Couette Flow

Reid

1974

Stud Appl Math

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This paper is concerned with the derivation of asymptotic approximations to the solutions of the Orr‐Sommerfeld equation and its adjoint which are uniformly valid in a full neighborhood of the critical point. In the case of plane Couette flow, it is shown that the expansions can be expressed to all orders in terms of a restricted class of generalized Airy functions and that the slowly varying coefficients in these expansions can be determined from simple recurrence relations. The expansions can easily be related to the Laplace integral representation of the solutions but this is not essential in their derivation. The structure of the theory also suggests how it can be immediately generalized to provide “first approximations” for a more general class of velocity profiles. Detailed results are given in the special case when the wave number α → 0. In this limit, the characteristic equation can be reduced to an especially simple form which permits a convenient global description of the modal structure.

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“…This result is not however of primary importance as we shall explain in the next section. The first notable numerical attack was made by Gallagher & Mercer (1962). They carried out an extensive numerical investigation of the basic twodimensional Orr-Sommerfeld problem for moderate values of a and R such that aR was less than about 1000, and indeed found that the flow was stable for this parameter range.…”

Section: Introductionmentioning

confidence: 99%

On the stability of plane Couette flow to infinitesimal disturbances

Davey

1973

J. Fluid Mech.

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It has been conjectured for many years that plane Couette flow is stable to infinitesimal disturbances although this has never been proved. In this paper we use a, combination of asymptotic analysis and numerical computation to examine the associated Orr-Sommerfeld differential problem in a systematic manner. We obtain new evidence that the conjecture is, in all probability, correct. In particular we find that, at a fixed large value of the Reynolds number R, as in an experiment, if a disturbance of wavenumber α has a damping rate - αci then – ci has a minimum value of order R−½ when α is of order R½. We believe that this result may be an essential prerequisite to an understanding of the stability of plane Couette flow to finite-amplitude disturbances.

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On the behaviour of small disturbances in plane Couette flow

Cited by 23 publications

References 5 publications

Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation. Part 1. Plane Couette Flow

On the stability of plane Couette flow to infinitesimal disturbances

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