Journal ArticleDOI
A non-linear instability theory for a wave system in plane Poiseuille flow
Keith Stewartson,J. T. Stuart +1 more
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In this paper, the initial value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given for values of the Reynolds number slightly greater than the critical value, above which perturbation may grow.Abstract:
The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are et and e½(x+a1rt), where t is the time, x the distance in the direction of flow, e the growth rate of linearized theory and (−a1r) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory.read more
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Pattern formation outside of equilibrium
Michael Cross,P. C. Hohenberg +1 more
TL;DR: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented in this article, with emphasis on comparisons between theory and quantitative experiments, and a classification of patterns in terms of the characteristic wave vector q 0 and frequency ω 0 of the instability.
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On three-dimensional packets of surface waves
A. Davey,Keith Stewartson +1 more
TL;DR: In this article, the authors used the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wavepacket of wavenumber k on water of finite depth.
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Nonlinear Modulation of Gravity Waves
Hidenori Hasimoto,Hiroaki Ono +1 more
TL;DR: In this paper, a nonlinear plane wave solution to this equation is found to correspond to the so-called Stokes wave, and the linear stability of this plane wave is essentially determined by the sign of the product of two coefficients in this equation, yielding Benjamin and Whitham's criterion.
Journal ArticleDOI
Energy growth of three-dimensional disturbances in plane poiseuille flow
TL;DR: In this article, the authors considered the development of a small 3D disturbance in plane Poiseuille flow and derived the response from individual (and damped) Orr-Sommerfeld modes.
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Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation.
TL;DR: In this paper, the interaction of slightly overlapping solitary pulses (SP's) is considered in the cubic nonlinear Schrodinger equation with small pumping and dissipation terms, and in the quintic Ginzburg-Landau equation, where small perturbing terms render the asymptotic wave form of a SP spatially oscillating.
References
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Finite bandwidth, finite amplitude convection
Alan C. Newell,John Whitehead +1 more
TL;DR: In this paper, a continuous finite bandwidth of modes can be incorporated into the description of post-critical Rayleigh-Benard convection by the use of slowly varying (in space and time) amplitudes.
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Transition in circular couette flow
TL;DR: In this article, two distinct kinds of transition have been identified in Couette flow between rotating cylinders: the Taylor motion (periodic in the axial direction) and a pattern of travelling waves in the circumferential direction.
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On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow
TL;DR: In this paper, the authors considered the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small.
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A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability
TL;DR: In this paper, the frequency and amplification rates for a disturbance growing with respect to time are compared with those of a spatially growing wave having the same wave number, and it is shown that the frequencies are equal to a high order of approximation.