Stability of plane poiseuille flow of a highly elastic liquid

Ho, Teh C.; Denn, Morton M.

doi:10.1016/0377-0257(77)80048-7

Cited by 120 publications

(94 citation statements)

References 14 publications

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“…The scheme used is a shooting method, in which a value of ω is guessed and the Newton-Raphson method is used to find the true value. The equations are integrated inwards from each wall and the jump conditions at the interface provide the dispersion relation through a determinant condition as introduced by Ho & Denn [42].…”

Section: Numerical Resultsmentioning

confidence: 99%

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Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson–Segalman fluids

Wilson

Fielding

2006

Journal of Non-Newtonian Fluid Mechanics

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We consider the linear stability of shear banded planar Couette flow of the JohnsonSegalman fluid, with and without the addition of stress diffusion to regularise the equations. In particular, we investigate the linear stability of an initially onedimensional "base" flow, with a flat interface between the bands, to two-dimensional perturbations representing undulations along the interface. We demonstrate analytically that, for the linear stability problem, the limit in which diffusion tends to zero is mathematically equivalent to a pure (non-diffusive) Johnson-Segalman model with a material interface between the shear bands, provided the wavelength of perturbations being considered is long relative to the (short) diffusion lengthscale.For no diffusion, we find that the flow is unstable to long waves for almost all arrangements of the two shear bands. In particular, for any set of fluid parameters and shear stress there is some arrangement of shear bands that shows this instability. Typically the stable arrangements of bands are those in which one of the two bands is very thin. Weak diffusion provides a small stabilising effect, rendering extremely long waves marginally stable. However, the basic long-wave instability mechanism is not affected by this, and where there would be instability as wavenumber k → 0 in the absence of diffusion, we observe instability for moderate to long waves even with diffusion. This paper is the first full analytical investigation into an instability first documented in the numerical study of [1]. Authors prior to that work have either happened to choose parameters where long waves are stable or used slightly different constitutive equations and Poiseuille flow, for which the parameters for instability appear to be much more restricted.We identify two driving terms that can cause instability: one, a jump in N 1 , as reported previously by Hinch et al. [2]; and the second, a discontinuity in shear rate. The mechanism for instability from the second of these is not thoroughly understood. Preprint submitted to Elsevier Science 3 May 2006We discuss the relevance of this work to recent experimental observations of complex dynamics seen in shear-banded flows.

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Section: Numerical Resultsmentioning

confidence: 99%

“…s ii , ω = kω, then the governing equations within each fluid, from (37)(38)(39)(40)(41)(42)(43), become, for the force balance:…”

Section: Wall Boundary Conditionsmentioning

confidence: 99%

Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson–Segalman fluids

Wilson

Fielding

2006

Journal of Non-Newtonian Fluid Mechanics

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“…We solve the eigenvalue problem for general n using the numerical method of Ho and Denn [11]. The aim is to follow the behaviour of roots of 3 in the complex plane as n (or k or W) varies.…”

Section: Intermediate Wavelengths: Numerical Resultsmentioning

confidence: 99%

Instability of channel flow of a shear-thinning White–Metzner fluid

Wilson

Rallison

1999

Journal of Non-Newtonian Fluid Mechanics

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We consider the inertialess planar channel flow of a White±Metzner (WM) fluid having a power-law viscosity with exponent n. The case n 1 corresponds to an upper-convected Maxwell (UCM) fluid. We explore the linear stability of such a flow to perturbations of wavelength k À1 . We find numerically that if n < n c % 0.3 there is an instability to disturbances having wavelength comparable with the channel width. For n close to n c , this is the only unstable disturbance. For even smaller n, several unstable modes appear, and very short waves become unstable and have the largest growth rate. If n exceeds n c , all disturbances are linearly stable. We consider asymptotically both the long-wave limit which is stable for all n, and the shortwave limit for which waves grow or decay at a finite rate independent of k for each n.The mechanism of this elastic shear-thinning instability is discussed. #

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“…For essentially all studied visco-elastic models, laminar plane Couette flow is linearly stable (Gorodtsov and Leonov, 1967, Renardy and Renardy, 1986, Renardy, 1992, Wilson et al, 1999 (note the exception (Grillet et al, 2002)). In the case of pipe flow, the linear stability was demonstrated numerically by Ho and Denn (Ho and Denn, 1978) for any value of the Weissenberg and Reynolds numbers. Therefore, it has become common knowledge that the parallel shear flows of fluids obeying simple visco-elastic models (UCM, Oldroyd-B, etc.)…”

Section: Introductionmentioning

confidence: 99%

Subcritical Instabilities in Plane Couette Flow of Visco-Elastic Fluids

Morozov

Saarloos

2005

Fluid Mechanics and Its Applications

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A non-linear stability analysis of plane Couette flow of the Upper-Convected Maxwell model is performed. The amplitude equation describing time-evolution of a finite-size perturbation is derived. It is shown that above the critical Weissenberg number, a perturbation in the form of an eigenfunction of the linearized equations of motion becomes subcritically unstable, and the threshold value for the amplitude of the perturbation decreases as the Weissenberg number increases.

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Stability of plane poiseuille flow of a highly elastic liquid

Cited by 120 publications

References 14 publications

Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson–Segalman fluids

Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson–Segalman fluids

Instability of channel flow of a shear-thinning White–Metzner fluid

Subcritical Instabilities in Plane Couette Flow of Visco-Elastic Fluids

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