Analysis of New Phenomena in Shear Flow of Non-Newtonian Fluids

Malkus, David S.; Nohel, John A.; Plohr, Bradley J.

doi:10.1137/0151044

Cited by 60 publications

(42 citation statements)

References 12 publications

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“…Instead, in a numerical study of shear banding in such a model, the shear stress in the steady banded state depends strongly on the startup history [21][22][23][24], and can have a stress anywhere in the range T 1 < T b < T 2 in figure 1. This conflicts notably with experiment, which consistently reveals a highly reproducible banding stress.…”

Section: Introductionmentioning

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 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: Introductionmentioning

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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“…A mathematical proof for the qualitative behaviour of the fluid remains for further research. Due to the integrodifferential character of our equations, an analogy with the methods used by Malkus et al [2] is not possible. However, we expect that a continued study of e.g.…”

Section: Discussionmentioning

confidence: 99%

“…Since in industrial practice this spurt effect distorts the extrudate by forming a pattern of irregularities at its surface, a good estimate of the critical value of the pressure gradient (or the associated critical stationary volumetric flow rate) beyond which spurt occurs, is of great practical value. The aspect which distinguishes our approach from that of [2], where a differential equation is used, is that we analysed a nonlinear viscoelastic constitutive equation containing a memory integral, leading to a nonlinear integrodifferential equation. Recapitulating our main results we proved by analytical means that…”

Section: Discussionmentioning

confidence: 99%

“…In two papers [1] and [2], Malkus et al analyzed a striking novel phenomenon in shearing flows of non-Newtonian fluids called the 'spurt' phenomenon. They associated spurt with a material property of the polymeric fluid in contrast to the widely accepted explanation of spurt being the failure of the fluid to adhere to the wall ('wall slip').…”

Section: Introductionmentioning

confidence: 99%

“…To describe the elastic response of the dissolved polymer, we have exploited a so-called Kaye-Bernstein-Kearsly-Zapas (KBKZ)-model (see [4, p. 141]) with an extra viscous term, due to the small-molecule solvent. As a generalisation of [2] we consider the three-dimensional axisymmetric shear flow through a pipe (Poiseuille flow). Since in simple shearing the second normal stress difference may be neglected, we use Wagner's modification (see [4, p. 209]) of the KBKZ-model.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Transient behaviour and stability points of the Poiseuille flow of a KBKZ-fluid

Aarts

Ven

1995

J Eng Math

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The Poiseuille flow of a KBKZ-fluid, being a nonlinear viscoelastic model for a polymeric fluid, is studied. The flow starts from rest and especially the transient phase of the flow is considered. It is shown that under certain conditions the steady flow equation has three different equilibrium points. The stability of these points is investigated. It is proved that two points are stable, whereas the remaining one is unstable, leading to several peculiar phenomena such as discontinuities in the velocity gradient near the wall of the pipe ('spurt') and hysteresis. Our theoretical results are confirmed by numerical calculations of the velocity gradient.

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Dissipative Feedback Synthesis for a Singularly Perturbed Model of a Piston Driven Flow of a Non-Newtonian Fluid

Ševčovič

1997

Math. Meth. Appl. Sci.

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The limiting behaviour of solutions of a system of singularly perturbed equations is studied. The goal is to construct a dissipative feedback control synthesis that stabilizes the prescribed output functional along trajectories of solutions. The results are applied to a singularly perturbed Johnson–Sagelman–Oldroyd model of shearing motions of a piston driven flow of a non‐Newtonian fluid. © 1997 by B.G. Teubner Stuttgart‐John Wiley & Sons, Ltd.

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Analysis of New Phenomena in Shear Flow of Non-Newtonian Fluids

Cited by 60 publications

References 12 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

Transient behaviour and stability points of the Poiseuille flow of a KBKZ-fluid

Dissipative Feedback Synthesis for a Singularly Perturbed Model of a Piston Driven Flow of a Non-Newtonian Fluid

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