Stability analysis of constitutive equations for polymer melts in viscometric flows

Grillet, Anne; Bogaerds, A.C.B.; Peters, Gwm Gerrit; Baaijens, F.P.T.

doi:10.1016/s0377-0257(02)00005-8

Cited by 55 publications

(61 citation statements)

References 41 publications

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“…Since the base state solution does not vary along streamlines, the classical approach involves expanding the disturbances into spatially periodic perturbations along these streamlines and the resulting 1D Generalized Eigenvalue problem (GEVP) can be solved (see e.g. Renardy and Renardy [1986], Ganpule and Khomami [1999], Grillet et al [2002]). A second approach involves tracking of the perturbed flow in time for a periodic flow domain of length L (figure 1) using, for instance, a Finite Element Method (FEM) [Brown et al, 1993, Grillet et al, 2002.…”

Section: Methodsmentioning

confidence: 99%

“…Renardy and Renardy [1986], Ganpule and Khomami [1999], Grillet et al [2002]). A second approach involves tracking of the perturbed flow in time for a periodic flow domain of length L (figure 1) using, for instance, a Finite Element Method (FEM) [Brown et al, 1993, Grillet et al, 2002.…”

Section: Methodsmentioning

confidence: 99%

“…In order to study the mechanism or the critical conditions for the onset of a viscoelastic instability, the extra stress should be related to the kinematics of the flow. The choice of this constitutive relation will have a major impact on the results of the stability analysis [Grillet et al, 2002]. Motivated by the excellent quantitative agreement of the Pom-Pom constitutive predictions and dynamical experimental data [Inkson et al, 1999, Graham et al, 2001, Verbeeten et al, 2001, we use the approximate differential form of the XPP model to capture the rheological behavior of the fluid.…”

Section: Problem Definitionmentioning

confidence: 99%

“…Apart from providing accurate predictions for the steady base flows, this model should also be able to capture the essential dynamics of the polymer melt. Grillet et al [2002] showed that this is a nontrivial task since generally accepted models for polymer melts can behave very different in terms of their stability characteristics while their rheological properties are similar. Their analysis proved that simple shear flows of the exponential Phan-Thien Tanner (PTT) model has a limiting Weissenberg number above which a discrete eigenmode becomes unstable.…”

Section: Introductionmentioning

confidence: 99%

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Stability analysis of polymer shear flows using the eXtended Pom–Pom constitutive equations

Bogaerds

Grillet

Peters

et al. 2002

Journal of Non-Newtonian Fluid Mechanics

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The stability of polymer melt shear flows is explored using the recently proposed eXtended Pom-Pom (XPP) model of Verbeeten et al. [2001]. We show that both the planar Couette and the planar Poiseuille flow are stable for 'small' disturbances. From our 1-dimensional eigenvalue analysis and 2-dimensional finite element calculations, excellent agreement is obtained with respect to the rate of slowest decay. For a single mode of the XPP model, the maximum growth of a perturbation is fully controlled by the rightmost part of the continuous eigenspectrum. We show that the local fourth order ordinary differential equation has three regular singular points that appear in the eigenspectrum of the Couette flow. Two of these spectra are branch cuts and discrete modes move in and out of these continuous spectra as the streamwise wavelength is varied. An important issue that is addressed is the error that is contained in the rightmost set of continuous modes. We observe that the XPP model behaves much better as compared to, for example, the Upper Convected Maxwell (UCM) model in terms of approximating the eigenvalues associated with the singular eigenfunctions. We demonstrate that this feature is extended to the 2D computations of a periodic channel which means that the requirements on the spatial grid are much less restrictive, resulting in the possibility to use multi mode simulations to perform realistic stability analysis of polymer melts. Also, it is shown that inclusion of a nonzero second normal stress difference has a strong stabilizing effect on the linear stability of both planar shear flows.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Problem Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability analysis of polymer shear flows using the eXtended Pom–Pom constitutive equations

Bogaerds

Grillet

Peters

et al. 2002

Journal of Non-Newtonian Fluid Mechanics

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“…The only results available are on the linear stability of these flows. 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.…”

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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