During the last two decades, a large number of flow calculations in planar and axisymmetric contractions (also known as slit-die or die entries) have been performed. It is no exaggeration to say that this type of geometry is probably the one most thoroughly investigated in viscoelastic flow calculations. Interest in die entry flow calculations is partly motivated by their relevance in processing procedures such as extrusion and injection molding. Apart from this practical aspect, the presence of a reentrant corner in the die entry geometry has been associated with non-convergence and other numerical difficulties collectively known as the HWNP (high Weissenberg number problem) which have made the contraction flow a popular benchmark problem.
Flows in planar or cylindrical contractions have the additional interest of possessing a strong elongational character in the region of the reentrant corner. Furthermore, the intensity of the elongational component of the flow varies smoothly from zero for the Poiseuille flow at the inlet to a maximum in the region of the die entry itself and then decreases back to zero for the Poiseuille flow downstream.
The importance of rheological characterization of polymer solutions and melts in elongational flows has been repeatedly stressed. It has been suggested that often poor quality of comparisons between simulations and experiments in complex flows can be at least partially attributed to missing or incomplete rheological characterization of the fluid under extensional flows. Unfortunately, measurements of material functions in purely elongational flows are often unreliable or even impossible. It has therefore been suggested that, apart from simple shear flows, complex flows should be used to characterize viscoelastic liquids. Point-wise measured stress and velocity data in a complex flow are used to determine the adequacy of constitutive equations. The geometries (confined cylinder and planar contraction) can therefore be considered as rheometers of sorts in which the quality of both a constitutive equation and its parameters (determined under simple shear flow) conditions is tested.
In spite of the substantial progress achieved in flow calculations, most current numerical methods are inherently unable to offer insights into molecular behavior: in traditional isothermal and incompressible calculations, the fluid is entirely and exclusively characterized by the relationship between stress and strain history expressed as a differential or integral constitutive equation (CE). Even for those CE's which allow a microscopic interpretation in terms of simple molecular models, the averaging procedure required to arrive at a macroscopic expression for the stress erases all microscopic (molecular) information. The primary variables involved in classical flow calculations are only velocity, pressure and stress; there is no way to retrieve any details of the lost molecular picture.
Furthermore, since the emphasis of the present work is on elongational flows, besides the choice of a suitable geometry, it is important to select a fluid with as realistic and non-trivial elongational behavior as possible. The patent inability of CE's like Oldroyd-B at describing realistc elongational behavior prompted us to use a fluid which admits of a molecular interpretation of finite extensibility. While retaining the conceptual simplicity of Hookean dumbbells, a molecular model with finite extensibility will display markedly different and more physically realistic behavior in the extensional part of the channel flow. The use of a FENE fluid, for which no closed-form constitutive equation exists, makes it necessary to resort to non-standard numerical methods (CONNFFESSIT) to solve the viscoelastic flow problem. Approximations and linearizations like the FENE-P and FENE-CR models are clearly less adequate: although they set limits on the value of the maximum average molecular extension, there is nothing in them preventing individual molecules from extending to and arbitrarily large extension. 
for LaTeX users @article{MLaso1999-,
author = {M. Laso and M. Picasso and H. C. \"Ottinger},
title = {Calculation of flows with large elongational components: CONNFFESSIT calculation of the flow of a FENE fluid in a planar 10:1 contraction},
journal = {in: Flexible Polymer Chains in Elongational Flow, Eds. T.Q. Nguyen and H.-H. Kausch},
volume = {},
pages = {pp 101-136},
year = {1999}
}
\bibitem{MLaso1999-} M. Laso, M. Picasso, H.C. \"Ottinger,
Calculation of flows with large elongational components: CONNFFESSIT calculation of the flow of a FENE fluid in a planar 10:1 contraction,
in: Flexible Polymer Chains in Elongational Flow, Eds. T.Q. Nguyen and H.-H. Kausch {\bf} (1999) pp 101-136.MLaso1999-
M. Laso, M. Picasso, H.C. \"Ottinger
Calculation of flows with large elongational components: CONNFFESSIT calculation of the flow of a FENE fluid in a planar 10:1 contraction
in: Flexible Polymer Chains in Elongational Flow, Eds. T.Q. Nguyen and H.-H. Kausch,,1999,pp 101-136 |
mk@mat.ethz.ch
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