21 November 200721 November 2007 Phys. Sc. & Engin. Grad School, CardiffPhys. Sc. & Engin. Grad School, Cardiff

VISCOELASTIC FLUIDS: VISCOELASTIC FLUIDS: A BRIEF DESCRIPTION AND A BRIEF DESCRIPTION AND

SOME MAIN FEATURESSOME MAIN FEATURES

EXTRUDATE SWELLEXTRUDATE SWELL A viscoelastic fluid is extruded from a pipe. The stress A viscoelastic fluid is extruded from a pipe. The stress gradient at the die is responsible for the extensional gradient at the die is responsible for the extensional swelling. swelling. Capturing this gradient, as well as Capturing this gradient, as well as the free surface’s behaviour, is the free surface’s behaviour, is crucial. Extrusion processes in food crucial. Extrusion processes in food and manufacturing industries are and manufacturing industries are the main industrial applicationsthe main industrial applications

FILAMENT STRETCHINGFILAMENT STRETCHING A viscoelastic fluid is confined A viscoelastic fluid is confined between two plates. When these platesbetween two plates. When these plates are pulled apart, the fluid stretches. are pulled apart, the fluid stretches. Tracking the free surface and describing Tracking the free surface and describing the necking effect at the centre are the necking effect at the centre are the main challenges. the main challenges. This phenomenon is common in fibre spinning and industrial This phenomenon is common in fibre spinning and industrial processes involving thin films. processes involving thin films.

Once the velocity has been computed at all the Once the velocity has been computed at all the internal nodes, then the values on the free surfaces internal nodes, then the values on the free surfaces are extrapolated along the vertical gridlinesare extrapolated along the vertical gridlines. The node . The node is then relocated according to the new velocity by is then relocated according to the new velocity by mean of an ALE (Arbitrary Lagrangian mean of an ALE (Arbitrary Lagrangian Eulerian) approach to obtain the correspondingEulerian) approach to obtain the correspondingboundary conditions. boundary conditions. None of the None of the surrounding air nodes (surrounding air nodes (drydry nodes) nodes)is involved; that’s the reason is involved; that’s the reason for this approach beingfor this approach beingcalled “wet”.called “wet”.

The differential equations above have to be inserted in The differential equations above have to be inserted in a computer code to obtain simulations. This is done by a computer code to obtain simulations. This is done by mean of the Numerical Analysis; among different mean of the Numerical Analysis; among different choices we adopt Spectral Element Methods. First the choices we adopt Spectral Element Methods. First the domain is divided into (Spectral) elements.domain is divided into (Spectral) elements.

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Numerical Simulation of Free Surface Flow of Viscoelastic FluidsNumerical Simulation of Free Surface Flow of Viscoelastic FluidsGiancarlo Russo, Cardiff School of Mathematics (Giancarlo Russo, Cardiff School of Mathematics (supervised bysupervised by Prof. Tim Phillips) Prof. Tim Phillips)

Modelling the flow: the OLDROYD – B equation

(3) ( ( ) ( )) (1 )TWe u u u dt

The dynamics of the problems describedThe dynamics of the problems describedis modelled by mean of the differential is modelled by mean of the differential conservation laws for mass (2) and momentum (1); conservation laws for mass (2) and momentum (1); moreover, the constitutive equation is required to moreover, the constitutive equation is required to close the system. For Newtonian fluids this relation close the system. For Newtonian fluids this relation is linear, in our case the Oldroyd-B constitutive is linear, in our case the Oldroyd-B constitutive model (3) will describe a polymer solution.model (3) will describe a polymer solution.

From the continuous to the discrete: the Spectral Element Methods

On each spectral element a particular grid is On each spectral element a particular grid is then used to allow high order polynomials as then used to allow high order polynomials as basis functions. Such polynomials are shown in basis functions. Such polynomials are shown in the pictures above in 1-D and 2-D. the pictures above in 1-D and 2-D.

1-D expansion 2-D expansion

Flow past a Flow past a cylinder in a planar channel: contours plotcylinder in a planar channel: contours plot

Tracking the free surfaceTracking the free surface

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2My research project: two problems of industrial relevanceMy research project: two problems of industrial relevance

VIscoelastic fluids are the most abundant class of VIscoelastic fluids are the most abundant class of fluids on earth. In fact, all the fluids in nature are fluids on earth. In fact, all the fluids in nature are viscoelastic, even though some of them like water are viscoelastic, even though some of them like water are very well approximated by the (simple, linear) very well approximated by the (simple, linear) Newtonian law of viscosity. Newtonian law of viscosity. The main characteristic of The main characteristic of viscoelastic matter is a partialviscoelastic matter is a partialelastic recovery after aelastic recovery after astrain has been applied; this strain has been applied; this can sometimes result in not can sometimes result in not so obvious effects like theso obvious effects like therod-climbing shown here on rod-climbing shown here on the right. Since the polymers, which are highly elastic, the right. Since the polymers, which are highly elastic, became the most used fluids in industrial processes, became the most used fluids in industrial processes, the request for highly accurate simulations of the request for highly accurate simulations of viscoelastic flows grew dramatically.viscoelastic flows grew dramatically.