Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

On Swell ofviscoelastic fluids

Luca Brandt - MekanikChristophe Duwig - HPT

KTH

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Outline■ What is die swell ?

■ What does delayed die swell mean ?■ The important parameters.

■ Numerical simulations, deeper viewof the phenomena.

■ Understanding through most recentresults.

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Where can we notice dieswell effects ?

■ Extrusion of polymers (moltenplastics)

■ For any viscoelastic extrudedsolution (Cloitre, 1998), no changeof structure !

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Description of die swell

■ Swell of the jet

■ Lower rate of extrusion■ Effect of elasticity when the shear

stops: N1 is high, elastic recovery

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

What is delayed dieswell ?

■ Observed by increasing theextrusion velocity

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Important parameters

■ Weissenberg number :

■ Reynolds number :■ Viscoelastic Mach number:

■ Elasticity number:

en

en

d

VWe

λ=

ηρ enendV=Re

Wec

VM en ⋅== Re

Re/2

Wed

Een

==ρλη

dex

denVen

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Speed of shear wavesinto a fluid at rest

λρη

ρηλρ

σρηλσσ

=

=⋅+⋅−⋅

=

=⋅

=+

c so

0

: toLead

t)u(y,u :shear Pure

u :equation Momentum

:model Maxwell

jij,ti,

,ij

tyytt

ijtij

uuu

d

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Delayed die swellExperiments by Joseph, Matta & Chen (1987)

■ General phenomenon

■ Critical phenomenon■ Steady or unsteady (large

relaxation times λ)■ Less apparent: low λ, low η

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Critical conditions■ UC, D/d, L/d decrease with pipe

diameter■ Critical Mach number MC >1 at exit■ After the swell ML

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Post critical conditions■ L increases with u■ D/d increases and then decreases■ ML increases with u (less than 1)

■ ML, D/d, (L-L)/d decrease with d,strength of shock is greater forsmall d

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

High rates of extrusion■ Unsteady flows for large time of relaxation

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

High rates of extrusion■ Smoothing

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Effect of gravity■ Control of jet shape

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Sketch of delayed die swell

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Numerical methods forsimulations

(Delvaux and Crochet, 1990)■ Upper convected Maxwell fluid +

small viscous component to makethe numerical solution easier:Oldroy B fluid.

■ Calculations done with low valuesof E (stability !) ie no experimentalvalidation !

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Numerical methods forsimulations

(Delvaux and Crochet, 1990)

■ Investigate the change of typewhich is related to : T* definite positive = elliptic

vv* ⋅−+= ρλησ IT

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Meshes used■ Important to avoid artificial

(numeric) viscosity:■ Swell deform the fluid elements:

inclined ray to preserve a bettershape of the elements.

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Results : Influence ofthe retardation time

■ Swelling ratio decrease with λ■ Shape of the free surface is

insensitive: no effects in delay !

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Results : Influence ofinertia

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Results : Influence ofelasticity

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Results : Supercriticalregions - vorticity

contour lines■ Oldroyd B, E=0,03, M =2,3 to 5,9

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Results : Supercriticalregions - vorticity

contour lines■ Oldroyd B, E=0,03, M =2,3 to 5,9

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Giesekus fluid

ijtij,ijij GI :modelEquation =+

+ λσσσ

ηαλ

■ Non linear contribution in theMaxwell Model:

■ Influence on the results:

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Importance of den■ Experiments done with different

fluids (E changes)

Luca Brandt - Christophe Duwig, Non newtonian fluids 1999

Conclusion■ Delayed die swell may occur in any

viscoelastic solution■ It is a critical phenomenon■ Numerics show that inertia induces

a change from subcritical tosupercritical conditions

■ Critical Mach number depends onthe diameter, Mcr increases whenden decreases.

■ Contradictions for the effects of E.