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Flow birefringence of polymer melts: Calculation of velocity and temperature profiles in a cone-and-plate apparatus

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Summary

In this paper an analysis is given of the shear flow which occurs in a closed cone-and-plate rheometer, as is used for the measurement of the flow birefringence of polymer melts. The influences of shear rate and temperature on the fluid viscosity are taken into account by the use of a relation which combines the concept of the inelastic “power law fluid” with that of an exponential temperature dependence. Velocity and temperature profiles are calculated. Among other things it is shown, how viscous dissipation can disturb the flow field.

Zusammenfassung

In der vorliegenden Arbeit wird die Scherströmung analysiert, die in einem zur Messung der Strömungsdoppelbrechung polymerer Schmelzen dienenden geschlossenen Kegel-Platte-Rheometer auftritt. Die Abhängigkeit der Viskosität von der Schergeschwindigkeit und der Temperatur wird mit Hilfe einer Beziehung berücksichtigt, die das Modell der unelastischen Ostwald-De Waele-Flüssigkeit mit einer exponentiellen Temperatur-Abhängigkeit verbindet. Geschwindigkeits-und Temperaturprofile werden berechnet. Unter anderem wird gezeigt, wie die viskose Dissipation das Geschwindigkeitsfeld stören kann.

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Abbreviations

1:

unity tensor

b :

parameter describing the temperature dependence of viscosity

C p :

heat capacity at constant pressure

D, D ij :

rate of strain tensor

D/Dt :

substantial derivative

E a :

apparent flow activation energy

g :

gravitational acceleration

Gr:

Griffith number

I 1 :

first invariant of the rate of strain tensor

I 2 :

second invariant of the rate of strain tensor

I 3 :

third invariant of the rate of strain tensor

k :

thermal conductivity

n φ :

unit vector tangential to the linesψ andr = constant

n ψ :

unit vector tangential to the linesφ andr = constant

n r :

unit vector in the direction of the linesφ andψ = constant

p :

undetermined hydrostatic pressure

p :

stress tensor

q n :

nominal shear rate

q φψ, qφr :

velocity gradients

r :

radius

R :

gas constant

R i :

inner radius of the ringshaped test channel

R 0 :

outer radius of the ringshaped test channel

t :

time

T :

temperature

T R :

reference temperature

ΔT :

temperature differenceT− TR

V :

mass average velocity

V r, Vθ, Vφ :

components of velocity vector

α :

dimensionless parameter describing the shear rate dependence of the viscosity

ε :

normalized gap angleψ/ψ m

η 1 :

constant in power law

θ :

spherical coordinate, co-latitude

ρ :

dimensionless radiusr/R 0

ρ 0 :

mass per unit of volume

φ :

spherical coordinate, azimuthal angle

Φ :

dimensionless temperature

ψ :

angleψ = π/2 − θ (latitude)

ψ :

gap angle

ω :

angular velocity

ω r :

angular velocity of rotating plate

Ω :

dimensionless angular velocityω/ω r

∇ :

the nabla operator

References

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Author information

Authors and Affiliations

  1. Laboratory of Physical Chemistry, Delft University of Technology, Julianalaan 136, Delft 8, The Netherlands

    F. H. Gortemaker, B. de Cindio & H. Janeschitz-Kriegl

  2. Istituto di Principi di Ingegneria Chimica, University of Naples, Italy

    B. de Cindio

Authors
  1. F. H. Gortemaker
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  2. B. de Cindio
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  3. H. Janeschitz-Kriegl
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Additional information

The present investigation has been carried out under the auspices of the Netherlands Organization for the Advancement of Pure Research (Z. W. O.).

On leave from University of Naples; Research Fellow, Delft University of Technology.

With 7 figures

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Gortemaker, F.H., de Cindio, B. & Janeschitz-Kriegl, H. Flow birefringence of polymer melts: Calculation of velocity and temperature profiles in a cone-and-plate apparatus. Rheol Acta 15, 479–486 (1976). https://doi.org/10.1007/BF01530350

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  • DOI: https://doi.org/10.1007/BF01530350

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