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Inertia effects in rheometrical flow systems Part 2: The balance rheometer

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Summary

The flow field of a linear viscoelastic fluid in the balance rheometer, taking fluid inertia into account, has been studied theoretically and an exact solution is given. The flow field of a Newtonian fluid is included in this solution as a special case. The forces and couples on the hemispheres are readily deduced from this solution.

Zusammenfassung

Das Strömungsfeld einer linear-viskoelastischen Flüssigkeit im Képès-Rheometer wird unter Berücksichtigung der Flüssigkeitsträgheit theoretisch untersucht und eine exakte Lösung angegeben. Das Strömungsfeld einer newtonschen Flüssigkeit ergibt sich als ein Sonderfall dieser Lösung. Die auf die Halbkugeln ausgeübten Kräfte und Drehmomente lassen sich in einfacher Weise aus der Lösung ableiten.

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Abbreviations

a :

distance between rotation axis of orthogonal rheometer

b :

body force

b r ,b θ ,b φ :

physical components of the body force

c :

velocity of propagation of shear waves in a Hookean material

c * =c′ + ic″ :

complex velocity of propagation of shear waves in a linear viscoelastic fluid

e :

dilatation

h :

gap distance between hemispheres of balance rheometer resp. between plates of orthogonal rheometer

i :

\(\sqrt 1\)

j n (x):

spherical Bessel-function of the first kind and ordern

k * =k′ − ik″ :

complex wave number

k :

-(ρω/2η)1/2

k e :

wave number in a Hookean material

n :

non-negative integer

r :

spherical polar coordinate

r 0 :

radius of plates of orthogonal rheometer

r 1 :

radius of inner hemisphere of balance rheometer

r 2 :

radius of outer hemisphere of balance rheometer

t :

time

u r ,u θ ,u φ :

physical components of displacement

v r ,v θ ,v φ :

physical components of velocity

x, y, z :

cartesian coordinates

y n (z):

spherical Bessel-function of the second kind and ordern

A, B, C :

integration constants

F :

force

G * =G′ + iG″ :

complex shear modulus

G :

shear modulus of a Hookean material

J 1/2(z):

Bessel-function of the first kind and order 1/2

M :

couple

P µ v (x):

associated Legendre-function (spherical harmonic) of the first kind, degreev and orderµ

P n (x):

Legendre polynomial of degreen

Q µ v (x):

associated Legendre-function (spherical harmonic) of the second kind, degreev and orderµ

S :

strain

Y 1/2(z):

Bessel-function of the second kind and order 1/2

γ :

r 32 /(r 32 −r 31 )

δ :

loss angle

ε :

angle between rotation axis of hemispheres

η :

viscosity

θ :

spherical polar coordinate

λ :

wave length

Λ :

second Lamé-function

ρ :

density

τ :

shear stress

φ :

spherical polar coordinate

ω :

angular velocity

ω r ,ω θ ,ω φ :

physical components of the spin tensor

References

  1. Waterman, H. A. Rheol. Acta15, 444 (1976).

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  2. Walters, K. J. Fluid Mech.40, 191 (1970).

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  3. Bowen, C. W., Phys. D. Thesis, University of Wales (1973).

  4. Landau, L. D., E. M. Lifshitz, Fluid Mechanics, Vol. 6 of Course of Theoretical Physics, p. 68 (1963).

  5. Abbott, T. N. G., G. W. Bowen, K. Walters J. Phys. D.4, 190 (1971).

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  6. Abramovitz, M., J. A. Stegun, Handbook of Mathematical Functions (New York).

  7. Képès, A., unpublished. Manufactured commercially by Contraves, A. G., Switzerland.

  8. Rheometrics, Inc., Union, N. J., For a detailed description of this instrument see e.g.Macosko, C. W., Phys. D. Thesis, Dept. Chem. Eng., Princeton University (1970).

  9. Waterman, H. A. J. Phys. D.3, 290 (1970).

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  10. Abbott, T. N. G., K. Walters J. Fluid Mech.40, 205 (1970).

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Authors and Affiliations

  1. Department of Applied Physics, Twente University of Technology, Enschede, The Netherlands

    H. A. Waterman

Authors
  1. H. A. Waterman
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With 4 figures

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Waterman, H.A. Inertia effects in rheometrical flow systems Part 2: The balance rheometer. Rheol Acta 15, 612–622 (1976). https://doi.org/10.1007/BF01524747

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  • DOI: https://doi.org/10.1007/BF01524747

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