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Flow of inelastic and viscoelastic fluids past a sphere

I. Drag coefficient in creeping and boundary-layer flows

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Summary

This is essentially an engineering study undertaken with a view to providing drag coefficient correlations for the motion of a solid sphere in inelastic and viscoelastic fluids over a wide range of Reynolds numbers. An approximate closed form analytical solution has been obtained for the motion of a solid sphere in powerlaw fluids and this solution has been verified experimentally. The high Reynolds number flow was analysed theoretically using boundary layer theory; the results were used to separate form and skin friction. Viscoelastic fluids showed “drag reduction” at high Reynolds number. About 300 experimental data points were used to obtain drag coefficient correlations over a wide range of material and flow parameters.

Zusammenfassung

Dies stellt im wesentlichen eine ingenieurwissenschaftliche Untersuchung dar mit dem Ziel, für die Bewegung einer festen Kugel in unelastischen und viskoelastischen Flüssigkeiten in einem weiten Reynolds-Zahl-Bereich Korrelationen für den Widerstandsbeiwert zu ermitteln. Eine Näherungslösung für die Bewegung einer Kugel in der Potenz-Gesetz-Flüssigkeit wird in geschlossener analytischer Form erhalten und experimentell verifiziert. Die Strömung für hohe Reynolds-Zahlen wird theoretisch mit Hilfe der Grenzschicht-Theorie analysiert, die Ergebnisse werden dazu verwendet, Form- und Wandreibung voneinander zu trennen. Viskoelastische Flüssigkeiten ergeben bei hohen Reynolds-Zahlen „Widerstandsverminderung“. Für die Erstellung der Widerstandsbeiwert-Korrelationen werden ungefähr 300 Meßwerte aus einem weiten Bereich der Stoff- und Fließparameter verwendet.

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Abbreviations

A :

Material parameter used in eqs. [65] and [66],\(\frac{{dynes sec^b }}{{cm^2 }}\)

a :

Radius of sphere used in eqs. [30] and [31]

a′,b′,c′,d′,e′ :

Constants used in eq. [40]

b :

Material parameter used in eqs. [65] and [66], dimensionless

C D :

Drag coefficient

CDb.L :

Drag coefficient for boundary layer flow

C f.L :

Local (skin) drag coefficient

CDVE :

Drag coefficient for viscoelastic liquid

C D INEL :

Drag coefficient for viscoinelastic liquid

C f :

Skin drag

C f.ar :

Average (skin) drag coefficient

E :

Differential operator

F :

Function used in eq. [33]

f 1, f2, f3 :

Functions used in eqs. [51] and [61]

i, j :

Integers 1, 2, 3

K :

Consistency index,\(\frac{{dynes sec^n }}{{cm^2 }}\)

n :

Flow behaviour index

P :

Dimensionless pressure defined in eq. [5]

p :

Pressure

r′ :

Dimensionless radius used in eq. [34]

r :

Radial distance from the sphere centre

R :

Sphere radius

Reow :

Reynolds number\(\left( {\frac{{\left( {2R} \right)^n U_\infty ^{2 - n} \rho }}{K}} \right)\)

U f :

Velocity at the edge of the boundary layer around the sphere

U ∞ :

Characteristic velocity

u, v :

Velocity components alongx andy respectively

X n :

Function ofn (1)

x ′n :

Function ofn (1)

Wi:

Weissenberg number

α, β :

Constants used in eq. [67]

δ * :

Displacement thickness

η :

y/δ

\(\dot \gamma \) :

Effective shear rate\(\left( { = \frac{U}{{2R}}} \right)\)

ξ :

Dimensionless radial distance

ρ :

Density

Π :

Second invariant of rate of deformation tensors

τ 0 :

Local shearing stress on the body

τ 12 :

Shear stress

τ 11 − τ22 :

Primary normal stress difference

τ, Δ :

Extra stress tensor

θ :

Momentum thickness

θ, φ :

Spherical co-ordinates

ψ :

Stream function

*:

Dimensionless quantity with respect to some characteristic dimension

References

  1. Astarita, G., G. Apuzzo, G. Marrucci AIChE J.16, 538 (1970).

    Google Scholar 

  2. Adachi, K., N. Yoshioka, K. Yamamoto Chem. Eng. Sci.28, 2033 (1975).

    Google Scholar 

  3. Bizzell, G. D., J. C. Slattery Chem. Eng. Sci.17, 777 (1962).

    Google Scholar 

  4. Broadbent, J. M., B. Mena Chem. Eng. J.8, 11 (1974).

    Google Scholar 

  5. Caswell, B., W. H. Schwarz J. Fluid Mech.13, 417 (1962).

    Google Scholar 

  6. Dallon, D. S., Ph. D. Thesis, Univ. of Utah (1967).

  7. Fararoui, A., R. C. Kintner Trans. Soc. Rheology5, 369 (1961).

    Google Scholar 

  8. Giesekus, H. Rheol. Acta1, 404 (1961);3, 59 (1963).

    Google Scholar 

  9. Hirose, T., M. Moo-Young Can. J. Chem. Eng.47, 265 (1969).

    Google Scholar 

  10. Hopke, S. W., J. C. Slattery AIChE J.16, 317 (1970).

    Google Scholar 

  11. Jones, J. R., M. L. Lewis ZAMP19, 746 (1968).

    Google Scholar 

  12. Kale, D. D., Ph. D. Thesis, University of Salford (1973).

  13. Kale, D. D., R. A. Mashelkar, J. Ulbrecht Rheol. Acta14, 631 (1975).

    Google Scholar 

  14. Kato, H., M. Tachibana, K. Oikawa Bull. JSME15, 1556 (1972).

    Google Scholar 

  15. Kelkar, J. V., R. A. Mashelkar, J. Ulbrecht Trans. Instn. Chem. Engrs.50, 343 (1972).

    Google Scholar 

  16. Kelkar, J. V., R. A. Mashelkar, J. Ulbrecht J. Appl. Poly. Sci.17, 3069 (1973).

    Google Scholar 

  17. Lang, T. A., H. V. L. Patrick, ASME Winter Annual Meeting (New York 1966).

  18. Leslie, F. N., R. I. Tanner Quart. J. Mech. App. Math.14, 36 (1961).

    Google Scholar 

  19. Luikov, A. V., Progress in Heat and Mass Transfer, Vol.4, p. 23 (New York 1971).

  20. Mashelkar, R. A., G. V. Devarajan Trans. Instn. Chem. Engrs.54, 108 (1976).

    Google Scholar 

  21. Nakano, Y., C. Tien AIChE J.14, 145 (1968).

    Google Scholar 

  22. Nakano, Y., C. Tien AIChE J.16, 569 (1970).

    Google Scholar 

  23. Rajeswari, G. K., S. L. Rathna ZAMP13, 43 (1962).

    Google Scholar 

  24. Ruszczycky, M. A. Nature206, 614 (1965).

    Google Scholar 

  25. Sanders, J. V. Intern. Shipbuilding Prog.14, 140 (1967).

    Google Scholar 

  26. Slattery, J. C., R. B. Bird Chem. Eng. Sci.16, 231 (1961).

    Google Scholar 

  27. Srivastava, A. C., M. K. Maiti Phys. Fluids9, 462 (1966).

    Google Scholar 

  28. Tomita, Y. Bull. Jap. Soc. Mech. Engrs.2, 469 (1959).

    Google Scholar 

  29. Turian, R. M. AIChE J.13, 999 (1967).

    Google Scholar 

  30. Ultman, J. S., M. M. Denn Chem. Eng. J.2, 81 (1971).

    Google Scholar 

  31. White, A. Nature211, 1390 (1966).

    Google Scholar 

  32. White, A. Nature212, 277 (1966).

    Google Scholar 

  33. Wasserman, M. L., J. C. Slattery AIChE J.10, 383 (1964).

    Google Scholar 

  34. Wallick, G. C., J. G. Savins, D. R. Arterburn Phys. Fluids5, 367 (1962).

    Google Scholar 

  35. Yoshioka, N., R. Nakamura Kogaku Kagaku4, 130 (1966).

    Google Scholar 

  36. Zana, E., G. Tiefenbruck, L. G. Leal Rheol. Acta14, 891 (1975).

    Google Scholar 

  37. Ziegenhagen, A. J., R. B. Bird, M. W. Johnson Trans. Soc. Rheol.5, 47 (1961).

    Google Scholar 

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Authors and Affiliations

  1. Department of Chemical Engineering, University of Salford, M 5 4 WT, Salford, England

    A. Acharya, R. A. Mashelkar & J. Ulbrecht

Authors
  1. A. Acharya
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  2. R. A. Mashelkar
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  3. J. Ulbrecht
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With 13 figures and 7 tables

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Acharya, A., Mashelkar, R.A. & Ulbrecht, J. Flow of inelastic and viscoelastic fluids past a sphere. Rheol Acta 15, 454–470 (1976). https://doi.org/10.1007/BF01530348

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  • DOI: https://doi.org/10.1007/BF01530348

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