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Creeping flow past a sphere of a Reiner-Rivlin fluid

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Summary

A variational principle for incompressible Reiner-Rivlin fluids proposed by Bird has been applied to creeping flow past a sphere of a fluid with constant coefficients of viscosity η and normal-stressη c . The drag coefficientf is found to be given byf = C/Re, whereC =C(η c v ∞/ηR). Available data in the literature for (apparently) Newtonian fluids show no indication of a non-zero normal-stress coefficient.

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Abbreviations

A, B, D, K s ,G p :

constants in general trial function for stream function (23)

C :

function ofRn defined by (28)

d ij :

rate-of-deformation tensor defined by (2)

E, F, G :

constants in trial function for stream function (26)

f :

friction factor defined by (13)

F d :

drag force which fluid exerts on sphere in the positive z direction

J :

variational functional defined by (8)

p :

pressure

r :

spherical coordinate

R :

radius of sphere

Re :

dimensionless Reynolds number defined as (2Rv ∞ρ/η).

Rn :

dimensionless group defined as (η c v ∞/ηR)

t ij :

stress tensor

v i :

velocity vector

v τ v θ :

physical components of the velocity vector in ther and0 directions

v z :

hysical component of the velocity vector in the positivez direction

v ∞ :

magnitude of vz at a large distance from the sphere

V :

used to indicate integration over the volume of the system as in (8)

x :

dimensionless radiusr/R

z :

rectangular coordinate

β:

spherical coordinate defined as π/2 — θ

δ i i :

Kronecker delta = 0(i ≠j) = 1(i =j)

η,η c :

functions of II, III defined by (3)

η0,η1:

constants defined by (6)

θ, ϕ:

spherical coordinates

ρ:

density

τ ij :

viscous portion of the stress tensor

ψ:

stream function defined by (16)

References

  1. Arnold, H. D., Phil. Mag.22 (1911) 755.

    Google Scholar 

  2. Biot, M. A., J. Aero. Sci.24 (1957) 857.

    MATH  MathSciNet  Google Scholar 

  3. Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, John Wiley, New York, 1960.

    Google Scholar 

  4. Bird, R. B., Chem. Engr. Sci.6 (1957) 123.

    Article  Google Scholar 

  5. Bird, R. B., private communication, April, 1960.

  6. Bird, R. B., Physics of Fluids3 (1960) 539.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Coleman, B. D. and W. Noll, Annals N.Y. Acad. Sci.89 (1961) 672.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Foster, R. D., M. S. Thesis, Northwestern University, 1961.

  9. Gupta, S. C, Appl. sci. Res.A 10 (1961) 85.

    Article  MATH  Google Scholar 

  10. Gupta, S. C, Appl. sci. Res.A 10 (1961) 229.

    Article  MATH  Google Scholar 

  11. Hill, R., J. Mech. Phys. Solids5 (1956) 66.

    Article  MATH  ADS  Google Scholar 

  12. Hill, R., Quart. J. Mech. Appl. Math.9 (1956) 313.

    Article  MATH  Google Scholar 

  13. Johnson Jr. M. W., Mathematics Research Center Technical Report 208, University of Wisconsin, November, 1960.

  14. Johnson Jr. M. W., Physics of Fluids3 (1960) 871.

    Article  ADS  Google Scholar 

  15. Lamb, H., Hydrodynamics, Dover, New York, 1945.

    Google Scholar 

  16. Leigh, D. C, Physics of Fluids5 (1962) 501.

    Article  MATH  ADS  Google Scholar 

  17. Liebster, H., Ann. d. Phys.82 (1927) 541.

    Article  Google Scholar 

  18. Markovitz, H., Trans. Soc. Rheol.1 (1957) 37.

    Article  Google Scholar 

  19. Pawlowski, J., Koiloid Z.138 (1954) 6.

    Article  Google Scholar 

  20. Reiner, M., Amer. J. Math.67 (1945) 350.

    Article  MATH  MathSciNet  Google Scholar 

  21. Rivlin, R. S., Nature160 (1947) 611.

    Article  MathSciNet  ADS  Google Scholar 

  22. Rivlin, R. S., Proc. Roy. Soc.A 193 (1948) 260.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Schechter, R. S., A.I. Ch. E. J.7 (1961) 445.

    Google Scholar 

  24. Schiller, L., Handbuch der Experimentalphysik, Band 4, 2 Teil, s. 337, Akademi-sche Verlagsgesellschaft, Leipzig, 1932.

  25. Schmiedel, J., Phys. Z.29 (1928) 593.

    Google Scholar 

  26. Slattery, J. C A.I. Ch. E. J.8 (1962) 663.

    Google Scholar 

  27. Sparrow, E. M., and R. Siegel, Trans. A.S.M.E.,81 (1959) 157.

    Google Scholar 

  28. Stewart, W. E., A. I. Ch. E. J.8 (1962) 425.

    Google Scholar 

  29. Tomita, Y., Bull. J. Soc. Mech. Engrs.2 (1959) 469.

    MathSciNet  Google Scholar 

  30. Truesdell, C., J. Rational Mech. Analysis1 (1952) 125.

    MathSciNet  Google Scholar 

  31. Truesdell, C, Trans. Soc. Rheol.4 (1960) 9.

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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Authors and Affiliations

  1. Department of Chemical Engineering, Northwestern University, Evanston, Illinois, U.S.A.

    Robert D. Foster & John C. Slattery

Authors
  1. Robert D. Foster
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  2. John C. Slattery
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Additional information

Based on M. S. thesis of Robert D. Foster

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Foster, R.D., Slattery, J.C. Creeping flow past a sphere of a Reiner-Rivlin fluid. Appl. sci. Res. 12, 213–222 (1963). https://doi.org/10.1007/BF03184973

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  • DOI: https://doi.org/10.1007/BF03184973

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