Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-20T15:37:20.554Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher-order theory of the Weissenberg effect

Published online by Cambridge University Press:  19 April 2006

J. Yoo ,
D. D. Joseph  and
G. S. Beavers
J. Yoo
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis Present address: Department of Mechanical Engineering, Seoul National University, Korea.
D. D. Joseph
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis
G. S. Beavers
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis
Get access Rights & Permissions [Opens in a new window]

Abstract

The higher-order theory of the Weissenberg effect is developed as a perturbation of the state of rest. The perturbation is given in powers of the angular frequency Ω of the rod and the solution is carried out to O(Ω4). The perturbation induces a slow-motion expansion of the stress into Rivlin-Ericksen tensors in combinations which are completely characterized by five viscoelastic parameters. The effects of each of the material parameters may be computed separately and their overall effect by superposition. The values of the parameters may be determined by measurement of the torque, surface angular velocity and height of climb. Such measurements are reported here for several different sample fluids. Good agreement between the third-order theory and measured values of the velocity is reported. Secondary motions which appear at are computed using biorthogonal series. The analysis predicts the surprising fact that secondary motions run up the climbing bubble against gravity and intuition.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 92 , Issue 3 , 12 June 1979 , pp. 529 - 590
Copyright
© 1979 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beavers, G. S. & Joseph, D. D. 1975 The rotating rod viscometer. J. Fluid Mech. 69, 475–511.Google Scholar
Coleman, B. D. & Markovitz, H. 1964 Normal stress effects in second-order fluids. J. Appl. Phy. 35, 1–9.Google Scholar
Coleman, B. D. & Noll, W. 1960 An approximation theorem for functionals with applications in continuum mechanics. Arch. Rat. Mech. Anal. 6, 355–370.Google Scholar
Giesekus, von H. 1961 Einige Bemerkungen zum Fliessverhalten elasto-viskoser Flüssigkeiten in stationären Schlichtströmungen. Rheol. Acta 1, 404–413.Google Scholar
Hoffman, A. H. & Gottenberg, W. G. 1973 Determination of the material functions for a simple fluid from a study of the climbing effect. Trans. Soc. Rheol. 17, 465–485.Google Scholar
Joseph, D. D. 1974 Slow motion and viscometric motion: stability and bifurcation of the rest state of a simple fluid. Arch. Rat. Mech. Anal. 56, 99–157.Google Scholar
Joseph, D. D. 1977 The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part 1. SIAM J. Appl. Math. 33, 337–347.Google Scholar
Joseph, D. D., Beavers, G. S. & Fosdick, R. L. 1973 The free surface on a liquid between cylinders rotating at different speeds. Part 2. Arch. Rat. Mech. Anal. 49, 381–401.Google Scholar
Joseph, D. D. & Fosdick, R. L. 1973 The free surface on a liquid between cylinders rotating at different speeds. Part 1. Arch. Rat. Mech. Anal. 49, 321–380.Google Scholar
Langlois, W. E. & Rivlin, R. S. 1963 Slow steady-state flow of visco-elastic fluids through non-circular tubes. Rendiconti di Matematica 22, 169–185.Google Scholar
Peter, S. & Noetzel, W. 1959 Über einige Experimente zur Frage der Existenz des Weissenberg-Effektes. Ann. Physik. Chem. 21, 422–431.Google Scholar
Sattinger, D. H. 1976 On the free surface of a viscous fluid motion. Proc. Roy. Soc. A 349, 183–204.Google Scholar
Saville, D. A. & Thompson, D. W. 1969 Secondary flows associated with the Weissenberg effect. Nature 223, 391–392.Google Scholar
Truesdell, C. 1974 The meaning of viscometry in fluid dynamics. Ann. Rev. Fluid Mech. 6, 111–146.Google Scholar
Truesdell, C. & Noll, W. 1965 The non-linear field theories of mechanics. Handbuch der Physik, vol. III/3. Springer.
Yoo, J. Y. 1977 Fluid motion between two cylinders rotating at different speeds. Ph.D. thesis, University of Minnesota.
Yoo, J. Y. & Joseph, D. D. 1978 Stokes flow in a trench between concentric cylinders. SIAM J. Appl. Math. 34, 247–285.Google Scholar