Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-18T17:07:21.698Z Has data issue: false hasContentIssue false
  • English
  • Français

Interfacial shapes between two superimposed rotating simple fluids

Published online by Cambridge University Press:  20 April 2006

H. A. Tieu ,
D. D. Joseph  and
G. S. Beavers
H. A. Tieu
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota 55455 Present address: Goodyear Tire and Rubber Company, Akron, Ohio.
D. D. Joseph
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota 55455
G. S. Beavers
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota 55455
Get access Rights & Permissions [Opens in a new window]

Abstract

The interfacial shape of two immiscible simple fluids in a vertical cylinder which oscillates about its axis is investigated using the theory of domain perturbations. The perturbation stresses are expressed by integrals over the history of the deformation. At first order the azimuthal velocity field satisfies the requirements of continuity in velocity and shear stresses across the interface. At second order the solution consists of a mean part and a time-periodic part varying at twice the frequency of the cylinder. The mean problem is inverted for the mean secondary flow, pressure and interfacial shape. Experimental data for two polymeric oils (TLA227 and STP) show qualitative agreement with theoretical predictions for the mean interfacial shapes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 145 , August 1984 , pp. 11 - 70
Copyright
© 1984 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

Abramowitz, M. & Stegun, I. A. 1970 Handbook of Mathematical Functions. Dover.
Beavers, G. S. & Joseph, D. D. 1975 The rotating rod viscometer. J. Fluid Mech. 69, 475.Google Scholar
Beavers, G. S. & Joseph, D. D. 1977 Novel Weissenberg effects. J. Fluid Mech. 81, 265.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1977 Dynamics of Polymeric Liquids, vol. I. Wiley.
Joseph, D. D. 1976 Stability of Fluid Motions, vol. II. Springer.
Joseph, D. D. 1977 A new separation of variables theory for problems of Stokes flow and elasticity. In Proc. Symp. on Trends in Applications of Pure Mathematics to Mechanics, Kozubnik, Poland.
Joseph, D. D. & Beavers, G. S. 1977 Free surface problems in rheological fluid mechanics. Rheol. Acta 16, 169.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 II. Arch. Rat. Mech. Anal. 49, 381.Google Scholar
Joseph, D. D. & Fosdick, R. L. 1973 The free surface on a liquid between cylinders rotating at different speeds. Part I. Arch. Rat. Mech. Anal. 49, 321.Google Scholar
Kantorovich, L. V. & Krylov, V. I. 1958 Approximate Methods of Higher Analysis. Interscience.
Kolpin, B. E. D., Beavers, G. S. & Joseph, D. D. 1980 Free surface on a simple fluid between cylinders undergoing torsional oscillations. IV. Oscillating rods. J. Rheol. 24, 719.Google Scholar
Tieu, A. H. 1983 Second-order effects in complex rheological flows. Ph.D. thesis, University of Minnesota.
Trogdon, S. A. & Joseph, D. D. 1980 The stick-slip problem for a round jet. I. Large surface tension. Rheol. Acta 19, 404.Google Scholar
Watson, G. N. 1958 A Treatise on the Theory of Bessel Functions. Cambridge University Press.
Yoo, J. Y. 1977 Fluid motion between 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. Maths 34, 247.Google Scholar