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Cellular flows of a viscous liquid that partly fills a horizontal rotating cylinder

Published online by Cambridge University Press:  21 April 2006

T. Brooke Benjamin  and
S. K. Pathak
T. Brooke Benjamin
Affiliation:
Mathematical Institute, 24/29 St Giles, Oxford OX1 3LB, UK
S. K. Pathak
Affiliation:
Department of Civil Engineering, University of Roorkee, Roorkee U.P., India
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Abstract

This theoretical and experimental investigation inquires into the various steady and unsteady motions that are possible when a highly viscous liquid partly fills a closed circular cylinder rotated about its horizontal axis at constant angular velocity. Fillings leaving an air bubble in the range roughly 10-20% by volume provide the most lively variety of observable phenomena.

The full hydrodynamic problem is too complicated to be amenable to quantitative theoretical treatment, except by numerical analysis which is not yet available; but the abstract qualitative theory developed in § 2 appears to capture all the essentials of experimentally found behaviour. An analogous finite-dimensional system, such as would be presented by a close finite-element approximation, is used to illuminate principles governing the order of multiple solutions and their stability. Then the connection between the full problem and the analogue is demonstrated. Finally a simple argument is outlined confirming the observed stability of the motion at small rates of rotation.

The experiments are described in § 3 and their results presented in § 4. For various values of the cylinder's aspect ratio, estimated singularities of the time-independent solution set are recorded as several-branched graphs of ωv/gR versus volume fraction filled by liquid (ω is the angular velocity of the container, R its radius and v the kinematic viscosity of the liquid). The experimental observations are discussed in § 5.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 183 , October 1987 , pp. 399 - 420
Copyright
© 1987 Cambridge University Press

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References

Balmer, R. T. 1970 The hydrocyst – a stability phenomenon in continuum mechanics. Nature 227, 600–601.Google Scholar
Benjamin, T. B. 1976 Applications of Leray-Schauder degree theory to problems of hydrodynamic stability. Math. Proc. Camb. Phil. Soc. 79, 373–392.Google Scholar
Benjamin, T. B. 1978a Bifurcation phenomena in steady flows of a viscous liquid. I. Theory. Proc. R. Soc. Lond. A 359, 1–26.Google Scholar
Benjamin, T. B. 1978b Bifurcation phenomena in steady flows of a viscous liquid. II. Experiments. Proc. R. Soc. Lond. A 359, 27–43.Google Scholar
Benjamin, T. B. 1987 Hamiltonian theory for motions of bubbles in an infinite liquid. J. Fluid Mech. 181, 349–379.Google Scholar
Benjamin, T. B. & Mullin, T. 1982 Anomalous modes in the Taylor experiment. Proc. R. Soc. Lond. A 377, 221–249.Google Scholar
Benjamin, T. B. & Olver, P. J. 1982 Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185.Google Scholar
Cliffe, K. A. & Mullin, T. 1985 A numerical and experimental study of anomalous modes in the Taylor experiment. J. Fluid Mech. 153, 243–258.Google Scholar
Debler, W. R. & Yih, C.-S. 1962 Formation of rings in a liquid film attached to the inside of a rotating cylinder. J. Aerospace Sci. 29, 364.Google Scholar
Guckenheimer, J. 1981 On a codimension two bifurcation. In Proc. Symp. on Dynamical Systems and Turbulence, Univ. of Warwick 1980. Lecture Notes in Mathematics, vol. 898, pp. 99–142. Springer.
Karweit, M. J. & Corrsin, S. 1975 Observations of cellular patterns in a partly filled, horizontal, rotating cylinder. Phys. Fluids 18, 111–112.Google Scholar
Keller, J. B., Rubenfeld, L. A. & Molyneux, J. E. 1967 Extremum principles for slow viscous flows with applications to suspensions. J. Fluid Mech. 30, 97–125.Google Scholar
Korteweg, D. J. 1883 On a general theorem of the stability of the motion of a viscous fluid. Phil. Mag. (5) 16, 112–118.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press. (Dover edition 1945.)
Lloyd, N. G. 1978 Degree theory. Cambridge University Press.
Moffatt, H. K. 1977 Behaviour of a viscous film on the outer surface of a rotating cylinder. J. Méc. 16, 651–673.Google Scholar
Mullin, T., Cliffe, K. A. & Pfister, G. 1987 Unusual time-dependent phenomena in the Taylor-Couette experiment at moderately low Reynolds numbers. Phys. Rev. Lett. 58, 2212–2215.Google Scholar
Orr, F. M. & Scriven, L. E. 1978 Rimming flow: a numerical simulation of steady, viscous, free-surface flow with surface tension. J. Fluid Mech. 84, 145–165.Google Scholar
Phillips, O. M. 1960 Centrifugal waves. J. Fluid Mech. 7, 340–352.Google Scholar
Pritchard, W. G. 1986 Instability and chaotic behaviour in a free-surface flow. J. Fluid Mech. 165, 1–60.Google Scholar
Ruschak, K. J. & Scriven, L. E. 1976 Rimming flow of liquid in a rotating horizontal cylinder. J. Fluid Mech. 76, 113–125.Google Scholar
Skalak, R. 1970 Extensions of extremum principles for slow viscous flows. J. Fluid Mech. 42, 527–548.Google Scholar