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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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