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Physics of Fluids / Volume 24 / Issue 2 / ARTICLES / Laminar Flows

Phys. Fluids 24, 023602 (2012); http://dx.doi.org/10.1063/1.3682714 (21 pages)

Draining of a thin film on the wall of a conical container set into rapid rotation about its vertical axis

Marius Ungarish1 and John D. Sherwood2

1Department of Computer Science, Technion, Haifa 32000, Israel
2Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

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(Received 14 September 2011; accepted 19 January 2012; published online 14 February 2012)

A theory for how rotation modifies the draining of a thin fluid film on the surface of a conical container is developed. Rapid rotation, imposed instantaneously, creates an Ekman layer that pumps fluid to the outer edge of the container. This fluid is flung out of the vessel, rather than being transferred to the interior of the container. In consequence, fluid in the vessel but outside the Ekman layer does not spin up and continues to drain towards the container base. The net result is that draining is enhanced by the outward Ekman flux and finishes in finite time (when only the thin Ekman layer remains). The governing equations can be solved by the method of characteristics, to within numerical quadrature. The amount of fluid pumped outwards (rather than draining towards the base of the container) depends upon a single parameter that characterizes the ratio of the Ekman flux to gravitational draining.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. FORMULATION
    1. Some dimensionless numbers
    2. Flow regime influenced by centrifugal-Coriolis accelerations
  3. THE MODEL
    1. Governing equations
    2. The fan
  4. THE CASE φ < 1
    1. Dominance of the boundary condition at the rim
    2. Arrival of the h = 0 front
    3. Profiles of h
    4. Volume of the fluid film
    5. End of the draining: The K i = 0 characteristic
  5. THE CASE φ > 1
  6. MORE COMPLICATED SCENARIOS
    1. Non-uniform initial film thickness h ( z )
    2. Rotation rate Ω( t ) varying with time t
  7. CONCLUDING REMARKS

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KEYWORDS, PACS, and IPC

Keywords

boundary layers, confined flow, film flow, integration, liquid films, numerical analysis, pumps, rotational flow

PACS

International Patent Classification (IPC)

  • F04

    Positive-displacement machines for liquids; Pumps for liquids or elastic fluids

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3682714

PUBLICATION DATA

ISSN

1070-6631 (print)  
1089-7666 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    J. D. Sherwood, “Optimal shapes for best draining,” Phys. Fluids 21, 113102 (2009)PHFLE6000021000011113102000001.

    R. V. Craster and O. K. Matar, “Dynamics and stability of thin liquid films,” Rev. Mod. Phys. 81, 1131 (2009).

    L. A. Dávalos-Orozco and F. H. Busse, “Instability of a thin film flowing on a rotating horizontal or inclined plane,” Phys. Rev. E 65, 026312 (2002).

    J. A. van de Konijnenberg and G. J. F. van Heijst, “Nonlinear spin-up in a circular cylinder,” Phys. Fluids 7, 2989 (1995)PHFLE6000007000012002989000001.

    P. A. Nikrityuk, M. Ungarish, K. Eckert, and R. Grundmann, “Spin-up of a liquid metal flow driven by a rotating magnetic field in a finite cylinder: a numerical and an analytical study,” Phys. Fluids 17, 067101 (2005)PHFLE6000017000006067101000001.


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