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Levitation of a cylinder by a thin viscous film

Published online by Cambridge University Press:  26 April 2021

Mohit P. Dalwadi Open the ORCID record for Mohit P. Dalwadi [Opens in a new window] ,
Radu Cimpeanu Open the ORCID record for Radu Cimpeanu [Opens in a new window] ,
Hilary Ockendon Open the ORCID record for Hilary Ockendon [Opens in a new window] ,
John Ockendon Open the ORCID record for John Ockendon [Opens in a new window]  and
Tom Mullin Open the ORCID record for Tom Mullin [Opens in a new window]
Mohit P. Dalwadi*
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK
Radu Cimpeanu
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK Mathematical Institute, University of Warwick, Zeeman Building, CoventryCV4 7AL, UK Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
Hilary Ockendon
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK
John Ockendon
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK
Tom Mullin
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK Linacre College, University of Oxford, OxfordOX1 3JA, UK
*
†Email address for correspondence: [email protected]
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Abstract

When a horizontal cylinder is placed on a vertically moving belt coated with a thin layer of viscous fluid, experiments reveal that, at a specific belt velocity, the cylinder can be levitated at a fixed height while rotating around its own axis at an a priori unknown rate. We develop and solve a model for this experiment, using a combination of asymptotic analysis and direct numerical simulation. We obtain a relationship between the belt speed and cylinder rotation rate, which we successfully compare with experimental results.

JFM classification

Boundary Layers: Boundary layer structure Interfacial Flows (free surface): Thin films Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 917 , 25 June 2021 , A28
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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Dalwadi et al. supplementary movie 1

A real-time video (24 FPS) of the cylinder levitation experiment described in Section 2 and shown in Figure 1, as viewed from the front. The aluminium cylinder has a diameter of 13 mm and a length of 15 mm. The belt width is 26.5 mm and the film thickness is 0.3 mm.

Download Dalwadi et al. supplementary movie 1(Video)
Video 5.5 MB

Dalwadi et al. supplementary movie 2

A real-time video (24 FPS) of the cylinder levitation experiment described in Section 2 and shown in Figure 1, as viewed from the side. The aluminium cylinder has a diameter of 8 mm and a length of 15 mm. The belt width is 26.5 mm and the film thickness is 0.3 mm.

Download Dalwadi et al. supplementary movie 2(Video)
Video 1.7 MB

Dalwadi et al. supplementary movie 3

Animation summary of the direct numerical simulation (DNS) framework designed for the inner problem (described in subsections 4.2 and 4.3) accompanying figure 7 of the manuscript. Illustrated are, in sequence, the two fluids with explanatory notes, followed by the pressure and velocity field norm, and finally the horizontal and vertical velocity component contributions. The relevant dimensionless flow parameters are $Ca = 2.0$, $U = 1.0$, and $\Omega = 0.6$, with the underlying dimensional quantities described in section 2 as part of the experimental set-up.

Download Dalwadi et al. supplementary movie 3(Video)
Video 5.7 MB