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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: dalwadi@maths.ox.ac.uk
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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

REFERENCES

Bico, J., Ashmore-Chakrabarty, J., McKinley, G.H. & Stone, H.A. 2009 Rolling stones: the motion of a sphere down an inclined plane coated with a thin liquid film. Phys. Fluids 21 (8), 082103.CrossRefGoogle Scholar
Brackbill, J.U., Kothe, D.B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.Google Scholar
Coyle, D.J., Macosko, C.W. & Scriven, L.E. 1986 Film-splitting flows in forward roll coating. J. Fluid Mech. 171, 183–207.Google Scholar
Coyne, J.C. & Elrod, H.G. Jr. 1970 Conditions for the rupture of a lubricating film. Part I: theoretical model. Trans. ASME J. Lubr. Technol. 92, 451–456.CrossRefGoogle Scholar
Coyne, J.C. & Elrod, H.G. Jr. 1971 Conditions for the rupture of a lubricating film. Part II: new boundary conditions for Reynolds equation. Trans. ASME J. Lubr. Technol. 93, 156–167.CrossRefGoogle Scholar
Duffy, B.R. & Wilson, S.K. 1999 Thin-film and curtain flows on the outside of a rotating horizontal cylinder. J. Fluid Mech. 394, 29–49.Google Scholar
Eggers, J., Kerswell, R.R. & Mullin, T. 2013 Balancing a cylinder on a thin vertical layer of viscous fluid. Phys. Rev. E 87 (6), 065001.CrossRefGoogle ScholarPubMed
Fitt, A.D., Kozyreff, G. & Ockendon, J.R. 2004 Inertial levitation. J. Fluid Mech. 508, 165–174.CrossRefGoogle Scholar
Hinch, E.J. & Lemaitre, J. 1994 The effect of viscosity on the height of disks floating above an air table. J. Fluid Mech. 273, 313–322.CrossRefGoogle Scholar
Landau, L. & Levich, B. 1988 Dragging of a liquid by a moving plate. In Dynamics of Curved Fronts (ed. P. Pelcé), pp. 141–153. Elsevier Inc.Google Scholar
Levich, B. & Landau, L. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. USSR 17, 42–54.Google Scholar
Lhuissier, H., Tagawa, Y., Tran, T. & Sun, C. 2013 Levitation of a drop over a moving surface. J. Fluid Mech. 733, R4.CrossRefGoogle Scholar
de Maleprade, H., Soto, D., Quéré, D., Hinch, E.J., Baier, T., Schür, M.T. & Hardt, S. 2018 Air-propelled, herringbone-textured platelets. Phys. Rev. Fluids 3 (10), 104101.Google Scholar
Mullin, T., Ockendon, H. & Ockendon, J.R. 2020 Levitation by thin viscous layers. J. Fluid Mech. 888, A17.CrossRefGoogle Scholar
Pearson, J.R.A. 1960 The instability of uniform viscous flow under rollers and spreaders. J. Fluid Mech. 7 (4), 481–500.CrossRefGoogle Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228 (16), 5838–5866.Google Scholar
Popinet, S. 2020 Basilisk flow solver and PDE library. http://basilisk.fr/, accessed: 2020-04-21.Google Scholar
Ruschak, K.J. 1982 Boundary conditions at a liquid/air interface in lubrication flows. J. Fluid Mech. 119, 107–120.CrossRefGoogle Scholar
Ruschak, K.J. 1985 Coating flows. Annu. Rev. Fluid Mech. 17 (1), 65–89.Google Scholar
Savage, M.D. 1977 Cavitation in lubrication. Part 1. On boundary conditions and cavity – fluid interfaces. J Fluid Mech 80 (4), 743–755.CrossRefGoogle Scholar
Sawaguchi, E., Matsuda, A., Hama, K., Saito, M. & Tagawa, Y. 2019 Droplet levitation over a moving wall with a steady air film. J. Fluid Mech. 862, 261–282.Google Scholar
Stieber, W. 1933 Des Schwimmlager: hydrodinamische Theorie des Gleitlagers. VDI-Verlag.Google Scholar
Swift, H.W. 1932 The stability of lubricating films in journal bearings. In Minutes of the Proceedings of the Institution of Civil Engineers, vol. 233, pp. 267–288. Thomas Telford-ICE Virtual Library.Google Scholar
Taroni, M., Breward, C.J.W., Howell, P.D. & Oliver, J.M. 2012 Boundary conditions for free surface inlet and outlet problems. J. Fluid Mech. 708, 100–110.CrossRefGoogle Scholar
Weinstein, S.J. & Ruschak, K.J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.CrossRefGoogle Scholar
Weymouth, G.D. & Yue, D.K.P. 2010 Conservative volume-of-fluid method for free-surface simulations on Cartesian-grids. J. Comput. Phys. 229 (8), 2853–2865.CrossRefGoogle Scholar

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