Three-dimensional coating and rimming flow: a ring of fluid on a rotating horizontal cylinder

Handle URI:
http://hdl.handle.net/10754/600014
Title:
Three-dimensional coating and rimming flow: a ring of fluid on a rotating horizontal cylinder
Authors:
Leslie, G. A.; Wilson, S. K.; Duffy, B. R.
Abstract:
The steady three-dimensional flow of a thin, slowly varying ring of Newtonian fluid on either the outside or the inside of a uniformly rotating large horizontal cylinder is investigated. Specifically, we study 'full-ring' solutions, corresponding to a ring of continuous, finite and non-zero thickness that extends all of the way around the cylinder. In particular, it is found that there is a critical solution corresponding to either a critical load above which no full-ring solution exists (if the rotation speed is prescribed) or a critical rotation speed below which no full-ring solution exists (if the load is prescribed). We describe the behaviour of the critical solution and, in particular, show that the critical flux, the critical load, the critical semi-width and the critical ring profile are all increasing functions of the rotation speed. In the limit of small rotation speed, the critical flux is small and the critical ring is narrow and thin, leading to a small critical load. In the limit of large rotation speed, the critical flux is large and the critical ring is wide on the upper half of the cylinder and thick on the lower half of the cylinder, leading to a large critical load. We also describe the behaviour of the non-critical full-ring solution and, in particular, show that the semi-width and the ring profile are increasing functions of the load but, in general, non-monotonic functions of the rotation speed. In the limit of large rotation speed, the ring approaches a limiting non-uniform shape, whereas in the limit of small load, the ring is narrow and thin with a uniform parabolic profile. Finally, we show that, while for most values of the rotation speed and the load the azimuthal velocity is in the same direction as the rotation of the cylinder, there is a region of parameter space close to the critical solution for sufficiently small rotation speed in which backflow occurs in a small region on the upward-moving side of the cylinder. © 2013 Cambridge University Press.
Citation:
Leslie GA, Wilson SK, Duffy BR (2013) Three-dimensional coating and rimming flow: a ring of fluid on a rotating horizontal cylinder. Journal of Fluid Mechanics 716: 51–82. Available: http://dx.doi.org/10.1017/jfm.2012.509.
Publisher:
Cambridge University Press (CUP)
Journal:
Journal of Fluid Mechanics
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
29-Jan-2013
DOI:
10.1017/jfm.2012.509
Type:
Article
ISSN:
0022-1120; 1469-7645
Sponsors:
The first author (G. A. L.) gratefully acknowledges the financial support of the United Kingdom Engineering and Physical Sciences Research Council (EPSRC) via a Doctoral Training Account (DTA) research studentship. Part of this work was undertaken while the corresponding author (S. K. W.) was a Visiting Fellow in the Department of Mechanical and Aerospace Engineering, School of Engineering and Applied Science, Princeton University, USA, and part of it was undertaken while he was a Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM), Mathematical Institute, University of Oxford, UK. This publication was based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).
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Full metadata record

DC FieldValue Language
dc.contributor.authorLeslie, G. A.en
dc.contributor.authorWilson, S. K.en
dc.contributor.authorDuffy, B. R.en
dc.date.accessioned2016-02-28T06:34:22Zen
dc.date.available2016-02-28T06:34:22Zen
dc.date.issued2013-01-29en
dc.identifier.citationLeslie GA, Wilson SK, Duffy BR (2013) Three-dimensional coating and rimming flow: a ring of fluid on a rotating horizontal cylinder. Journal of Fluid Mechanics 716: 51–82. Available: http://dx.doi.org/10.1017/jfm.2012.509.en
dc.identifier.issn0022-1120en
dc.identifier.issn1469-7645en
dc.identifier.doi10.1017/jfm.2012.509en
dc.identifier.urihttp://hdl.handle.net/10754/600014en
dc.description.abstractThe steady three-dimensional flow of a thin, slowly varying ring of Newtonian fluid on either the outside or the inside of a uniformly rotating large horizontal cylinder is investigated. Specifically, we study 'full-ring' solutions, corresponding to a ring of continuous, finite and non-zero thickness that extends all of the way around the cylinder. In particular, it is found that there is a critical solution corresponding to either a critical load above which no full-ring solution exists (if the rotation speed is prescribed) or a critical rotation speed below which no full-ring solution exists (if the load is prescribed). We describe the behaviour of the critical solution and, in particular, show that the critical flux, the critical load, the critical semi-width and the critical ring profile are all increasing functions of the rotation speed. In the limit of small rotation speed, the critical flux is small and the critical ring is narrow and thin, leading to a small critical load. In the limit of large rotation speed, the critical flux is large and the critical ring is wide on the upper half of the cylinder and thick on the lower half of the cylinder, leading to a large critical load. We also describe the behaviour of the non-critical full-ring solution and, in particular, show that the semi-width and the ring profile are increasing functions of the load but, in general, non-monotonic functions of the rotation speed. In the limit of large rotation speed, the ring approaches a limiting non-uniform shape, whereas in the limit of small load, the ring is narrow and thin with a uniform parabolic profile. Finally, we show that, while for most values of the rotation speed and the load the azimuthal velocity is in the same direction as the rotation of the cylinder, there is a region of parameter space close to the critical solution for sufficiently small rotation speed in which backflow occurs in a small region on the upward-moving side of the cylinder. © 2013 Cambridge University Press.en
dc.description.sponsorshipThe first author (G. A. L.) gratefully acknowledges the financial support of the United Kingdom Engineering and Physical Sciences Research Council (EPSRC) via a Doctoral Training Account (DTA) research studentship. Part of this work was undertaken while the corresponding author (S. K. W.) was a Visiting Fellow in the Department of Mechanical and Aerospace Engineering, School of Engineering and Applied Science, Princeton University, USA, and part of it was undertaken while he was a Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM), Mathematical Institute, University of Oxford, UK. This publication was based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherCambridge University Press (CUP)en
dc.subjectcontact linesen
dc.subjectinterfacial flows (free surface)en
dc.subjectthin filmsen
dc.titleThree-dimensional coating and rimming flow: a ring of fluid on a rotating horizontal cylinderen
dc.typeArticleen
dc.identifier.journalJournal of Fluid Mechanicsen
dc.contributor.institutionUniversity of Strathclyde, Glasgow, United Kingdomen
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
kaust.grant.numberKUK-C1-013-04en
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