This version is available at https://strathprints.strath.ac.uk/42349/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.Under consideration for publication in J. Fluid Mech. 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 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 ...
This version is available at https://strathprints.strath.ac.uk/33826/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output. AbstractA comprehensive description is obtained of the two-dimensional steady gravity-driven flow with prescribed volume flux of a thin film of Newtonian fluid with temperature-dependent viscosity on a stationary horizontal cylinder. When the cylinder is uniformly hotter than the surrounding atmosphere (positive thermoviscosity) the effect of increasing the heat transfer to the surrounding atmosphere at the free surface is to increase the average viscosity and hence reduce the average velocity within the film, with the net effect that the film thickness (and hence the total fluid load on the cylinder) is increased to maintain the fixed volume flux of fluid. When the cylinder is uniformly colder than the surrounding atmosphere (negative thermoviscosity) the opposite occurs. Increasing the heat transfer at the free surface from weak to strong changes the film thickness everywhere (and hence the load, but not the temperature or the velocity) by a constant factor which depends only on the specific viscosity model considered. The effect of increasing the thermoviscosity is always to increase the film thickness and hence the load. In the limit of strong positive thermoviscosity the velocity is small and uniform outside a narrow boundary layer near the cylinder leading to a large film thickness, while in the limit of strong negative thermoviscosity the velocity increases from zero at the cylinder to a large value at the free surface leading to a small film thickness. * Author to whom correspondence should be addressed.
SummaryA comprehensive description is obtained of steady thermoviscous (i.e. with temperature-dependent viscosity) coating and rimming flow on a uniformly rotating horizontal cylinder which is uniformly hotter or colder than the surrounding atmosphere. It is found that, as in the corresponding isothermal problem, there is a critical solution with a corresponding critical load (which depends, in general, on both the Biot number B and the thermoviscosity number V ) above which no "fullfilm" solutions corresponding to a continuous film of fluid covering the entire outside or inside of the cylinder exist. The effect of thermoviscosity on both the critical solution and the full-film solution with a prescribed load is described. In particular, there are no full-film solutions with a prescribed load M for any value of B when M ≥f −1/2 Mc 0 for positive V and when M > Mc 0 for negative V , wheref is a monotonically decreasing function of V and Mc 0 ≃ 4.44272 is the critical load in the constant-viscosity case. It is also found that when the prescribed load satisfies M < 1.50315 there is a narrow region of the B-V parameter plane in which backflow occurs.
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