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Inertial effects at moderate Reynolds number in thin-film rimming flows driven by surface shear

Kay, E.D.; Hibberd, S.; Power, H.

Authors

E.D. Kay

S. Hibberd

H. Power

Abstract

In this paper, we study two-dimensional thin-film flow inside a stationary circular cylinder driven by an imposed surface shear stress. Modelling is motivated by a need to understand the cooling and film dynamics provided by oil films in an aero-engine bearing chamber characterised by conditions of very high surface shear and additional film mass flux from oil droplets entering the film through the surface. In typical high-speed operation, film inertial effects can provide a significant leading-order mechanism neglected in existing lubrication theory models. Inertia at leading-order is included within a depth-averaged formulation where wall friction is evaluated similar to hydraulic models. This allows key nonlinear inertial effects to be included while retaining the ability to analyse the problem in a mathematically tractable formulation and compare with other approaches. In constructing this model, a set of simplified mass and momentum equations are integrated through the depth of the film yielding a spatially one-dimensional depth-averaged formulation of the problem. An a priori assumed form of velocity profile is needed to complete the system. In a local Stokes flow analysis, a quadratic profile is the exact solution for the velocity field though it must be modified when inertial effects become important. Extension of the velocity profile to a cubic profile is selected enabling specification of a wall friction model to include the roughness of the cylinder wall. A modelling advantage of including the inertia term, relevant to the applications considered, is that a smooth progression in solution can be obtained between cases of low Reynolds number corresponding to lubrication theory, and high Reynolds number corresponding to uniform rimming-flow. Importantly, we also investigate the effect of inertia on some typical solutions from other studies and present a greater insight to existing and new film solutions which arise from including inertia effects.

Journal Article Type Article
Journal Physics of Fluids
Print ISSN 1070-6631
Electronic ISSN 1089-7666
Publisher AIP Publishing
Peer Reviewed Peer Reviewed
Volume 25
Issue 10
Institution Citation Kay, E., Hibberd, S., & Power, H. (in press). Inertial effects at moderate Reynolds number in thin-film rimming flows driven by surface shear. Physics of Fluids, 25(10), doi:10.1063/1.4825134
DOI https://doi.org/10.1063/1.4825134
Publisher URL http://aip.scitation.org/doi/10.1063/1.4825134
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf