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Similarity theory of lubricated Hertzian contacts
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We consider a heavily loaded, lubricated contact between two elastic bodies at relative speed U, such that there is substantial elastic deformation. As a result of the interplay between hydrodynamics and non-local elasticity, a fluid film develops between the two solids, whose thickness scales as U 3/5. The film profile h is selected by a universal similarity solution along the upstream inlet. Another similarity solution is valid at the outlet, which exhibits a local minimum in the film thickness. The two solutions are connected by a hyperbolic problem underneath the contact. Our asymptotic results for a soft sphere pressed against a hard wall are shown to agree with both experiment and numerical simulations.
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