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Mathematical Problems in Engineering
Volume 2010, Article ID 963983, 15 pages
Research Article

A New Approach for Estimating the Friction in Thin Film Lubrication

1Narvik University College, 8505 Narvik, Norway
2Department of Mathematics, Luleå University of Technology, 971 87 Luleå, Sweden

Received 10 September 2010; Accepted 11 December 2010

Academic Editor: K. Vajravelu

Copyright © 2010 Dag Lukkassen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An important problem in the theory of lubrication is to model and analyze the effect of surface roughness on, for example, the friction and load carrying capacity. A direct numerical computation is often impossible since an extremely fine mesh is required to resolve the surface roughness. This suggests that one applies some averaging technique. The branch in mathematics which deals with this type of questions is known as homogenization. In this paper we present a completely new method for computing the friction. The main idea is that we study the variational problem corresponding to the Reynolds equation. We prove that the homogenized variational problem is closely related to the homogenized friction. Finally we use bounds on the homogenized Lagrangian to derive bounds for the friction. That these bounds can be used to efficiently compute the friction is demonstrated in a typical example.