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On the various aspects of the thin film equation in hydrodynamic lubrication when the roughness occurs. (English) Zbl 0649.76013
Applications of multiple scaling in mechanics, Proc. Int. Conf., Paris 1986, Rech. Math. Appl. 4, 14-30 (1987).
[For the entire collection see Zbl 0625.00020.]
In this paper the authors have studied in a deterministic way, the asymptotic behaviour of the Stokes equation, when both the roughness spacing \(\epsilon\) and the gap height \(\eta\) tend to zero by assuming a periodic roughness. The limiting cases of \(\epsilon\) tending to zero faster, slower or at the same rate as \(\eta\) are discussed. It is shown that all three limiting equations are of the Reynolds type but different. The height of the roughness has no influence on the qualitative aspect of these equations. In the case \(\lambda =\eta /\epsilon\) tending to zero the results of Christensen can be used with confidence for small roughness spacing.
Reviewer: J.Prakash

MSC:
76D08 Lubrication theory
35Q99 Partial differential equations of mathematical physics and other areas of application