Abstract
Porous metal bearings are widely used in small and micro devices. To compute the pressure one has to solve the Reynolds equation coupled with the Laplace equation. We show that it is possible to give to the relevant boundary value problem a variational formulation. We show that the pressure of the film in a porous bearing is less than that of the corresponding non-porous bearing.
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Notes
Taken here equal to zero in a suitable scale.
The conditions (12) are valid in the non-porous case. They are justified by the Lewy–Stampacchia theorem (see [6], and [7, p. 223]). For, the solution of the variational inequality which gives the unilateral solution is globally of class \(C^{1,\alpha }\). Hence, on the side of the free boundary where \(p=0\), we have \(p_{\theta }=0\), \(p_y=0\). This implies, by continuity, (12). The conditions valid on the free boundary in the porous case, are an open questions, see Sect. 6.
These constant can easily reintroduced with minor notational changes
To verify this relation, recall that \(\int _0^\pi \sin (mx)\sin (nx)dx\) is equal to \(\frac{\pi }{2}\) if \(m=n\) and equal to 0 if \(m\ne n\).
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Several pertinent and useful remarks made by the Referee have been used to improve this paper.
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Cimatti, G. A variational principle for the system of P.D.E. of porous metal bearings. Meccanica 56, 1079–1086 (2021). https://doi.org/10.1007/s11012-021-01322-6
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DOI: https://doi.org/10.1007/s11012-021-01322-6