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DERIVATION OF THE REYNOLDS EQUATION FOR LUBRICATION OF A ROTATING SHAFT

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*A. DUVNJAK and E. MARUSIC-PALOKA*

**Address.** A. Duvnjak, Faculty of Electrical Engineering and Computing,
University of Zagreb, Department of Applied Mathematics, Unska 3, 10000
Zagreb, CROATIA
E. Marusic-Paloka, Department of Mathematics, University of Zagreb,
Bijenicka 30, 10000 Zagreb, CROATIA

**E-mail:** emarusic@math.hr

**Abstract.** In this paper, using the asymptotic expansion, we prove
that the Reynolds lubrication equation is an approximation of the full
Navier--Stokes equations in thin gap between two coaxial cylinders in relative
motion. Boundary layer correctors are computed. The error estimate in terms
of domain thickness for the asymptotic expansion is given. The corrector
for classical Reynolds approximation is computed.

**AMSclassification.** 35B25, 35B40, 76D08

**Keywords.** Lubrication, Reynolds equation, Navier-Stokes system,
lower-dimensional approximation