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lubrication; Reynolds equation; Navier-Stokes system; lower-dimensional approximation
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.
[1] Assemien A., Bayada G., Chambat M.: Inertial Effects in the Asymptotic Behavior of a Thin Film Flow. Asymptotic.Anal., 9(1994), 177–208. MR 1295293 | Zbl 0813.35076
[2] Bayada G., Chambat M.: The Transition Between the Stokes Equations and the Reynolds Equation: A Mathematical Proof. Appl. Math. Optim., 14 (1986), 73–93. MR 0826853 | Zbl 0701.76039
[3] Bourgeat A., Marušić-Paloka E.: Loi d’écoulement non linéaire entre deux plaques ondulées. C.R.Acad.Sci.Paris, Série I , t.321 (1995), 1115–1120. MR 1360583 | Zbl 0841.76082
[4] Bourgeat A., Marušić-Paloka E.: Nonlinear Effects for Flow in Periodically Constricted Channel Caused by High Injection Rate. Math.Models Methods Appl.Sci., Vol 8, No 3 (1998), 379–405. MR 1624867 | Zbl 0920.76082
[5] Capriz G.: On the Vibrations of Shaft Rotating on Lubricated Bearings. Ann. Math. Pure. Appl., 50(1960), 223–248. MR 0115378
[6] Cimatti G.: A Rigorous Justification of the Reynolds Equation. Quart. Appl. Math., XLV (4) (1987), 627–644. MR 0917014 | Zbl 0661.76028
[7] Elrod H. G.: A Derivation of the Basic Equations for Hydrodynamics Lubrication with a Fluid Having Constant Properties. Quart. Appl. Math. 27 (1960), 349–385. MR 0109552
[8] Galdi G. P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, I, II. Springer–Verlag, Berlin, 1994. MR 2808162
[9] Iosifyan G. A., Oleinik O. A.: O povedenii na beskonečnosti rešenij elliptičeskih uravnenij vtorogo porjadka v oblastjah s nekompaktnoj granicej. Mat. Sb., 112, 4(8) (1980), 588–610. MR 0587039
[10] Hughes T. J. R., Marsden J. E.: A Short Course in Fluid Mechanics. Publish or Perish, Boston, 1976. MR 0468526 | Zbl 0329.76001
[11] Marušić-Paloka E.: The Effects of Torsion and Flexion for a Fluid Flow Through a Curved Pipe. to appear in Appl. Math. Optim. MR 1851740
[12] Nazarov S.A.: Asymptotic solution of the Navier-Stokes problem on the flow of a thin layer of fluid. Siberian Math.J., 31 (1990) 2, 296–307. MR 1065588 | Zbl 0712.76037
[13] Reynolds O.: On the Theory of Lubrication and its Application to Beauchamp Tower’s Experiment. Phil. Trans. Roy. Soc. London, A 117 (1886), 157–234.
[14] Wannier G. H.: A Contribution to the Hydrodynamics of Lubrication. Quart. Appl. Math., 8 (1950), 1–32. MR 0037146 | Zbl 0036.25804
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