The transition between the Stokes equations and the Reynolds equation: A mathematical proof. (English) Zbl 0701.76039

Bayada, Guy; Chambat, Michèle

Appl. Math. Optimization 14, 73-93 (1986).

Summary: The Reynolds equation is used to calculate the pressure distribution in a thin layer of lubricant film between two surfaces. Using the asymptotic expansion in the Stokes equations, we show the existence of singular perturbation phenomena whenever the two surfaces are in relative motion. We prove that the Reynolds equation is an approximation of the Stokes equations and that the kind of convergence is strongly related with the boundary conditions on the velocity field.

MSC:

76D07	Stokes and related (Oseen, etc.) flows
35Q30	Navier-Stokes equations

Keywords:

Reynolds equation; thin layer of lubricant film; Stokes equations

References:

[1]	Bayada G (1985) Use of homogenization method in the analysis of the roughness for 2-D lubricated bearing, Proc 10th LEEDS-LYON Symp IPC Science Technology Press
[2]	Bayada G, Chambat M (1984) Existence and uniqueness for a lubrication problem with non regular conditions on the free boundary, Boll Un Math It (6) 3-B:543-557 · Zbl 0612.35026
[3]	Cameron A (1966) The principles of lubrication, Longmans, London · Zbl 0139.45203
[4]	Capriz G (1960) On the vibrations of shaft rotating on lubricated bearings, Ann Math Pure Appl 50:223-248 · Zbl 0097.19203
[5]	Cimatti G (1983) How the Reynolds equation is related to the Stokes equations, Appl Math Optim 10:267-274 · Zbl 0538.76038
[6]	Dridi H (1982) Comportement asymptotique des ?quations de Navier-Stokes dans des domaines ?aplatis?, Bull Sc Math 106:369-385 · Zbl 0512.35015
[7]	Dumont T (1978) D?composition par projection de certains probl?mes aux limites elliptiques non lin?aires, Thesis Univ Lyon 1, France
[8]	Elrod HG (1959) A derivation of the basic equations for hydrodynamic lubrication with a fluid having constant properties, Quart Appl Math 17-4:349-359 · Zbl 0087.41101
[9]	Fantino B, Frene J, Godet M (1971) Conditions d’utilisation de l’?quation de Reynolds en m?canique des films minces, Ct Rd Acad Sc Paris 262A:155-165
[10]	Girault V, Raviart PA (1981) Finite element approximation of the Navier-Stokes equations, Rvd Reprint ed Springer Verlag, Berlin · Zbl 0441.65081
[11]	Johnson R (1983) Power-law creep of a material being compressed between parallel plates: a singular perturbation problem, J Eng Math 18:105-117 · Zbl 0571.76032
[12]	Mutuli S, Bonneau D, Frene J (1985) Velocity measurements in the inlet zone of a hydrodynamic contact, Proc 10th LEEDS-LYON Symp IPC Science Technology Press
[13]	Phan-Thien N, Atkinson JD (1982) On the effects of homogeneous Reynolds roughness in a two dimensional slider bearing with exponential film thickness, ASME J Lub Tech 104:220-226
[14]	Reynolds O (1886) On the theory of lubrication and its application to M Beauchamp Tower’s experiments, Phil Trans Roy Soc London A117:157-234 · JFM 18.0946.04
[15]	Temam R (1979) Navier-Stokes equations, North Holland, Amsterdam · Zbl 0426.35003
[16]	Tuck EO, Bentwich M (1983) Sliding sheets: lubrication with comparable viscous and inertia forces, J Fluid Mech 135:51-69 · Zbl 0571.76027