Numerical computation of free boundary problems in elastohydrodynamic lubrication. (English) Zbl 0851.73058

Summary: An alternative algorithm has been developed for computing the behavior of thin fluid films in two elastohydrodynamic lubrication problems. The presence of elasticity, lubrication, and cavitation leads to a nonlinear coupled system of partial differential equations. The hydrodynamic part of both problems is governed by the well-known Reynolds equation combined with the cavitation model of Elrod-Adams, which motivates the appearance of a free boundary. Elastic deformations are taken into account by means of the Hertz equation in rolling ball contact problems, or the elastic hinged plate biharmonic equation is used in the case of journal-bearing devices with thin bearing. A numerical method decoupling the hydrodynamic part of the problem and the elastic one is presented. This method also involves an upwind scheme to discretize the lubrication model and finite element approximations.


74S05 Finite element methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76D08 Lubrication theory
Full Text: DOI


[1] Dowson, D.; Higginson, G. R., Elastohydrodynamic Lubrication (1966), Pergamon Press
[2] Bayada, G.; Chambat, M., Sur quelques modélisations de la zone de cavitation en lubrification hydrodynamique, J. Theor. Appl. Mech., 5, 5, 703-729 (1986) · Zbl 0621.76030
[3] Oden, J. T.; Kikuchi, N., Theory of variational inequalities with applications to problems of flow through porous media, Int. J. Eng. Sci., 18, 1173-1284 (1980) · Zbl 0444.76069
[4] Oden, J. T.; Wu, S. R., Existence of solutions to the Reynolds equation of elastohydrodynamic lubrication, Int. J. Eng. Sci., 23, 207-215 (1985) · Zbl 0619.76043
[5] Oden, J. T.; Wu, S. R., A note on some mathematical studies on elastohydrodynamic lubrication, Int. J. Eng. Sci., 25, 681-690 (1987) · Zbl 0622.76039
[6] Rodrigues, J. F., Remarks on the Reynolds problem of elastohydrodynamic lubrication, Eur. J. Appl. Math., 4, 83-96 (1993) · Zbl 0768.76016
[7] Hu, B., A quasivariational inequality arising in elastohydrodynamics, SIAM J. Math. Anal., 21, 18-36 (1990) · Zbl 0718.35101
[8] Durany, J.; Vázquez, C., Mathematical analysis of an elastohydrodynamic lubrication problem with cavitation, Appl. Anal., 1, 135-142 (1994) · Zbl 0841.35133
[9] Bayada, G.; Durany, J.; Vázquez, C., Existence of solution for a lubrication problem in elastic journal bearing devices with thin bearing, Math. Meth. Appl. Sci., 18, 225-266 (1995) · Zbl 0820.35110
[10] Johnson, K. L., Contact Mechanics (1985), Cambridge University Press · Zbl 0599.73108
[11] Cimatti, G., Existence and uniqueness for nonlinear Reynolds equations, Int. J. Eng. Sci., 24, 5, 827-834 (1986) · Zbl 0624.76090
[12] Glowinski, R.; Pironneau, O., Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem, SIAM Rev., 21, 167-212 (1979) · Zbl 0427.65073
[13] Douglas, J.; Russell, T., Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference, SIAM J. Numer. Anal., 19, 5, 871-885 (1982) · Zbl 0492.65051
[14] Pironneau, O., On the transpport-diffusion algorithm and its application to Navier-Stokes equations, Numer. Math., 38, 309-332 (1982) · Zbl 0505.76100
[15] Bermúdez, A.; Durany, J., Numerical solution of cavitation problems in lubrication, Comp. Meth. Appl. Mech Eng., 75, 457-466 (1989) · Zbl 0687.76030
[16] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Applications (1980), Academic Press · Zbl 0457.35001
[17] Bermúdez, A.; Moreno, C., Duality methods for solving variational inequalities, Comp. Math. Appl., 7, 43-58 (1981) · Zbl 0456.65036
[18] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland · Zbl 0445.73043
[19] Ciarlet, P. G.; Raviart, P. A., A mixed finite element method for the biharmonic equation, (de Boor, C., Mathematical Aspects of Finite Elements in Partial Differential Equations (1974), Academic Press), 125-145 · Zbl 0337.65058
[20] Balasundaram, S.; Bhattacharyya, P. K., A mixed finite element method for the Dirichlet problem of fourth order elliptic operators with variable coefficients, (Kawai, T., Finite Element Methods in Flow Problems (1982), Tokyo Univ. Press: Tokyo Univ. Press Tokyo) · Zbl 0508.76006
[21] Vázquez, C., Análisis Matemático y Resolución Numérica de Problemas de Lubricación con Cavitación, (Ph.D. Thesis (1992), University of Santiago: University of Santiago Spain)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.