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An elastohydrodynamic coupled problem between a piezoviscous Reynolds equation and a hinged plate model. (English) Zbl 0879.73044
We prove the existence of solution for a mathematical model governing the displacement of piezoviscous thin fluid films between elastic and rigid surfaces. The hydrodynamic part is governed by the Reynolds lubrication equation combined with the cavitation free-boundary model of Elrod-Adams and with the Barus law for piezoviscous fluids. The elastic deformation of one of the lubricated surfaces is modelled by the hinged plate biharmonic equation where the fluid pressure is treated as an external force. An iterative algorithm is suggested which decouples the hydrodynamic and elastic parts. The method also includes finite element approximations and upwind technique to discretize the lubrication model.

##### MSC:
 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74K20 Plates 76D08 Lubrication theory 35Q72 Other PDE from mechanics (MSC2000)
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##### References:
 [1] R. A. ADAMS, 1975, Sobolev Spaces. Academic Press. Zbl0314.46030 MR450957 · Zbl 0314.46030 [2] S. ALVAREZ, 1986, Problemas de Frontera Libre en Teoría de Lubrificatión.Thesis, University Complutense of Madrid, Spain. [3] S. BALASUNDARAM and P. K. BHATTACHARYYA, 1982, A mixed finite element method for the Dirichlet problem of fourth order elliptic operators with variable coefficients. In : Finite Element Methods in Flow Problems (Ed. T. Kawai) Tokyo Univ. Press. Zbl0508.76006 · Zbl 0508.76006 [4] G. BAYADA, M. CHAMBAT, 1984, Existence and uniqueness for a lubrication problem with nonregular conditions on the free boundary. Boll. UMI. 6, 3B, 543-557. Zbl0612.35026 MR762718 · Zbl 0612.35026 [5] G. BAYADA, M. CHAMBAT, 1986, The transition between the Stokes equation and the Reynolds equation : a mathematical proof. Appl. Math. Opt., 14, 73-93. Zbl0701.76039 MR826853 · Zbl 0701.76039 · doi:10.1007/BF01442229 [6] G. BAYADA, M. CHAMBAT, 1986, Sur quelques modélisations de la zone de cavitation en lubrification hydrodynamique. J. of Theor. and Appl. Mech., 5, 703-729. Zbl0621.76030 MR878123 · Zbl 0621.76030 [7] G. BAYADA, J. DURANY, C. VAZQUEZ, 1995, Existence of solution for a lubrication problem in elastic journal bearing devices with thin bearing. Math. Meth. in the Appl. Sc., 18, 255-266. Zbl0820.35110 MR1319998 · Zbl 0820.35110 · doi:10.1002/mma.1670180402 [8] G. BAYADA, M. EL ALAOUI, C. VÁZQUEZ, 1996, Existence of solution for elastohydrodynamic piezoviscous lubrication problems with a new model of cavitation. Eur. J. of Appl. Math., 7, 63-73. Zbl0856.76013 MR1381799 · Zbl 0856.76013 · doi:10.1017/S0956792500002205 [9] A. BERMÚDEZ, J. DURANY, 1989, Numerical solution of cavitation problems in lubrication. Comp. Meth. in Appl. Mech. and Eng., 75, 457-466. Zbl0687.76030 MR1035758 · Zbl 0687.76030 · doi:10.1016/0045-7825(89)90041-8 [10] A. BERMÚDEZ, C. MORENO, 1981, Duality methods for solving variational inequalities. Comp. Math, with Appl., 7, 43-58. Zbl0456.65036 MR593554 · Zbl 0456.65036 · doi:10.1016/0898-1221(81)90006-7 [11] A. CAMERON, 1981, Basic Lubrication Theory. Ellis Horwood. [12] M. CHIPOT, 1984, Variational Inequalities and Flow on Porous Media. Appl. Math. Sc. Series 52. Springer-Verlag. Zbl0544.76095 MR747637 · Zbl 0544.76095 [13] P. CIARLET, 1978, The Finite Element Method for Elliptic Problems. North-Holland. Zbl0383.65058 MR520174 · Zbl 0383.65058 [14] P. CIARLET, P. A. RAVIART, 1974, A mixed finite element method for the biharmonic equation. In : Mathematical Aspects of Finite Elements in Partial Differential Equations. (Ed. C. de Boor), Academic Press, 125-145. Zbl0337.65058 MR657977 · Zbl 0337.65058 [15] G. CIMATTI, 1983, How the Reynolds equation is related to the Stokes equation. Appl. Math. Opt., 10, 223-248. Zbl0538.76038 MR722490 · Zbl 0538.76038 · doi:10.1007/BF01448389 [16] G. CIMATTI, 1986, Existence and uniqueness for nonlinear Reynolds equations.Int. J. Eng. Sc. 24, N.5, 827-834. Zbl0624.76090 MR841923 · Zbl 0624.76090 · doi:10.1016/0020-7225(86)90116-3 [17] D. DOWSON, C. M. TAYLOR, 1979, Cavitation in bearings. Ann. Rev. Fluid Mech.,35-66. [18] J. DURANY, G. GARCIA, C. VAZQUEZ, 1996, A mixed Dirichlet-Neumann problem for a nonlinear Reynolds equation in elastohydrodynamic piezoviscous lubncation. Proc. of the Edinb. Math Soc., 39, 151-162. Zbl0857.35044 MR1375675 · Zbl 0857.35044 · doi:10.1017/S0013091500022860 [19] J. DURANY, G GARCÍA, C. VÁZQUEZ, 1996, Numerical computation of free boundary problems in elastohydrodynamic lubrication Appl. Math. Mod, 20, 104-113. Zbl0851.73058 · Zbl 0851.73058 · doi:10.1016/0307-904X(95)00091-W [20] J. DURANY, C. VAZQUEZ, 1994, Mathematical analysis of an elastohydrodynamic lubrication problem with cavitation Appl. Anal. Vol. 53, N. 1, 135-142. Zbl0841.35133 MR1379190 · Zbl 0841.35133 · doi:10.1080/00036819408840250 [21] V. GlRAULT, P. A. RAVIART, 1979, Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics, 749, Springer. Zbl0413.65081 MR548867 · Zbl 0413.65081 · doi:10.1007/BFb0063447 [22] R. GLOWINSKI, O. PIRONNEAU, 1979, Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem SIAM Review, 21, 167-212. Zbl0427.65073 MR524511 · Zbl 0427.65073 · doi:10.1137/1021028 [23] D. KINDERLEHRER, G. STAMPACCHIA, 1980, An Introduction to Variational Inequalities and their Applications Academic Press. Zbl0457.35001 MR567696 · Zbl 0457.35001 [24] O. REYNOLDS, 1986, On the theory of lubrication and its applications to M. Beauchamp Tower’s experiments Phil. Trans. Roy. Soc. London, A117, 157-234. JFM18.0946.04 · JFM 18.0946.04
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