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Mathematical analysis of some new Reynolds-rod elastohydrodynamic models. (English) Zbl 1068.74018
Summary: In this paper, some new elastohydrodynamic Reynolds-rod models are posed to obtain the existence of solution (the lubricant pressure and the elastic rod displacement). More precisely, a sign restriction on fluid pressure for cavitation modelling and different unilateral conditions on the rod displacement associated with a rigid structure coating are formulated in terms of coupled variational inequalities. The particular hinged or clamped boundary conditions on the rod displacement require different techniques to prove the existence of solution. Besides nearly linear coupled problems, two nonlinear rod problems including curvature effects are analysed. Mainly, regularity results and $$L^{\infty}$$ estimates for the solution of variational inequalities and fixed-point theorems lead to the existence results for the various coupled models.

##### MSC:
 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) 74G40 Regularity of solutions of equilibrium problems in solid mechanics 49J40 Variational inequalities 76D08 Lubrication theory
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##### References:
 [1] Dowson, Elastohydrodynamic Lubrication (1966) [2] Bayada, The transition between the Stokes equation and the Reynolds equation. A mathematical proof, Applied Mathematics and Optimization 14 pp 73– (1986) · Zbl 0701.76039 · doi:10.1007/BF01442229 [3] Cimatti, How the Reynolds equation is related to the Stokes equation, Applied Mathematics and Optimization 10 pp 223– (1983) · Zbl 0538.76038 · doi:10.1007/BF01448389 [4] Bayada, Sur quelques modélisations de la zone de cavitation en lubrification hydrodynamique, Journal of Theoretical and Applied Mechanics 5 pp 703– (1986) · Zbl 0621.76030 [5] Bayada, Existence of solution for elastohydrodynamic piezoviscous lubrication problems with a new model for cavitation, European Journal of Applied Mathematics 7 pp 63– (1996) · Zbl 0856.76013 · doi:10.1017/S0956792500002205 [6] Durany, Numerical computation of free boundary problems in elastohydrodynamic lubrication, Applied Mathematical Modelling 20 pp 104– (1996) · Zbl 0851.73058 · doi:10.1016/0307-904X(95)00091-W [7] Oden, Existence of solutions to the Reynolds equation of elastohydrodynamic lubrication, International Journal of Engineering Science 23 pp 207– (1985) · Zbl 0619.76043 · doi:10.1016/0020-7225(85)90075-8 [8] Oden, A note on some mathematical studies in elastohydrodynamic lubrication, International Journal of Engineering Science 25 pp 681– (1987) · Zbl 0622.76039 · doi:10.1016/0020-7225(87)90057-7 [9] Bayada, Existence of a solution for a lubrication problem in elastic journal-bearing devices with thin bearing, Mathematical Methods in the Applied Sciences 18 pp 255– (1995) · Zbl 0820.35110 · doi:10.1002/mma.1670180402 [10] Cimatti, Existence and uniqueness for non linear Reynolds equation, International Journal of Engineering Science 24 (5) pp 827– (1986) · Zbl 0624.76090 · doi:10.1016/0020-7225(86)90116-3 [11] Durany, An elastohydrodynamic coupled problem between a piezoviscous Reynolds equation and a hinged plate model, Mathematical Modelling and Numerical Analysis 31 pp 495– (1997) · Zbl 0879.73044 · doi:10.1051/m2an/1997310404951 [12] Arregui, Finite element solution of a Reynolds-Koiter coupled problem for the elastic journal bearing, Computer Methods in Applied Mechanics and Engineering 190 pp 2051– (2001) · Zbl 1013.74019 · doi:10.1016/S0045-7825(00)00221-8 [13] Kinderlehrer, An Introduction to Variational Inequalities (1980) · Zbl 0457.35001 [14] Gilbarg, Elliptic Partial Differential Equations of Second Order (1998) [15] Goeleven, On the one-dimensional nonlinear elastohydrodynamic lubrication, Bulletin of the Australian Mathematical Society 50 pp 353– (1994) · Zbl 0819.35151 · doi:10.1017/S0004972700013484
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