# On the Reynolds Equation and the Load Problem in Lubrication: Literature Review and Mathematical Modelling

• Hassán Lombera Rodríguez
• J. Ignacio Tello
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 200)

## Abstract

In this chapter, we provide a literature review concerning the theory of hydrodynamic lubrication, especially applied to journal bearings. The device consists of an external cylinder surrounding a rotating shaft, both separated by a lubricant to prevent contact. In particular, we derive the fluid film thickness model for journal bearings, considering both the parallel and the misaligned case. The hydrodynamic Reynolds equation with cavitation phenomenon, through both Reynolds and Elrod-Adams models are fully derived in this chapter. Subsequently, we pose two suitable variational formulations for the hydrodynamic problem considering both cavitation models. In addition, we present the admissible range of misalignment angle projections for prescribed values of the shaft eccentricity and angular coordinate. Finally, we properly state the problem of a loaded misaligned journal bearing for stationary regime, considering the balance of force and torque components involved.

## Keywords

Reynolds equation Hydrodynamic lubrication Journal bearing Misalignment Cavitation Inverse problem

## References

1. 1.
Abass, B.A., Sahib, M.M.: Effect of bearing compliance on thermo-hydrodynamic lubrication of high speed misaligned journal bearing lubricated with bubbly oil. Ind. Eng. Lett 3, 48–60 (2013)Google Scholar
2. 2.
Álvarez, S.J.: Problemas de frontera libre en teoría de lubrificación. Ph.D. thesis. Universidad Complutense de Madrid (1986)Google Scholar
3. 3.
Asanabe, S., Akakoski, M., Asai, R.: Theoretical and experimental investigation on misaligned journal bearing performance. Ind. Lubr. Tribol. 23(6), 208 (1971)Google Scholar
4. 4.
Bayada, G., Chambat, M.: Sur quelques modélizations de la zone de cavitation en lubrification hydrodynamique. J. Theor. Appl. Mech. 5(5), 703–729 (1986)
5. 5.
Bayada, G., Chambat, M.: The transition between the Stokes equation and the Reynolds equation: a mathematical proof. Appl. Math. Opt. 14(1), 73–93 (1986)
6. 6.
Bayada, G., Chambat, M., Vázquez, C.: Characteristics method for the formulation and computation of a free boundary cavitation problem. J. Comput. Appl. Math. 98(2), 191–212 (1998)
7. 7.
Bayada, G., Martin, S., Vázquez, C.: An average flow model of the Reynolds roughness including a mass-flow preserving cavitation model. J. Tribol-T. ASME. 127(4), 793–802 (2005)Google Scholar
8. 8.
Bayada, G., Martin, S., Vázquez, C.: Homogenization of a nonlocal elastohydrodynamic lubrication problem: a new free boundary model. Math. Mod. Meth. Appl. S. 15(12), 1923–1956 (2005)
9. 9.
Bayada, G., Martin, S., Vázquez, C.: Homogneisation du modèle d’Elrod-Adams hydrodynamique. Asymptotic. Anal. 44, 75–110 (2005)
10. 10.
Bayada, G., Martin, S., Vázquez, C.: Two-scale homogenization of a hydrodynamic Elrod-Adams model. Asymptotic. Anal. (2005)Google Scholar
11. 11.
Bayada, G., Martin, S., Vázquez, C.: Micro-roughness effects in (elasto) hydrodynamic lubrication including a mass-flow preserving cavitation model. Tribol. Int. 39(12), 1707–1718 (2006)Google Scholar
12. 12.
Bayada, G., Vázquez, C.: A survey on mathematical aspects of lubrication problems. Boletín SeMA 39, 37–74 (2007)
13. 13.
Bermúdez, A., Durany, J.: Numerical solution of cavitation problems in lubrication. Comput. Methods Appl. Mech. Eng. 75, 455–466 (1989)
14. 14.
Bermúdez, A., Moreno, C.: Duality methods for solving variational inequalities. Comp. Math. Appl. 7, 43–58 (1981)
15. 15.
Blum, C., Vallès, M.Y., Blesa, M.J.: An ant colony optimization algorithm for DNA sequencing by hybridization. Comput. Oper. Res. 35(11), 3620–3635 (2008)
16. 16.
Boedo, S., Booker, J.F.: Classical bearing misalignment and edge loading: a numerical study of limiting cases. J. Tribol-T. ASME. 126(3), 535–541 (2004)Google Scholar
17. 17.
Boucherit, H., Lahmar, M., Bou-Said, B.: Misalignment effects on steady-state and dynamic behaviour of compliant journal bearings lubricated with couple stress fluids. Lubr. Sci. 20(3), 241–268 (2008)Google Scholar
18. 18.
Bouyer, J., Fillon, M.: An experimental analysis of misalignment effects on hydrodynamic plain journal bearing performances. J. Tribol. 124(2), 313–319 (2002)Google Scholar
19. 19.
Bouyer, J., Fillon, M.: Improvement of the THD performance of a misaligned plain journal bearing. J. Tribol. 125(2), 334–342 (2003)Google Scholar
20. 20.
Brito, F.P.: Thermohydrodynamic performance of twin groove journal bearings considering realistic lubricant supply conditions: a theoretical and experimental study. Ph.D. thesis. Universidade do Minho (2009)Google Scholar
21. 21.
Calvo, N., Durany, J., Vázquez, C.: Comparación de algoritmos numéricos en problemas de lubricación hidrodinámica con cavitación en dimensión uno. Rev. Int. Metod. Numer. 13(2), 185–209 (1997)Google Scholar
22. 22.
Capriz, G., Cimatti, G.: Free boundary problems in the theory of hydrodynamic lubrication: a survey. In: Fasano, A., Primicerio, M. (eds.) Free Boundary Problems: Theory and Applications, vol. 2. Research Notes in Mathematics 79. Pitman, pp. 613–635 (1983)Google Scholar
23. 23.
Chelouah, R., Siarry, P.: Enhanced continuous tabu search: an algorithm for optimizing multiminima functions. In: Meta-Heuristics. Advances and Trends in Local Search Paradigms for Optimization (Chap. 4), pp. 49–61. Springer, Berlin (1999)Google Scholar
24. 24.
Chelouah, Rachid, Siarry, Patrick: A continuous genetic algorithm designed for the global optimization of multimodal functions. J. Heuristics 6(2), 191–213 (2000)
25. 25.
Christopherson, D.G.: A new mathematical method for the solution of film lubrication problems. Inst. Mech. Engrs. J. Proc. 146, 126–135 (1941)
26. 26.
Cimatti, G.: On certain nonlinear problems arising in the theory of lubrication. Appl. Math. Opt. 11(1), 227–245 (1984)
27. 27.
Cimatti, G.: Existence and uniqueness for nonlinear Reynolds equations. Int. J. Eng. Sci. 24(5), 827–834 (1986)
28. 28.
Ciuperca, I.S., Hafidi, I., Jai, M.: Singular perturbation problem for the incompressible Reynolds equation. Electr. J. Differ. Equ. 2006(83), 1–19 (2006)
29. 29.
Ciuperca, I.S., Jai, M., Tello, J.I.: On the existence of solutions of equilibria in lubricated journal bearings. SIAM J. Math. Anal. 40(6), 2316–2327 (2009)
30. 30.
Ciuperca, I.S., Jai, M., Tello, J.I.: Equilibrium analysis for a mass-conserving model in presence of cavitation. Nonlinear Anal.: R. World Appl. 35, 250–264 (2017)
31. 31.
Ciuperca, I.S., Tello, J.I.: Lack of contact in a lubricated system. Q. Appl. Math. 69(2), 357–378 (2011a)
32. 32.
Ciuperca, I.S., Tello, J.I.: On a variational inequality on elasto-hydrodynamic lubrication. J. Math. Anal. Appl. 383(2), 597–607 (2011b)
33. 33.
Cryer, C.W.: The method of Christopherson for solving free boundary problems for infinite journal bearings by means of finite differences. Math. Comput. 25(115), 435–443 (1971)
34. 34.
Cuvelier, C.: A free boundary problem in hydrodynamic lubrication including surface tension. In: Cabannes, H., Holt, M., Rusanov, V. (eds.) Sixth International Conference on Numerical Methods in Fluid Dynamics. Lecture Notes in Physics, vol. 90, pp. 143–148. Springer, Berlin (1979)Google Scholar
35. 35.
Das, S., Guha, S.K., Chattopadhyay, A.K.: On the steady-state performance of misaligned hydrodynamic journal bearings lubricated with micropolar fluids. Tribol. Int. 35(4), 201–210 (2002)Google Scholar
36. 36.
Díaz, J.I., Tello, J.I.: A note on some inverse problems arising in lubrication theory. Differ. Integral. Equ. 17, 583–592 (2004)
37. 37.
Díaz, M., Lombera, H., et al.: An approach for assembly sequence planning based on MAX-MIN Ant System. IEEE Latin Am. Trans. 13(4), 907–912 (2015)Google Scholar
38. 38.
Dorigo, M.: Optimization, learning and natural algorithms. Italian. Ph.D. thesis. Dipartimento di Elettronica, Politecnico di Milano, Milan, Italy (1992)Google Scholar
39. 39.
Dorigo, M., Maniezzo, V., Colorni, A.: Ant system: optimization by a colony of cooperating agents. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 26(1), 29–41 (1996)Google Scholar
40. 40.
Dowson, D.: A generalized Reynolds equation for fluid film lubrication. Int. J. Mech. Sci. 4, 159–170 (1962)Google Scholar
41. 41.
Dubois, G.B., Mabie, H.H., Ocvirk, F.W.: Experimental investigation of oil film pressure distribution for misaligned plain bearings. Technical Report. National Advisory Committee for Aeronautics, KittyHawk, NC, USA (1951)Google Scholar
42. 42.
Durany, J., García, G., Vázquez, C.: Numerical simulation of a lubricated hertzian contact problem under imposed load. Finite. Elem. Anal. Des. 38(7), 645–658 (2002)
43. 43.
Durany, J., Pereira, J., Varas, F.: Numerical solution to steady and transient problems in thermohydrodynamic lubrication using a combination of finite element, finite volume and boundary element methods. Finite. Elem. Anal. Des. 44(11), 686–695 (2008)
44. 44.
Durany, J., Pereira, J., Varas, F.: Dynamical stability of journal-bearing devices through numerical simulation of thermohydrodynamic models. Tribol. Int. 43, 1703–1718 (2010)Google Scholar
45. 45.
Durany, J., Vázquez, C.: Numerical approach of lubrication problems in journal bearing devices with axial supply. Numer. Methods Eng. 92, 839–844 (1992)Google Scholar
46. 46.
El Alaoui Talibi, M., Bayada, G.: Une méthode du type caractéristique pour la résolution d’un problème de lubrification hydrodynamique en régime transitoire. ESAIM-Math. Model. Num. 25(4), 395–423 (1991)
47. 47.
Elrod, H.G., Adams, M.L.: A computer program for cavitation and starvation problems. Cavitation and related phenomena in lubrication: proceedings of the 1st Leeds-Lyon Symposium on Tribology. Mech. Eng. Publ. pp. 37–42 (1975)Google Scholar
48. 48.
Frêne, J., Nicolas, D., et al.: Hydrodynamic Lubrication: Bearings and Thrust Bearings, vol. 33. Elsevier, Amsterdam (1997)
49. 49.
Gambardella, L.M., Taillard, É., Agazzi, G.: MACS-VRPTW: a multiple ant colony system for vehicle routing problems with time windows. In: Corne, D., Dorigo, M., Glover, F. (eds.) New Ideas in Optimization (Chap. 5), pp. 63–76. McGraw-Hill, London (1999)Google Scholar
50. 50.
Garcia-Najera, A., Bullinaria, J.A.: Extending ACOR to solve multi-objective problems. In: Proceedings of the UK Workshop on Computational Intelligence, London (2007)Google Scholar
51. 51.
Gómez-Mancilla, J., Nosov, V.: Short journal bearing with misaligned axes. In: Proceedings 1st International Symposium on Control of Rotating Machinery (2001)Google Scholar
52. 52.
Gómez-Mancilla, J., Nosov, V.: Perturbed pressure field solution for misaligned short journal bearings. In: Proceedings 9th International Symposium on Transport Phenomena and Rotating Machinery. ISROMAC-9, Hawaii, EU (2002)Google Scholar
53. 53.
Guha, S.K.: Analysis of steady-state characteristics of misaligned hydrodynamic journal bearings with isotropic roughness effect. Tribol. Int. 33(1), 1–12 (2000)
54. 54.
He, Z., Zhang, J., et al.: Misalignment analysis of journal bearing influenced by asymmetric deflection, based on a simple stepped shaft model. J. Zhejiang Univ. Sci. A 13(9), 647–664 (2012)Google Scholar
55. 55.
Jang, J.Y., Khonsari, M.M.: Design of bearings on the basis of thermohydrodynamic analysis. In: Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, vol. 218, No. 5, pp. 355–363. Sage, Beverley Hills (2004)Google Scholar
56. 56.
Jang, J.Y., Khonsari, M.M.: On the characteristics of misaligned journal bearings. Lubricants 3(1), 27–53 (2015)Google Scholar
57. 57.
Jang, J.Y., Khonsari, M.M., Bair, S.: On the elastohydrodynamic analysis of shear-thinning fluids. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 463, No. 2088, pp. 3271–3290. The Royal Society (2007)Google Scholar
58. 58.
Khonsari, M.M., Booser, E.R.: Applied Tribology: Bearing Design and Lubrication, 2nd edn. Wiley, New York (2008). ISBN: 978-0-470-05711-7Google Scholar
59. 59.
Kumar, P., Khonsari, M.M.: Traction in EHL line contacts using free-volume pressureviscosity relationship with thermal and shear-thinning effects. J. Tribol. 131(1), 011503 (2009)Google Scholar
60. 60.
Leguizamón, G., Coello, C.A.: A study of the scalability of ACOR for continuous optimization problems. Technical report, The Evolutionary Computation Group at CINVESTAV-IPN (2010)Google Scholar
61. 61.
Leguizamón, G., Coello, C.A.: An alternative ACOR algorithm for continuous optimization problems. In: Swarm Intelligence: 7th International Conference, ANTS 2010, vol. 6234, pp. 48–59. Lecture Notes in Computer Science, Brussels, Belgium. Springer, Berlin (2010)Google Scholar
62. 62.
Liao, T.: Improved ant colony optimization algorithms for continuous and mixed discrete continuous optimization problems. MA thesis. Université Libre de Bruxelles, Belgium (2011)Google Scholar
63. 63.
Liao, T., Montes de Oca, M.A., et al.: An incremental ant colony algorithm with local search for continuous optimization. In: Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation, pp. 125–132. ACM, New York (2011)Google Scholar
64. 64.
Liao, T., Stützle, T., et al.: A unified ant colony optimization algorithm for continuous optimization. Euro. J. Oper. Res. 234(3), 597–609 (2014)
65. 65.
66. 66.
Liu, G.: Dual variational principles for the free-boundary problem of cavitated bearing lubrication. In: Gao, D.Y. (ed.) Complementarity, Duality and Symmetry in Nonlinear Mechanics: Proceedings of the IUTAM Symposium, vol. 6, pp. 179–189. Springer Science & Business Media (2004)Google Scholar
67. 67.
Lombera, H., Tello, J.I.: A numerical approach to solve an inverse problem in lubrication theory. RACSAM. Rev. R. Acad. A. 108(2), 617–631 (2014)
68. 68.
Martin, S.: Influence of multiscale roughness patterns in cavitated flows: applications to journal bearings. Math. Probl. Eng. 2008, 1–26 (2008)
69. 69.
Maru, M.M., Tanaka, D.K.: Consideration of stribeck diagram parameters in the investigation on wear and friction behavior in lubricated sliding. J. Braz. Soc. Mech. Sci. Eng. 29(1), 55–62 (2007)Google Scholar
70. 70.
McKee, S.A., McKee, T.R.: Pressure distribution in oil films of journal bearings. ASME 54, 149–165 (1932)Google Scholar
71. 71.
Nikolakopoulos, P.G., Papadopoulos, C.A.: Non-linearities in misaligned journal bearings. Tribol. Int. 27(4), 243–257 (1994)Google Scholar
72. 72.
Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer Science + Business Media, LLC, Berlin (2006)
73. 73.
Ocvirk, F.W., DuBois, G.B.: Analytical derivation and experimental evaluation of shortbearing approximation for full journal bearings. Technical Report 1157. NACA (1953)Google Scholar
74. 74.
Osman, T.A.: Misalignment effect on the static characteristics of magnetized journal bearing lubricated with ferrofluid. Tribol. Lett. 11(3–4), 195–203 (2001)Google Scholar
75. 75.
Pedraza, G., Díaz, M., Lombera, H.: An approach for assembly sequence planning by genetic algorithms. IEEE Latin Am. Trans. 14(5), 2066–2071 (2016)Google Scholar
76. 76.
Pierre, I., Bouyer, J., Fillon, M.: Thermohydrodynamic study of misaligned plain journal bearings-comparison between experimental data and theoretical results. Appl. Mech. Eng. 7(3), 949–960 (2002)Google Scholar
77. 77.
Pierre, I., de France, E., et al.: Thermohydrodynamic behavior of misaligned plain journal bearings: theoretical and experimental approaches. Tribol. Trans. 47(4), 594–604 (2004)Google Scholar
78. 78.
Piggott, R.J.S.: Bearings and lubrication. Bearing troubles traceable to design can be avoided by engineering study. Mech. Eng. 64, 259 (1942)Google Scholar
79. 79.
Piniganti, L.: A Survey of Tabu search in combinatorial optimization. MA thesis. University of Nevada, Las Vegas (2014)Google Scholar
80. 80.
Pinkus, O., Bupara, S.S.: Analysis of misaligned grooved journal bearings. J. Tribol. 101(4), 503–509 (1979)Google Scholar
81. 81.
Pinkus, O., Sternlicht, B.: Theory of Hydrodynamic Lubrication. McGraw-Hill, New York (1961)
82. 82.
Reynolds, O.: On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, including an experimental determination of the viscosity of olive oil. Phil. Trans. Roy. Soc. A 117, 157–234 (1886)
83. 83.
Rodrigues, J.F.: Remarks on the Reynolds problem of elastohydrodynamic lubrication. Eur. J. Appl. Math. 4(01), 83–96 (1993)
84. 84.
Rohde, S.M., McAllister, G.T.: A variational formulation for a class of free boundary problems arising in hydrodynamic lubrication. Int. J. Eng. Sci. 13(9–10), 841–850 (1975)
85. 85.
Sharma, S.C., Jain, S.C., Nagaraju, T.: Combined influence of journal misalignment and surface roughness on the performance of an orifice compensated non-recessed hybrid journal bearing. Tribol. Trans. 45(4), 457–463 (2002)Google Scholar
86. 86.
Siarry, Patrick, Berthiau, Gérard, et al.: Enhanced simulated annealing for globally minimizing functions of many-continuous variables. ACM Trans. Math. Softw. (TOMS) 23(2), 209–228 (1997)
87. 87.
Socha, K.: Ant colony optimization for continuous and mixed-variable domains. PhD thesis. Université Libre de Bruxelles, Belgium (2008)Google Scholar
88. 88.
Socha, K., Dorigo, M.: Ant colony optimization for continuous domains. Euro. J. Oper. Res. 185(3), 1155–1173 (2008)
89. 89.
Srinivas, M., Patnaik, L.M.: Genetic algorithms: a survey. Computer 27(6), 17–26 (1994)Google Scholar
90. 90.
Stachowiak, G., Batchelor, A.W.: Engineering Tribology. Butterworth-Heinemann, Oxford (2013)Google Scholar
91. 91.
Stieber, W.: Das Schwimmlager. Technical Report Ver. Dtsch. Ing. Berlin (1933)Google Scholar
92. 92.
Stützle, T., Hoos, H.: MAX-MIN Ant System. Futur. Gener. Comput. Syst. 16(8), 889–914 (2000)
93. 93.
Sun, J., Deng, M., et al.: Thermohydrodynamic lubrication. Analysis of misaligned plain journal bearing with rough surface. J. Tribol. 132(1), 011704 (2010)Google Scholar
94. 94.
Sun, J., Zhu, X., et al.: Effect of surface roughness, viscosity-pressure relationship and elastic deformation on lubrication performance of misaligned journal bearings. Ind. Lubr. Tribol. 66(3), 337–345 (2014)Google Scholar
95. 95.
Swift, H.W.: The stability of lubricating films in journal bearings. J. Inst. Civ. Eng. 233(Part 1), 267–288 (1932)Google Scholar
96. 96.
Thomsen, K., Klit, P.: Improvement of journal bearing operation at heavy misalignment using bearing flexibility and compliant liners. Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 226(8), 651–660 (2012)Google Scholar
97. 97.
Vijayaraghavan, D., Keith, T.G.: Effect of cavitation on the performance of a grooved misaligned journal bearing. Wear 134(2), 377–397 (1989)Google Scholar
98. 98.
Vijayaraghavan, D., Keith, T.G.: Analysis of a finite grooved misaligned journal bearing considering cavitation and starvation effects. J. Tribol-T. ASME. 112(1), 60–67 (1990)Google Scholar
99. 99.
Xu, G., Zhou, J., et al.: Research on the static and dynamic characteristics of misaligned journal bearing considering the turbulent and thermohydrodynamic effects. J. Tribol. 137(2), 024504 (2015)Google Scholar
100. 100.
Zhang, Y., Wang, S., Ji, G.: A comprehensive survey on particle swarm optimization algorithm and its applications. Math. Probl. Eng. 2015 (2015)Google Scholar

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## Authors and Affiliations

1. 1.Centro de Informática IndustrialUniversidad de las Ciencias InformáticasLa HabanaCuba