Two-dimensional Hydrodynamic Pad Bearing Design Optimization for Maximum Load Capacity under the Navier Slip Boundary Condition on its Stationary Surface

Document Type : Research Paper

Authors
Ihssane Messoude 1
Samir Aghzer 1 , 2
1 Laboratory of Mechanics, Engineering and Innovation, National Higher School of Electricity and Mechanics, Hassan II University of Casablanca, Morocco
2 Higher School of Technology, Hassan II University of Casablanca, Morocco
10.22055/jacm.2026.48813.5515
Abstract
This work focuses on the influence of the Navier slip/adhesion boundary condition on the hydrodynamic performances of a two-dimensional non-conventional pad bearing. The modified Reynolds equation governing the hydrodynamic behavior of such a bearing, in which the slip condition is assumed to be partially or fully applied on its stationary surface, is established. The effect induced by the convergence ratio, slip coefficient, and slip zone extent and position on the pad bearing’s performances, including load capacity, friction force on the moving surface, and required lubrication flow rate, is evaluated and discussed. The numerical results show that introducing the slip/adhesion condition significantly affects the hydrodynamic performances of a non-conventional pad bearing. Notably, it can support a load even in a divergent configuration or with parallel surfaces. By applying a constrained optimization procedure based on the modified Hooke and Jeeves method, which proves to be highly effective, the optimal values of the convergence ratio, the slip coefficient, and the slip zone extent and position that maximize the load capacity are determined. The optimal slip zone is shaped as a rectangle located at the hydrodynamic contact inlet. A comparative analysis between the optimal non-conventional pad bearing and the conventional configuration is carefully conducted. The impact of the slip/adhesion condition on the slip velocity fields and the flow velocity distribution within the lubricant film is also investigated and discussed. Finally, comparisons with results from the literature confirm the relevance of this study.
Keywords
Hydrodynamic pad bearing
Navier slip boundary condition
Modified Reynolds equation
Slip coefficient
Optimization
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Dadouche, A., Etude des phénomènes thermiques dans les butées hydrodynamiques, Thèse de Doctorat de l’Université de Poitiers, France, 1998.
[2] Frene, J., Nicolas, D., Degueurce, B., Berthe, D., Godet, M., Lubrification hydrodynamique. Paliers et Butées, Collection des Etudes et Recherche d’Electricité de France, Eyrolles, Paris, 1990.
[3] Liravi, M., Pakzad, H., Moosavi, A., Nouri-Borujerdi, A., A comprehensive review on recent advances in superhydrophobic surfaces and their applications for drag reduction, Progress in Organic Coatings, 140, 2020, 105537. 
[4] Navier, C., Mémoire sur les lois du mouvement des fluides, Mémoires de l’Académie Royale des Sciences de l’Institut de France, 1822, 389-440. 
[5] Maxwell, J. C., On stresses in rarified gases arising from inequalities of temperature, Philosophical Transactions of the Royal Society of London, 170, 1879, 231-256.
[6] Yang, K., Tang, J., Huang, J., Zhang, H., Chen, H., Xiong, Y., Chen, H., Hydrophobic and tribological properties of biomimetic interfaces, Coatings, 14(5), 2024, 529.
[7] Thompson, P. A., Troian, S. M., A general boundary condition for liquid flow at solid surfaces, Nature, 389, 1997, 360-362. 
[8] Tabeling, P., Phénomènes de glissement à l’interface liquide-solide, Comptes Rendus Physique, 5(5), 2004, 531–537. 
[9] Vinogradova, O. I., Slippage of water over hydrophobic surfaces, International Journal of Mineral Processing, 56(1), 1999, 31-60. 
[10] Charlaix, E., Cottin-Bizonne, C., Crassoux, J., Leroy, S., Restagno, F., Steingerger, A., Propriétés hydrodynamiques au voisinage d’une surface, Reflets de la Physique, 22, 2011, 14-18. 
[11] Sochi, T., Slip at Fluid-Solid Interface, Polymer Reviews, 51(4), 2011, 309-340. 
[12] Pit, R., Hervet, H., Léger, L., Friction and slip of a simple liquid at a solid surface, Tribology Letters, 7, 1999, 147-152. 
[13] Henry, C. L., Craig, V. S. J., Measurement of no-slip and slip boundary conditions in confined Newtonian fluids using atomic force microscopy, Physical Chemistry Chemical Physics, 11(41), 2009, 9514-9521. 
[14] Spikes, H., Granick, S., Equation for slip of simple liquids at smooth solid surfaces, Langmuir, 19(12), 2003, 5065-5071. 
[15] Spikes, H. A., The half-wetted bearing. Part 1: extended Reynolds equation, Proceedings of the Institution of Mechanical Engineers, Part J:  Journal of Engineering Tribology, 217(1), 2003, 1-14. 
[16] Wu, C. W., Ma, G. J., Zhou, P., Low friction and high load support capacity of slider bearing with a mixed slip surface, Journal of Tribology, 128(4), 2006, 904-907. 
[17] Messoude, I., Aghzer, S., Numerical Study of the Effects of the Navier Slip Boundary Condition on the Hydrodynamic Performances of a Newtonian Fluid-lubricated Pad Bearing, Journal of Applied and Computational Mechanics, 11(3), 2025, 754-769. 
[18] Tauviqirrahman, M., Ismail, R., Jamari, J., Schipper, DJ., Friction reduction in lubricated-MEMS with complex slip surface pattern, Procedia Engineering, 68, 2013, 331-337. 
[19] Savio, D., Falk, K., Moseler, M., Slipping domains in water-lubricated microsystems for improved load support, Tribology International, 120, 2018, 269-279. 
[20] Lin, Q., Wei, Z., Tang, Y., Numerical study on shear flow in sliding bearing with partial slip surface, Procedia CIRP, 3, 2012, 197-202. 
[21] Salant, R. F., Fortier, A. E., Numerical analysis of a slider bearing with a heterogeneous slip/no-slip surface, Tribology Transactions, 47(3), 2004, 328-334.
[22] Zidane, I., Zahloul, H., Hajjam, M., Abbadeni, M., Modeling and analysis of the slip conditions in hydrodynamic lubrication, Arabian Journal for Science and Engineering, 39, 2014, 7199-7210. 
[23] Guo, L., Wong, P. L., Guo, F., Effects of viscosity and sliding speed on boundary slippage in thin film hydrodynamic lubrication, Tribology International, 107, 2017, 85–93. 
[24] Sun, B. W., Chen, L., Guo, L., Wang, W., Wong, P. L., Experimental Evidence on the Enhancement of Bearing Load Capacity by Localised Boundary Slip Effect, Tribology Letters, 69, 2021, 1-8.
[25] Rao, T. V. V. L. N., Analysis of single-grooved slider and journal bearing with partial slip surface, Journal of Tribology, 132(1), 2010, 014501. 
[26] Çam, M. Y., Giacopini, M., Dini, D., Biancofiore, L., A numerical algorithm to model wall slip and cavitation in two-dimensional hydrodynamically lubricated contacts, Tribology International, 184, 2023, 108444. 
[27] Wu, C., Performance of hydrodynamic lubrication journal bearing with a slippage surface, Industrial Lubrication and Tribology, 60(6), 2008, 293–298. 
[28] Cui, S., Zhang, C., Fillon, M., Gu, L., Optimization performance of plain journal bearings with partial wall slip, Tribology International, 145, 2020, 106137. 
[29] Xie, Z. L., Ta, N., Rao, Z. S., The lubrication performance of water lubricated bearing with consideration of wall slip and inertial force, Journal of Hydrodynamics, Ser. B, 29(1), 2017, 52-60. 
[30] Tauviqirrahman, M., Ismail, R., Schipper, D. J., Optimization of the complex slip surface and its effect on the hydrodynamic performance of two-dimensional lubricated contacts, Computers and Fluids, 79, 2013, 27–43. 
[31] Zhang, H., Liu, Y., Dai, S., Li, F., Dong, G., Optimization of boundary slip region on bearing sliders to improve tribological performance, Tribology International, 168, 2022, 107446. 
[32] Zhang, W., Zhu, B., Optimization design for slip/no-slip configuration of hydrophobic sliding bearings using Monte Carlo search, Tribology International, 178(A), 2023, 108034. 
[33] Minhas, M. S., Awasthi, R. K., Influence of pentagonal texture on performance of slider bearing, International Journal of Advance Research and Innovation, 4(4), 2016, 669-675. 
[34] Singh, D., Singh, N., Awasthi, R. K., Effect of surface texture parameters on the performance of finite slider bearing, Materials Today: Proceedings, 5(9), 2018, 19349-19358. 
[35] Majumdar, B. C., Introduction to Tribology of Bearings, S. Chand and Co., New Delhi, 2008.
[36] Rao, S. S., Optimization: theory and applications, 2nd Ed., Wiley Eastern, New Delhi, 1984.
[37] Aghzer, S., Contribution à l’optimisation des paliers hydrodynamiques, Thèse de Doctorat de l’Université de Poitiers, France, 1998.
Journal of Applied and Computational Mechanics
Volume 12, Issue 3
Summer 2026
Pages 1124-1148