About a cavitation model including bubbles in thin film lubrication taking convection into account
Authors:
Guy Bayada and Ionel Sorin Ciuperca
Journal:
Quart. Appl. Math. 80 (2022), 237-257
MSC (2020):
Primary 35A01, 35B35, 35Q35, 76D08, 76B10
DOI:
https://doi.org/10.1090/qam/1609
Published electronically:
January 18, 2022
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Additional Information
Abstract: In lubrication problems, which concern thin film flow, cavitation has been considered as a fundamental element to correctly describe the characteristics of lubricated mechanisms. This cavitation model consists of a coupled problem between the compressible Reynolds PDE (that describes the flow) and the Rayleigh-Plesset ODE (that describes micro-bubbles evolution). Very few theoretical results exist in the mathematical literature about such couple problems. A complete form including bubbles convection is studied here. Local times existence results are proved based on the semi group theory. Stability theorems are obtained in a particular case.
References
- U. H. Augsdorfer, A. K. Evans, and D. P. Oxley, Thermal noise and the stability of single sonoluminescing bubbles, Phys. Rev. E. (2000), no. 61, 5278–5286.
- G. Bayada and M. Chambat, Analysis of a free boundary problem in partial lubrication, Quart. Appl. Math. 40 (1982/83), no. 4, 369–375. MR 693872, DOI 10.1090/S0033-569X-1983-0693872-8
- Guy Bayada, Sébastien Martin, and Carlos Vázquez, About a generalized Buckley-Leverett equation and lubrication multifluid flow, European J. Appl. Math. 17 (2006), no. 5, 491–524. MR 2296026, DOI 10.1017/S0956792506006668
- A. Bermúdez and J. Durany, Numerical solution of cavitation problems in lubrication, Proceedings of the Eighth International Conference on Computing Methods in Applied Sciences and Engineering (Versailles, 1987), 1989, pp. 457–466. MR 1035758, DOI 10.1016/0045-7825(89)90041-8
- M. Braun and W. Hannon, Cavitation formation and modelling for fluid film bearings: a review, J. Eng. Tribol. 224 (2010), 839–863.
- Christopher Earls Brennen, Cavitation and bubble dynamics, Cambridge University Press, New York, 2014. Corrected edition of the 1995 original. MR 3155605
- M. P. Brenner, S. Hilgenfeldt, and D. Lohse, Single-bubble sonoluminescence, Rev. Modern Phys. (2002), no. 74, 425–484.
- M. P. Brenner, D. Lohse, and T. F. Dupont, Bubble shape oscillations and the onset of sonoluminescence, Phys. Rev. Lett., 1995, no. 75, 954–957.
- Giovanni Cimatti, On a problem of the theory of lubrication governed by a variational inequality, Appl. Math. Optim. 3 (1976/77), no. 2-3, 227–242. MR 449171, DOI 10.1007/BF01441967
- G. Cimatti, A free boundary problem in the theory of lubrication, Internat. J. Engrg. Sci. 18 (1980), no. 5, 703–711. MR 660178, DOI 10.1016/0020-7225(80)90104-4
- D. Dowdon and C. Taylor, Cavitation in bearings, Annu. Rev. Fluid Mech. 11 (1979), 35–66.
- Donald A. Drew and Stephen L. Passman, Theory of multicomponent fluids, Applied Mathematical Sciences, vol. 135, Springer-Verlag, New York, 1999. MR 1654261, DOI 10.1007/b97678
- J. Franc, The Rayleigh-Plesset equation: a simple and powerful tool to understand various aspects of cavitation, Springer Vienna, Vienna, 2007, pp. 1–41.
- Tadahisa Funaki, Masashi Ohnawa, Yukihito Suzuki, and Satoshi Yokoyama, Existence and uniqueness of solutions to stochastic Rayleigh-Plesset equations, J. Math. Anal. Appl. 425 (2015), no. 1, 20–32. MR 3299647, DOI 10.1016/j.jmaa.2014.12.018
- J. Gehannin, M. Arghir, and O. Bonneau, Evaluation of Rayleigh-Plesset equation based cavitation models for squeeze film dampers, J. Tribol. 131 (2009), no. 2, 024501.
- T. Geike and V. Popov, A bubble dynamics based approach to the simulation of cavitation in lubricated contacts, J. Tribol. 131 (2009), no. 1, 011704.
- T. Geike and V. Popov, Cavitation within the framework of reduced description of mixed lubrication, Tribol. Int. 42 (2009), no. 1, 93–98.
- Robert Hakl, Pedro J. Torres, and Manuel Zamora, Periodic solutions of singular second order differential equations: upper and lower functions, Nonlinear Anal. 74 (2011), no. 18, 7078–7093. MR 2833696, DOI 10.1016/j.na.2011.07.029
- Robert Hakl, Pedro J. Torres, and Manuel Zamora, Periodic solutions to singular second order differential equations: the repulsive case, Topol. Methods Nonlinear Anal. 39 (2012), no. 2, 199–220. MR 2985878
- V. Hoang Man, M. Suan Li, P. Derreumaux, and P. H. Nguyen, Rayleigh-Plesset equation of the bubble stable cavitation in water: a nonequilibrium all-atom molecular dynamics simulation study, J. Chem. Phys. 148 (2018), no. 9.
- J. Holzfuss, Surface-wave instabilities, period doubling, and an approximate universal boundary of bubble stability at the upper threshold of sonoluminescence, Phys. Rev. E (2008), no. 77.
- Alfredo Jaramillo, Guy Bayada, Ionel Ciuperca, and Mohammed Jai, About a cavitation model including bubbles in thin film lubrication: a first mathematical analysis, European J. Appl. Math. 31 (2020), no. 5, 828–853. MR 4139913, DOI 10.1017/s0956792519000287
- A. Jaramillo and G. Buscaglia, A stable numerical strategy for Reynolds-Rayleigh-Plesset coupling, Tribol. Int. 130 (2019), 191–205.
- T. Kawase and T. Someya, Investigation into the oil film pressure distribution in dynamically loaded journal bearings, Trans. Japan Soc. Mech. Eng. Ser. C 51 (1985), no. 470, 2562–2570.
- David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
- A. Kubota, H. Kato, and H. Yamaguchi, A new modelling of cavitating flows: a numerical study of unsteady cavitation on a hydrofoil section, J. Fluid Mech. 240 (1992), 59–96.
- Lakshmi Burra and Fabio Zanolin, Non-singular solutions of a Rayleigh-Plesset equation under a periodic pressure field, J. Math. Anal. Appl. 435 (2016), no. 2, 1364–1381. MR 3429647, DOI 10.1016/j.jmaa.2015.11.016
- F. Meng, L. Zhang, and T. Long, Effect of Groove textures on the performances of gaseous bubble in the lubricant of journal bearing, J. Tribol. 139 (2016), no. 3, 031701.
- F. Migout, Etude théorique et expérimentale du changement de phase dans l’interface des garnitures mécaniques d’étanchéité, Ph.D. Thesis, University of Poitiers (France), 2010.
- S. Natsumeda and T. Someya, Paper iii(ii) negative pressures in statically and dynamically loaded journal bearings, Fluid Film Lubrication - Osborne Reynolds Centenary, Tribology Series, vol. 11, Elsevier, 1987, pp. 65–72.
- Masashi Ohnawa and Yukihito Suzuki, Mathematical and numerical analysis of the Rayleigh-Plesset and the Keller equations, Mathematical fluid dynamics, present and future, Springer Proc. Math. Stat., vol. 183, Springer, Tokyo, 2016, pp. 159–180. MR 3613761, DOI 10.1007/978-4-431-56457-7_{7}
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- M. W. Plesset, The dynamic of cavitation bubbles, J. Appl. Mech. 16 (1949), 277–282.
- G. H. Schnerr and J. Sauer, Physical and numerical modeling of unsteady cavitation dynamics, Fourth International Conference on Multiphase Flow, 2001.
- A. K. Singhal, H. Y. Li, M. M. Athavale, and Y. Jiang, Mathematical basis and validation of the full cavitation model, J. Fluids Eng. 124 (2001), no. 3, 617–624.
- T. A. Snyder, M. J. Braun, and K. C. Pierson, Two-way coupled Reynolds and Rayleigh-Plesset equations for a fully transient, multiphysics cavitation model with pseudo-cavitation, Tribol. Int. 93 (2016), 429–445.
- T. Someya, On the development of negative pressure in oil film and the characteristics of journal bearing, Meccanica, 38 (2003), no. 6, 643–658.
- L. Xia, Analysis of the integrability of the R.P. equation with Painlevé test and the Symmetry group, ArXiv, (1809 07479), 2018.
- C. Xing, M. J. Braun, and H. Li, A 3-dimensional Navier-Stokes based numerical model for squeeze film dampers. Part 1: effects of gaseous cavitation on pressure distribution and damping coefficients without consideration of inertia, Tribol. Trans. 52 (2009), 680–694.
- Y. Hao and A. Prosperetti, The effect of viscosity on the spherical stability of oscillating gas bubbles, Phys. Fluids 11 (1999), no. 6, 1309–1317. MR 1690510, DOI 10.1063/1.869996
- K. Yasui, Effects of thermal conduction on bubble dynamics near the sonoluminescence threshold, J. Acoust. Soc. Am. (1995), no. 98, 2772–2782.
- P. J. Zwart, A. G. Gerber, and T. Belarmi, A two-phase flow model for predicting cavitation dynamics, Fifth International Conference on Multiphase Flow, 2004.
References
- U. H. Augsdorfer, A. K. Evans, and D. P. Oxley, Thermal noise and the stability of single sonoluminescing bubbles, Phys. Rev. E. (2000), no. 61, 5278–5286.
- G. Bayada and M. Chambat, Analysis of a free boundary problem in partial lubrication, Quart. Appl. Math. 40 (1982/83), no. 4, 369–375. MR 693872, DOI 10.1090/qam/693872
- Guy Bayada, Sébastien Martin, and Carlos Vázquez, About a generalized Buckley-Leverett equation and lubrication multifluid flow, European J. Appl. Math. 17 (2006), no. 5, 491–524. MR 2296026, DOI 10.1017/S0956792506006668
- A. Bermúdez and J. Durany, Numerical solution of cavitation problems in lubrication, Proceedings of the Eighth International Conference on Computing Methods in Applied Sciences and Engineering (Versailles, 1987), 1989, pp. 457–466. MR 1035758, DOI 10.1016/0045-7825(89)90041-8
- M. Braun and W. Hannon, Cavitation formation and modelling for fluid film bearings: a review, J. Eng. Tribol. 224 (2010), 839–863.
- Christopher Earls Brennen, Cavitation and bubble dynamics, Cambridge University Press, New York, 2014. Corrected edition of the 1995 original. MR 3155605
- M. P. Brenner, S. Hilgenfeldt, and D. Lohse, Single-bubble sonoluminescence, Rev. Modern Phys. (2002), no. 74, 425–484.
- M. P. Brenner, D. Lohse, and T. F. Dupont, Bubble shape oscillations and the onset of sonoluminescence, Phys. Rev. Lett., 1995, no. 75, 954–957.
- Giovanni Cimatti, On a problem of the theory of lubrication governed by a variational inequality, Appl. Math. Optim. 3 (1976/77), no. 2-3, 227–242. MR 449171, DOI 10.1007/BF01441967
- G. Cimatti, A free boundary problem in the theory of lubrication, Internat. J. Engrg. Sci. 18 (1980), no. 5, 703–711. MR 660178, DOI 10.1016/0020-7225(80)90104-4
- D. Dowdon and C. Taylor, Cavitation in bearings, Annu. Rev. Fluid Mech. 11 (1979), 35–66.
- Donald A. Drew and Stephen L. Passman, Theory of multicomponent fluids, Applied Mathematical Sciences, vol. 135, Springer-Verlag, New York, 1999. MR 1654261, DOI 10.1007/b97678
- J. Franc, The Rayleigh-Plesset equation: a simple and powerful tool to understand various aspects of cavitation, Springer Vienna, Vienna, 2007, pp. 1–41.
- Tadahisa Funaki, Masashi Ohnawa, Yukihito Suzuki, and Satoshi Yokoyama, Existence and uniqueness of solutions to stochastic Rayleigh-Plesset equations, J. Math. Anal. Appl. 425 (2015), no. 1, 20–32. MR 3299647, DOI 10.1016/j.jmaa.2014.12.018
- J. Gehannin, M. Arghir, and O. Bonneau, Evaluation of Rayleigh-Plesset equation based cavitation models for squeeze film dampers, J. Tribol. 131 (2009), no. 2, 024501.
- T. Geike and V. Popov, A bubble dynamics based approach to the simulation of cavitation in lubricated contacts, J. Tribol. 131 (2009), no. 1, 011704.
- T. Geike and V. Popov, Cavitation within the framework of reduced description of mixed lubrication, Tribol. Int. 42 (2009), no. 1, 93–98.
- Robert Hakl, Pedro J. Torres, and Manuel Zamora, Periodic solutions of singular second order differential equations: upper and lower functions, Nonlinear Anal. 74 (2011), no. 18, 7078–7093. MR 2833696, DOI 10.1016/j.na.2011.07.029
- Robert Hakl, Pedro J. Torres, and Manuel Zamora, Periodic solutions to singular second order differential equations: the repulsive case, Topol. Methods Nonlinear Anal. 39 (2012), no. 2, 199–220. MR 2985878
- V. Hoang Man, M. Suan Li, P. Derreumaux, and P. H. Nguyen, Rayleigh-Plesset equation of the bubble stable cavitation in water: a nonequilibrium all-atom molecular dynamics simulation study, J. Chem. Phys. 148 (2018), no. 9.
- J. Holzfuss, Surface-wave instabilities, period doubling, and an approximate universal boundary of bubble stability at the upper threshold of sonoluminescence, Phys. Rev. E (2008), no. 77.
- Alfredo Jaramillo, Guy Bayada, Ionel Ciuperca, and Mohammed Jai, About a cavitation model including bubbles in thin film lubrication: a first mathematical analysis, European J. Appl. Math. 31 (2020), no. 5, 828–853. MR 4139913, DOI 10.1017/s0956792519000287
- A. Jaramillo and G. Buscaglia, A stable numerical strategy for Reynolds-Rayleigh-Plesset coupling, Tribol. Int. 130 (2019), 191–205.
- T. Kawase and T. Someya, Investigation into the oil film pressure distribution in dynamically loaded journal bearings, Trans. Japan Soc. Mech. Eng. Ser. C 51 (1985), no. 470, 2562–2570.
- David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
- A. Kubota, H. Kato, and H. Yamaguchi, A new modelling of cavitating flows: a numerical study of unsteady cavitation on a hydrofoil section, J. Fluid Mech. 240 (1992), 59–96.
- Lakshmi Burra and Fabio Zanolin, Non-singular solutions of a Rayleigh-Plesset equation under a periodic pressure field, J. Math. Anal. Appl. 435 (2016), no. 2, 1364–1381. MR 3429647, DOI 10.1016/j.jmaa.2015.11.016
- F. Meng, L. Zhang, and T. Long, Effect of Groove textures on the performances of gaseous bubble in the lubricant of journal bearing, J. Tribol. 139 (2016), no. 3, 031701.
- F. Migout, Etude théorique et expérimentale du changement de phase dans l’interface des garnitures mécaniques d’étanchéité, Ph.D. Thesis, University of Poitiers (France), 2010.
- S. Natsumeda and T. Someya, Paper iii(ii) negative pressures in statically and dynamically loaded journal bearings, Fluid Film Lubrication - Osborne Reynolds Centenary, Tribology Series, vol. 11, Elsevier, 1987, pp. 65–72.
- Masashi Ohnawa and Yukihito Suzuki, Mathematical and numerical analysis of the Rayleigh-Plesset and the Keller equations, Mathematical fluid dynamics, present and future, Springer Proc. Math. Stat., vol. 183, Springer, Tokyo, 2016, pp. 159–180. MR 3613761, DOI 10.1007/978-4-431-56457-7_7
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- M. W. Plesset, The dynamic of cavitation bubbles, J. Appl. Mech. 16 (1949), 277–282.
- G. H. Schnerr and J. Sauer, Physical and numerical modeling of unsteady cavitation dynamics, Fourth International Conference on Multiphase Flow, 2001.
- A. K. Singhal, H. Y. Li, M. M. Athavale, and Y. Jiang, Mathematical basis and validation of the full cavitation model, J. Fluids Eng. 124 (2001), no. 3, 617–624.
- T. A. Snyder, M. J. Braun, and K. C. Pierson, Two-way coupled Reynolds and Rayleigh-Plesset equations for a fully transient, multiphysics cavitation model with pseudo-cavitation, Tribol. Int. 93 (2016), 429–445.
- T. Someya, On the development of negative pressure in oil film and the characteristics of journal bearing, Meccanica, 38 (2003), no. 6, 643–658.
- L. Xia, Analysis of the integrability of the R.P. equation with Painlevé test and the Symmetry group, ArXiv, (1809 07479), 2018.
- C. Xing, M. J. Braun, and H. Li, A 3-dimensional Navier-Stokes based numerical model for squeeze film dampers. Part 1: effects of gaseous cavitation on pressure distribution and damping coefficients without consideration of inertia, Tribol. Trans. 52 (2009), 680–694.
- Y. Hao and A. Prosperetti, The effect of viscosity on the spherical stability of oscillating gas bubbles, Phys. Fluids 11 (1999), no. 6, 1309–1317. MR 1690510, DOI 10.1063/1.869996
- K. Yasui, Effects of thermal conduction on bubble dynamics near the sonoluminescence threshold, J. Acoust. Soc. Am. (1995), no. 98, 2772–2782.
- P. J. Zwart, A. G. Gerber, and T. Belarmi, A two-phase flow model for predicting cavitation dynamics, Fifth International Conference on Multiphase Flow, 2004.
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Additional Information
Guy Bayada
Affiliation:
Université de Lyon, CNRS, INSA de Lyon, Institut Camille Jordan UMR 5208, F-69621 Villeurbanne, France
MR Author ID:
32875
Email:
guy.bayada@gmail.com
Ionel Sorin Ciuperca
Affiliation:
Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan UMR 5208, F-69622 Villeurbanne, France
MR Author ID:
605812
Email:
ciuperca@math.univ-lyon1.fr
Keywords:
Cavitation modeling,
thin film lubrication,
Reynolds equation,
Rayleigh-Plesset equation
Received by editor(s):
June 7, 2021
Received by editor(s) in revised form:
December 14, 2021
Published electronically:
January 18, 2022
Article copyright:
© Copyright 2022
Brown University
