On derivation of compressible fluid systems on an evolving surface
Author:
Hajime Koba
Journal:
Quart. Appl. Math. 76 (2018), 303-359
MSC (2010):
Primary 37E35, 49S05, 37D35
DOI:
https://doi.org/10.1090/qam/1491
Published electronically:
October 13, 2017
MathSciNet review:
3769898
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Abstract: We consider the governing equations for the motion of compressible fluid on an evolving surface from both energetic and thermodynamic points of view. We employ our energetic variational approaches to derive the momentum equation of our compressible fluid systems on the evolving surface. Applying the first law of thermodynamics and the Gibbs equation, we investigate the internal energy, enthalpy, entropy, and free energy of the fluid on the evolving surface. We also study conservative forms and conservation laws of our compressible fluid systems on the evolving surface. Moreover, we derive the generalized heat and diffusion systems on an evolving surface from an energetic point of view. This paper gives a mathematical validity of the surface stress tensor determined by the Boussinesq-Scriven law. Using a flow map on an evolving surface and applying the Riemannian metric induced by the flow map are key ideas to analyze fluid flow on the evolving surface.
References
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- David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102–163. MR 271984, DOI https://doi.org/10.2307/1970699
- Renée Gatignol and Roger Prud’homme, Mechanical and thermodynamical modeling of fluid interfaces. World Scientific, Singapore, 2001. xviii,+248 pp. ISBN=9810243057.
- Morton E. Gurtin, Allan Struthers, and William O. Williams, A transport theorem for moving interfaces, Quart. Appl. Math. 47 (1989), no. 4, 773–777. MR 1031691, DOI https://doi.org/10.1090/qam/1031691
- Morton E. Gurtin, Eliot Fried, and Lallit Anand, The mechanics and thermodynamics of continua, Cambridge University Press, Cambridge, 2010. MR 2884384
- Jürgen Jost, Riemannian geometry and geometric analysis, 6th ed., Universitext, Springer, Heidelberg, 2011. MR 2829653
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- Hajime Koba and Kazuki Sato, Energetic variational approaches for non-Newtonian fluid systems, preprint, arXiv:1705.06956
- Yoshihiko Mitsumatsu and Yasuhisa Yano, Geometry of an incompressible fluid on a Riemannian manifold, Sūrikaisekikenkyūsho K\B{o}kyūroku 1260 (2002), 33–47 (Japanese). Geometric mechanics (Japanese) (Kyoto, 2002). MR 1930362
- Lars Onsager Reciprocal Relations in Irreversible Processes. I. Physical Review. (1931);37:405-109 DOI:https://doi.org/10.1103/PhysRev.37.405
- Lars Onsager Reciprocal Relations in Irreversible Processes. II. Physical Review. (1931);38:2265-79 DOI:https://doi.org/10.1103/PhysRev.38.2265
- L.E. Scriven, Dynamics of a fluid interface Equation of motion for Newtonian surface fluids. Chem. Eng. Sci. 12 (1960), 98–108.
- James Serrin, Mathematical principles of classical fluid mechanics, Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959, pp. 125–263. MR 0108116
- John C. Slattery, Momentum and moment-of-momentum balances for moving surfacesChemical Engineering Science, Volume 19, 1964, Pages 379–385.
- John C. Slattery, Leonard Sagis, and Eun-Suok Oh, Interfacial transport phenomena, 2nd ed., Springer, New York, 2007. MR 2284654
- Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
- J. W. Strutt, Some General Theorems relating to Vibrations, Proc. Lond. Math. Soc. 4 (1871/73), 357–368. MR 1575554, DOI https://doi.org/10.1112/plms/s1-4.1.357
- Michael E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1407–1456. MR 1187618, DOI https://doi.org/10.1080/03605309208820892
References
- Marc Arnaudon and Ana Bela Cruzeiro, Lagrangian Navier-Stokes diffusions on manifolds: variational principle and stability, Bull. Sci. Math. 136 (2012), no. 8, 857–881. MR 2995006, DOI https://doi.org/10.1016/j.bulsci.2012.06.007
- V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 319–361 (French). MR 0202082
- V. I. Arnol′d, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, [1989?]. Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein; Corrected reprint of the second (1989) edition. MR 1345386
- David E. Betounes, Kinematics of submanifolds and the mean curvature normal, Arch. Rational Mech. Anal. 96 (1986), no. 1, 1–27. MR 853973, DOI https://doi.org/10.1007/BF00251411
- Dieter Bothe and Jan Prüss, On the two-phase Navier-Stokes equations with Boussinesq-Scriven surface fluid, J. Math. Fluid Mech. 12 (2010), no. 1, 133–150. MR 2602917, DOI https://doi.org/10.1007/s00021-008-0278-x
- M. J. Boussinesq, Sur l’existence d’une viscosité seperficielle, dans la mince couche de transition séparant un liquide d’un autre fluide contigu, Ann. Chim. Phys. 29 (1913), 349–357.
- Philippe G. Ciarlet, An introduction to differential geometry with application to elasticity, J. Elasticity 78/79 (2005), no. 1-3, iv+215. With a foreword by Roger Fosdick. MR 2196098, DOI https://doi.org/10.1007/s10659-005-4738-8
- G. Dziuk and C. M. Elliott, Finite elements on evolving surfaces, IMA J. Numer. Anal. 27 (2007), no. 2, 262–292. MR 2317005, DOI https://doi.org/10.1093/imanum/drl023
- David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid., Ann. of Math. (2) 92 (1970), 102–163. MR 0271984, DOI https://doi.org/10.2307/1970699
- Renée Gatignol and Roger Prud’homme, Mechanical and thermodynamical modeling of fluid interfaces. World Scientific, Singapore, 2001. xviii,+248 pp. ISBN=9810243057.
- Morton E. Gurtin, Allan Struthers, and William O. Williams, A transport theorem for moving interfaces, Quart. Appl. Math. 47 (1989), no. 4, 773–777. MR 1031691, DOI https://doi.org/10.1090/qam/1031691
- Morton E. Gurtin, Eliot Fried, and Lallit Anand, The mechanics and thermodynamics of continua, Cambridge University Press, Cambridge, 2010. MR 2884384
- Jürgen Jost, Riemannian geometry and geometric analysis, 6th ed., Universitext, Springer, Heidelberg, 2011. MR 2829653
- Hajime Koba, Chun Liu, and Yoshikazu Giga, Energetic variational approaches for incompressible fluid systems on an evolving surface, Quart. Appl. Math. 75 (2017), no. 2, 359–389. MR 3614501, DOI https://doi.org/10.1090/qam/1452
- Hajime Koba and Kazuki Sato, Energetic variational approaches for non-Newtonian fluid systems, preprint, arXiv:1705.06956
- Yoshihiko Mitsumatsu and Yasuhisa Yano, Geometry of an incompressible fluid on a Riemannian manifold, Sūrikaisekikenkyūsho Kōkyūroku 1260 (2002), 33–47 (Japanese). Geometric mechanics (Japanese) (Kyoto, 2002). MR 1930362
- Lars Onsager Reciprocal Relations in Irreversible Processes. I. Physical Review. (1931);37:405-109 DOI:https://doi.org/10.1103/PhysRev.37.405
- Lars Onsager Reciprocal Relations in Irreversible Processes. II. Physical Review. (1931);38:2265-79 DOI:https://doi.org/10.1103/PhysRev.38.2265
- L.E. Scriven, Dynamics of a fluid interface Equation of motion for Newtonian surface fluids. Chem. Eng. Sci. 12 (1960), 98–108.
- James Serrin, Mathematical principles of classical fluid mechanics, Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959, pp. 125–263. MR 0108116
- John C. Slattery, Momentum and moment-of-momentum balances for moving surfacesChemical Engineering Science, Volume 19, 1964, Pages 379–385.
- John C. Slattery, Leonard Sagis, and Eun-Suok Oh, Interfacial transport phenomena, 2nd ed., Springer, New York, 2007. MR 2284654
- Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
- J. W. Strutt, Some General Theorems relating to Vibrations, Proc. London Math. Soc. (2) 4 (1871/73), 357–368. MR 1575554, DOI https://doi.org/10.1112/plms/s1-4.1.357
- Michael E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1407–1456. MR 1187618, DOI https://doi.org/10.1080/03605309208820892
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Additional Information
Hajime Koba
Affiliation:
Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyamacho, Toyonaka, Osaka, 560-8531, Japan
MR Author ID:
1013948
Email:
iti@sigmath.es.osaka-u.ac.jp
Received by editor(s):
July 19, 2017
Published electronically:
October 13, 2017
Additional Notes:
This work was partly supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers JP25887048 and JP15K17580.
Article copyright:
© Copyright 2017
Brown University