Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On singular limit equations for incompressible fluids in moving thin domains

Author: Tatsu-Hiko Miura
Journal: Quart. Appl. Math.
MSC (2010): Primary 35Q35, 35R01, 76M45; Secondary 76A20
DOI: https://doi.org/10.1090/qam/1495
Published electronically: December 8, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the incompressible Euler and Navier-Stokes equations in a three-dimensional moving thin domain. Under the assumption that the moving thin domain degenerates into a two-dimensional moving closed surface as the width of the thin domain goes to zero, we give a heuristic derivation of singular limit equations on the degenerate moving surface of the Euler and Navier-Stokes equations in the moving thin domain and investigate relations between their energy structures. We also compare the limit equations with the Euler and Navier-Stokes equations on a stationary manifold, which are described in terms of the Levi-Civita connection.

References [Enhancements On Off] (What's this?)

  • [1] Rutherford Aris, Vectors, tensors and the basic equations of fluid mechanics, reprint of the 1962 original ed., Mineola, NY: Dover Publications, 1989.
  • [2] 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
  • [3] V. I. Arnold, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295
  • [4] Marino Arroyo and Antonio DeSimone, Relaxation dynamics of fluid membranes, Phys. Rev. E (3) 79 (2009), no. 3, 031915, 17. MR 2497175, https://doi.org/10.1103/PhysRevE.79.031915
  • [5] John W. Barrett, Harald Garcke, and Robert Nürnberg, Stable numerical approximation of two-phase flow with a Boussinesq-Scriven surface fluid, Commun. Math. Sci. 13 (2015), no. 7, 1829-1874. MR 3393177, https://doi.org/10.4310/CMS.2015.v13.n7.a9
  • [6] 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, https://doi.org/10.1007/s00021-008-0278-x
  • [7] J. Boussinesq, Contribution à la théorie de l'action capillaire, avec extension des forces de viscosité aux couches superficielles des liquides et application notamment au lent mouvement vertical, devenu uniforme, d'une goutte fluide sphérique, dans un autre fluide indéfini et d'un poids spécifique différent, Ann. Sci. École Norm. Sup. (3) 31 (1914), 15-85 (French). MR 1509172
  • [8] Paolo Cermelli, Eliot Fried, and Morton E. Gurtin, Transport relations for surface integrals arising in the formulation of balance laws for evolving fluid interfaces, J. Fluid Mech. 544 (2005), 339-351. MR 2262061, https://doi.org/10.1017/S0022112005006695
  • [9] Bang-Yen Chen, Total mean curvature and submanifolds of finite type, 2nd ed., Series in Pure Mathematics, vol. 27, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. With a foreword by Leopold Verstraelen. MR 3362186
  • [10] G. Dziuk and C. M. Elliott, Finite elements on evolving surfaces, IMA J. Numer. Anal. 27 (2007), no. 2, 262-292. MR 2317005, https://doi.org/10.1093/imanum/drl023
  • [11] Gerhard Dziuk and Charles M. Elliott, Finite element methods for surface PDEs, Acta Numer. 22 (2013), 289-396. MR 3038698
  • [12] 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, https://doi.org/10.2307/1970699
  • [13] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
  • [14] Morton E. Gurtin, An introduction to continuum mechanics, Mathematics in Science and Engineering, vol. 158, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 636255
  • [15] Luan T. Hoang and George R. Sell, Navier-Stokes equations with Navier boundary conditions for an oceanic model, J. Dynam. Differential Equations 22 (2010), no. 3, 563-616. MR 2719921, https://doi.org/10.1007/s10884-010-9189-7
  • [16] Dragoş Iftimie, Geneviève Raugel, and George R. Sell, Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J. 56 (2007), no. 3, 1083-1156. MR 2333468, https://doi.org/10.1512/iumj.2007.56.2834
  • [17] T. Jankuhn, M. A. Olshanskii, and A. Reusken, Incompressible fluid problems on embedded surfaces: Modeling and variational formulations, ArXiv e-prints (2017).
  • [18] 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, https://doi.org/10.1090/qam/1452
  • [19] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1969 original; A Wiley-Interscience Publication. MR 1393941
  • [20] John M. Lee, Introduction to smooth manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR 2954043
  • [21] Marius Mitrea and Michael Taylor, Navier-Stokes equations on Lipschitz domains in Riemannian manifolds, Math. Ann. 321 (2001), no. 4, 955-987. MR 1872536, https://doi.org/10.1007/s002080100261
  • [22] Tatsu-Hiko Miura, Zero width limit of the heat equation on moving thin domains, Interfaces Free Bound. 19 (2017), no. 1, 31-77. MR 3665918, https://doi.org/10.4171/IFB/376
  • [23] Tatsu-Hiko Miura, Yoshikazu Giga, and Chun Liu, An energetic variational approach for nonlinear diffusion equations in moving thin domains, Hokkaido University Preprint Series in Math. #1101 (2017).
  • [24] Liviu I. Nicolaescu, Lectures on the geometry of manifolds, 2nd ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. MR 2363924
  • [25] I. Nitschke, A. Voigt, and J. Wensch, A finite element approach to incompressible two-phase flow on manifolds, J. Fluid Mech. 708 (2012), 418-438. MR 2975450, https://doi.org/10.1017/jfm.2012.317
  • [26] Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
  • [27] M. Prizzi, M. Rinaldi, and K. P. Rybakowski, Curved thin domains and parabolic equations, Studia Math. 151 (2002), no. 2, 109-140. MR 1917228, https://doi.org/10.4064/sm151-2-2
  • [28] Geneviève Raugel, Dynamics of partial differential equations on thin domains, Dynamical systems (Montecatini Terme, 1994) Lecture Notes in Math., vol. 1609, Springer, Berlin, 1995, pp. 208-315. MR 1374110, https://doi.org/10.1007/BFb0095241
  • [29] Geneviève Raugel and George R. Sell, Navier-Stokes equations on thin $ 3$D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993), no. 3, 503-568. MR 1179539, https://doi.org/10.2307/2152776
  • [30] L.E. Scriven, Dynamics of a fluid interface equation of motion for newtonian surface fluids, Chemical Engineering Science 12 (1960), no. 2, 98-108.
  • [31] 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, https://doi.org/10.1080/03605309208820892
  • [32] Michael E. Taylor, Partial differential equations II. Qualitative studies of linear equations, 2nd ed., Applied Mathematical Sciences, vol. 116, Springer, New York, 2011. MR 2743652
  • [33] R. Temam and M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996), no. 4, 499-546. MR 1401403
  • [34] R. Temam and M. Ziane, Navier-Stokes equations in thin spherical domains, Optimization methods in partial differential equations (South Hadley, MA, 1996) Contemp. Math., vol. 209, Amer. Math. Soc., Providence, RI, 1997, pp. 281-314. MR 1472301, https://doi.org/10.1090/conm/209/02772

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 35Q35, 35R01, 76M45, 76A20

Retrieve articles in all journals with MSC (2010): 35Q35, 35R01, 76M45, 76A20

Additional Information

Tatsu-Hiko Miura
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan
Email: thmiura@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/qam/1495
Received by editor(s): March 28, 2017
Published electronically: December 8, 2017
Article copyright: © Copyright 2017 Brown University

American Mathematical Society