- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Invited
  • Access by Xinjiang University

Role of singularities in hydrodynamics

J. Eggers

  • School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom

PDF

Phys. Rev. Fluids 3, 110503 – Published 21 November, 2018

DOI: https://doi.org/10.1103/PhysRevFluids.3.110503

Abstract

Some of the most interesting structures observed in hydrodynamics are best understood as singularities of the equations of fluid mechanics. Examples are drop formation in free-surface flow, shock waves in compressible gas flow, or vortices in potential flow. These examples show that singularities are characteristic for the tendency of the hydrodynamic equations to develop small-scale features spontaneously, starting from smooth initial conditions. As a result, new structures are created, which form the building blocks of more complicated flows. The mathematical structure of singularities is self-similar, and their characteristics are fixed by universal properties. We review recent developments in this field through the lens of one of the great scientific challenges of today: understanding the structure of turbulence.

View figure in article

Physics Subject Headings (PhySH)

Collections

This article appears in the following collection:

2018 Invited Papers

Physical Review Fluids publishes a collection of papers associated with the invited talks presented at the 70th Annual Meeting of the APS Division of Fluid Dynamics.

Article Text

References (56)

  1. L. Euler, Principes génereaux du mouvement des fluides, Mem. Acad. Sci. Berlin 11, 274 (1757).
  2. C. L. Navier, Sur les lois du mouvement des fluides, Mem. Acad. R. Sci. France 6, 389 (1827).
  3. O. Darrigol, Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl (Oxford University Press, Oxford, 2005).
  4. J. le Rond D'Alembert, Paradoxe proposé aux géometrès sur la résistance des fluides, in Opuscules mathématiques (Paris, 1768), Vol. 5, pp. 132–138.
  5. K. Stewartson, D'Alembert's paradox, SIAM Rev. 23, 308 (1981).
  6. H. Helmholtz, Über diskontinuierliche Flüssigkeitsbewegungen, Monatsber. Berlin Akad., 215 (1868).
  7. G. Kirchhoff, Zur Theorie freier Flüssigkeitsstrahlen, J. Reine Angew. Math. 70, 289 (1869).
  8. H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, J. Reine Angew. Math. 55, 25 (1858).
  9. S. G. Llewellyn Smith, How do singularities move in potential flow? Physica D 240, 1644 (2011).
  10. B. Riemann, Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 8, 43 (1860).
  11. R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves (Interscience Publishers, New York, 1948).
  12. P. E. Dimotakis, The mixing transition in turbulent flows, J. Fluid Mech. 409, 69 (2000).
  13. L. F. Richardson, Atmospheric diffusion shown in a distance-neighbor graph, Proc. R. Soc. London A 110, 709 (1926).
  14. A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds' numbers, Dokl. Akad. Nauk SSSR 30, 301 (1941).
  15. U. Frisch, Turbulence (Cambridge University Press, Cambridge, 1995).
  16. G. L. Eyink and K. R. Sreenivasan, Onsager and the theory of hydrodynamic turbulence, Rev. Mod. Phys. 78, 87 (2006).
  17. J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1, 171 (1948).
  18. H. K. Moffatt, Euler's disk and its finite-time singularity, Nature (London) 404, 833 (2000).
  19. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63, 193 (1934).
  20. T. A. Kowalewski, On the separation of droplets, Fluid Dyn. Res. 17, 121 (1996).
  21. J. Eggers, Nonlinear dynamics and breakup of free-surface flows, Rev. Mod. Phys. 69, 865 (1997).
  22. P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley, and S.-M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Phys. Rev. E 47, 4169 (1993).
  23. A. L. Bertozzi, M. P. Brenner, T. F. Dupont, and L. P. Kadanoff, Singularities and similarities in interface flows, in Trends and Perspectives in Applied Mathematics, Applied Mathematical Sciences Vol. 100, edited by L. Sirovich (Springer, New York, 1994), p. 155.
  24. J. Eggers, Universal Pinching of 3D Axisymmetric Free-Surface Flow, Phys. Rev. Lett. 71, 3458 (1993).
  25. J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial differential equations, Nonlinearity 22, R1 (2009).
  26. J. Eggers and M. A. Fontelos, Singularities: Formation, Structure, and Propagation (Cambridge University Press, Cambridge, 2015).
  27. J. Eggers and T. F. Dupont, Drop formation in a one-dimensional approximation of the Navier-Stokes equation, J. Fluid Mech. 262, 205 (1994).
  28. D. H. Peregrine, G. Shoker, and A. Symon, The bifurcation of liquid bridges, J. Fluid Mech. 212, 25 (1990)).
  29. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, New York, 1993).
  30. P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1981).
  31. P. Marmottant and E. Villermaux, On spray formation, J. Fluid Mech. 498, 73 (2004).
  32. W. C. Griffith and W. Bleakney, Shock waves in gases, Am. J. Phys. 22, 597 (1954).
  33. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1984).
  34. A. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics (Springer, New York, 1990).
  35. Y. Pomeau, M. Le Berre, P. Guyenne, and S. Grilli, Wave-breaking and generic singularities of nonlinear hyperbolic equations, Nonlinearity 21, T61 (2008).
  36. G. I. Barenblatt, Similarity Self-Similarity and Intermediate Asymptotics (Cambridge University Press, Cambridge, 1996).
  37. Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat-equations, Commun. Pure Appl. Math. 42, 845 (1989).
  38. J. Eggers and N. Suramlishvili, Singularity theory of plane curves and its applications, Eur. J. Mech. B 65, 107 (2017).
  39. J. Eggers and S. Grossmann, Effect of dissipation fluctuations on anomalous velocity scaling in turbulence, Phys. Rev. A 45, 2360 (1992).
  40. K. R. Sreenivasan, R. Ramshankar, and C. Meneveau, Mixing, entrainment and fractal dimensions of surfaces in turbulent flows, Proc. Roy. Soc. London A 421, 79 (1989).
  41. A. N. Kolmogorov, A refinement of previous hypothesis concerning the local structure of turbulence in a viscous incompressible fluid at high Reynold numbers, J. Fluid Mech. 13, 82 (1962).
  42. B. B. Mandelbrot, Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier, J. Fluid Mech. 62, 331 (1974).
  43. T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33, 1141 (1986).
  44. M. Tjahjadi, H. A. Stone, and J. M. Ottino, Satellite and subsatellite formation in capillary breakup, J. Fluid Mech. 243, 297 (1992).
  45. M. W. Choptuik, Universality and Scaling in Gravitational Collapse of a Massless Scalar Field, Phys. Rev. Lett. 70, 9 (1993).
  46. Y. Pomeau and D. Sciamarella, An unfinished tale of nonlinear PDEs: Do solutions of 3D incompressible Euler equations blow up in finite time? Physica D 205, 215 (2005).
  47. A. Pumir, B. I. Shraiman, and E. D. Siggia, Vortex morphology and Kelvin's theorem, Phys. Rev. A 45, R5351 (1992).
  48. M. P. Brenner, S. Hormoz, and A. Pumir, Potential singularity mechanism for the Euler equations, Phys. Rev. Fluids 1, 084503 (2016).
  49. R. K. Scott and D. G. Dritschel, Numerical Simulation of a Self-Similar Cascade of Filament Instabilities in the Surface Quasigeostrophic System, Phys. Rev. Lett. 112, 144505 (2014).
  50. M. C. Dallaston, M. A. Fontelos, D. Tseluiko, and S. Kalliadasis, Discrete Self-Similarity in Interfacial Hydrodynamics and the Formation of Iterated Structures, Phys. Rev. Lett. 120, 034505 (2018).
  51. W. W. Zhang and J. R. Lister, Similarity solutions for Van der Waals rupture of thin film on a solid substrate, Phys. Fluids 11, 2454 (1999).
  52. T. P. Witelski and A. J. Bernoff, Dynamics of three-dimensional thin film rupture, Physica D 147, 155 (2000).
  53. D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley, Wetting and spreading, Rev. Mod. Phys. 81, 739 (2009).
  54. P. G. Drazin, Nonlinear Systems (Cambridge University Press, Cambridge, 1992).
  55. https://www.youtube.com/watch?v=gWGLAAYdbbc
  56. J. Eggers, T. Grava, M. A. Herrada, and G. Pitton, Spatial structure of shock formation, J. Fluid Mech. 820, 208 (2017).

Outline

Information

Phys. Rev. Fluids 3, 110503– Published 21 November, 2018
  • Received 16 June 2018
Crossmark - Check for updates button

©2018 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation