Abstract
In the analysis of singular Euler flows, real or imagined, Lagrangian variables offer an attractive option, although it is well known that in many flow problems Lagrangian methods can be intractable. In other examples, Lagrangian variables are found to simplify the representation. We examine here some aspects of Lagrangian analysis of fluid flows, then focus on simple singular solutions of Euler’s equations having infinite kinetic energy. We use these solutions to explore the different forms taken by these flows in Eulerian and Lagrangian formulations. We then apply a Lagrangian construction to a set of singular flows in two dimensions, as examples of a general method. Finally, we comment on the analogous but far more difficult application of the method to singular Euler flows in three dimensions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ashurst, W.T., Kerstein, A.R., Kerr, R.M., & Gibson, C.H. (1987) Alignment of vorticity and scale gradient with strain rate in simulated Navier-Stokes turbulence. Phys. Fluids 30, 2343–2353.
Beale, J.T., Kato, T. & Majda, A.J. (1989) Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66.
Caflisch, R.E. (1993) Singularity formation for complex solutions of the 3D incompressible Euler equations. Physica D 67, 1–18.
Cantwell, B.J. (1984) Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids A 4, 782–793.
Childress, S. (1987) Nearly two-dimensional solutions of Euler’s equations. Phys. Fluids 30, 944–953.
Childress, S., Ierley, G.R., Spiegel, E.A., & Young, W.R. (1989) Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form. J. Fluid Mech. 203, 1–22.
Coddington, E.A. & Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill.
Constantin, P. (2000) The Euler equations and nonlocal conservative Riccati equations. Int. Math. Res. Notices 9, 455–465.
Constantin, P. (2000) Some open problems and research directions in the mathematical study of fluid dynamics. Preprint.
Dresselhaus, E. (1992) Material Element Stretching and Alignment in Turbulence. Ph.D. thesis, Columbia University.
Dresselhaus, E. & Tabor, M. 1991 The stretching and alignment of material elements in general flow fields. J. Fluid Mech. 236, 415–444.
Kato, T. (1967) On the classical solutions of the two-dimensional non-stationary Euler equation. Arch. Rat. Mech. Anal. 25, 188–200.
Lamb, H. (1932) Hydrodynamics. Dover Publications, Sixth edition.
Moffatt, H.K. (2000) The interaction of skewed vortex pairs: a model for blow-up of the Navier-Stokes equations. J. Fluid Mech. 409, 51–68.
Ohkitani, S.K. & Gibbon, J.D. (2000) Numerical study of singularity formation in a class of Euler and Navier-Stokes flows. Phys. Fluids 12, 3181–3194.
Rankine, W. J.M. (1863) On the exact form of waves near the surface of deep water. Trans. Roy. Soc. London 153, 127–138.
Stuart, J.T. (1987) Nonlinear Euler partial differential equations: singularities in their solution. In Proceedings in honor of C.C. Lin (ed. D.J. Benney et al.), pp. 81–95. World Scientific, Singapore.
Stuart, J.T. (1991) The Lagrangian picture of fluid motion and its implication for flow structures. IMA J. Appl. Math 46, 147–163.
Vieillefosse, P. (1984) Internal motions of a small element of fluid in an inviscid flow. Physica A 125, 150–162.
Wasow, W. (1965) Asymptotic Expansions of Ordinary Differential Equations. Interscience.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Childress, S. (2001). Euler Singularities from the Lagrangian Viewpoint. In: Ricca, R.L. (eds) An Introduction to the Geometry and Topology of Fluid Flows. NATO Science Series, vol 47. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0446-6_16
Download citation
DOI: https://doi.org/10.1007/978-94-010-0446-6_16
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-0207-6
Online ISBN: 978-94-010-0446-6
eBook Packages: Springer Book Archive