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Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form

Published online by Cambridge University Press:  26 April 2006

S. Childress
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10014, USA
G. R. Ierley
Affiliation:
Fluids Research Oriented Group, Department of Mathematical Sciences, Michigan Technological Institute, Houghton, MI 49931, USA
E. A. Spiegel
Affiliation:
Astronomy Department, Columbia University, New York, NY 10027, USA
W. R. Young
Affiliation:
Scripps Institution of Oceanography, A-021, University of California at San Diego, La Jolla, CA 92093, USA

Abstract

The time-dependent form of the classic, two-dimensional stagnation-point solution of the Navier-Stokes equations is considered. If the viscosity is zero, a class of solutions of the initial-value problem can be found in closed form using Lagrangian coordinates. These solutions exhibit singular behaviour in finite time, because of the infinite domain and unbounded initial vorticity. Thus, the blow-up found by Stuart in three dimensions using the stagnation-point form, also occurs in two. The singularity vanishes under a discrete, finite-dimensional ‘point vortex’ approximation, but is recovered as the number of vortices tends to infinity. We find that a small positive viscosity does not arrest the breakdown, but does strongly alter its form. Similar results are summarized for certain Boussinesq stratified flows.

Type
Research Article
Copyright
© 1989 Cambridge University Press

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References

Aref, H.: 1983 Integrable, chaotic and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345389.Google Scholar
Batchelor, G. K.: 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Drazin, P. G.: 1983 Solitons. Lond. Math. Soc. Lecture Notes Series, vol. 85. Cambridge University Press.
Gottlieb, D. & Orszag, S. A., 1977 Numerical Analysis of Spectral Methods. Theory and Applications. CBMS Regional Conference Series in Applied Mathematics, vol. 26, Society for Industrial and Applied Mathematics, Philadelphia.
Hölder, E.: 1933 Über die un beschränkte Fortsetzbarkeit einer stetigen abenen Bewegung in einer unbegretzen inkompressiblen Flüssigkeit. Math. Z. 37, 727738.Google Scholar
Howarth, X. X.: 1951 The boundary layer in three-dimensional flow. Part 2. The flow near a stagnation point. Phil. Mag. 42, 14331440.Google Scholar
Kato, T.: 1967 On the classical solution of the two-dimensional, non-stationary Euler equation. Arch. Rat. Mech. Anal. 25, 188200.Google Scholar
Lin, C. C.: 1958 Note on a class of exact solutions in magnetohydrodynamics. Arch. Rat. Mech. Anal. 1, 391395.Google Scholar
Stuart, J. T.: 1987 Nonlinear Euler partial differential equations: singularities in their solution. In Symposium to Honor C. C. Lin (ed. D. J. Benney, F. H. Shu & C. Yuan). World Scientific.
Wolibner, W.: 1933 Un theorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendent un temps infiniment long. Math. Z. 37, 698726.Google Scholar