Journal of Fluid Mechanics



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


S.  Childress a1, G. R.  Ierley a2, E. A.  Spiegel a3 and W. R.  Young a4
a1 Courant Institute of Mathematical Sciences, New York University, New York, NY 10014, USA
a2 Fluids Research Oriented Group, Department of Mathematical Sciences, Michigan Technological Institute, Houghton, MI 49931, USA
a3 Astronomy Department, Columbia University, New York, NY 10027, USA
a4 Scripps Institution of Oceanography, A-021, University of California at San Diego, La Jolla, CA 92093, USA

Article author query
childress s   [Google Scholar] 
ierley gr   [Google Scholar] 
spiegel ea   [Google Scholar] 
young wr   [Google Scholar] 
 

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.

(Published Online April 26 2006)
(Received February 16 1988)



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