Local and Global Perspectives in Fluid Dynamics

Conclusion

The smoothness, or alternatively the finite-time singularity, of the Navier-Stokes equations offers a challenge that will continue to make great demands on both analytical ingenuity and computational power. If, as computer simulations continue to indicate (Kerr 1997), a finite-time singularity does occur and if this is generic behavior, then of course we shall have to understand by what mechanism these putative singularities are resolved. Since the pressure gradient must also become unbounded as a singularity is approached, the incompressibility assumption, on which most analyses of this phenomenon are based, becomes no longer tenable. The infinite stress at a singularity can be relieved by cavitation in liquids, and by acoustic radiation in gases. The Japanese bath provides a congenial environment for the contemplation of such problems!

As I mentioned in my introductory remarks, this Congress exhibits a most fruitful interplay between local and global characteristics. I would maintain now that it also has some features of a finite-time singularity: there was certainly a decreasing length-scale as we all converged on Chicago; and the Organizers were acutely aware of a decreasing time-scale in the last few weeks of hectic preparations. We should be relieved that the Marriott Hotel has neither imploded nor exploded under the extreme pressure of activity that it has experienced this week. Now we are at the stage where the singularity must be resolved; we shall return to our homes around the world like acoustic pulses radiating from this source, and the locally acquired impact of this Congress will surely inform and inspire future research in theoretical and applied mechanics on a global scale. On that note, and like a finite-time singularity, I must bring this talk to a sudden end.