Otto LaPorte Lecture: Singularities in Fluid Dynamics

Singularities of the Navier-Stokes equations occur when some (possibly high-order) derivative of the velocity field is infinite at any point of a field of flow (or, in an evolving flow, becomes infinite at any point within a finite time). Such singularities can be mathematical but unphysical (as e.g. in two-dimensional flow near a sharp corner) in which case they can be 'resolved' by improving the physical model considered; or they can be physical but non-mathematical (as e.g. in the case of cusp singularities at a fluid/fluid interface) in which case resolution of the singularity may involve incorporation of additional physical effects; such examples will be briefly reviewed. The 'finite-time singularity problem' for the Navier-Stokes equations will then be discussed and a new analytical approach will be presented; here it will be shown that there is indeed a singularity of the 'physical but non-mathematical' type, in that, at sufficiently high Reynolds number, vorticity can be amplified by an arbitrarily large factor within a finite time. In this case, the singularity is resolved by three-dimensional vortex reconnection in a manner that admits analytical description. Implications for turbulence will be presented. [Part of this work has been in collaboration with Yoshifumi Kimura.]