Instabilities in Fluid Motion, Volume 46, Number 11

1358 NOTICES OF THE AMS VOLUME 46, NUMBER 11 O ur everyday life is full of examples of fluid motion, from the drama of tornadoes and hurricanes, the turbulent cascading flows in rivers, and the breaking of massive ocean waves to the undramatic familiarity of stirring a cup of tea. Inspired by observation, scientists have sought for centuries to understand and predict fluid behavior. Over two hundred years ago Euler formulated his celebrated equations that give the fundamental mathematical description of fluid motion. It is interesting to note that other areas of physics such as the theory of heat or the motion of waves advanced rapidly in the early nineteenth century because linear models were in good agreement with experiments and the mathematical tool of Fourier series was successfully developed to attack such linear equations. While almost all the partial differential equations in mathematical physics are nonlinear, a number such as the heat equation and the wave equation become linear in commonly occurring physical situations (e.g., the heat equation when the coefficient of thermal conductivity is independent of the temperature itself). In contrast, the process of understanding fluid behavior was much slower because of the crucial nonlinearity in the Euler equations. Only the simplest flows in which all fluid particles are moving along straight lines or around circles could be described by explicit solutions of the hydrodynamic equations. The theoretical solutions named Poiseuille flow in a circular pipe and rotating Couette flow between concentric circular cylinders were found in the mid-nineteenth century. However, by the end of the nineteenth century a striking experimental fact became clear: these flows were realizable in experiments only when they were sufficiently slow. Reynolds [1] gave a beautiful description that we reproduce Susan Friedlander is professor of mathematics at the University of Illinois at Chicago. Her research is partially supported by NSF grant DMS-99 70977. Her e-mail address is [email protected].