University of Washington
Seattle, Washington, U. S. A.
In examining the dynamics of any system we resort to the use of analogies from classical mechanics for understanding. In effect, this means that the sum (scalar or vector) of all forces that may be acting vanish and this establishes criteria for any equilibria. Likewise, small perturbation analysis is used to examine the stability of the equilibrium loci so established.
In many respects, this same approach for investigating problems in fluid mechanics is valid but not completely satisfactory. First of all, the systems involve partial differential equations and novel physical results ensue that require further investigation for understanding. Second, even the mathematics that is required for such analyses does not follow any standard pattern. And, lastly, the hopes and aspirations that were once thought possible to be ascertained from this basis have yet to be realized. A salient review of the history of this subject will be given with an emphasis to describing the variety of different problems that are presented in this field. Both the important physics as well as the mathematics will be discussed and up dated. The role of numerics will be established as well. Several specific prototypical fluid flow examples will be detailed in order to demonstrate the successes, the remaining needs, and the current status of reaching the hopes and goals needed for understanding both the equilibrium and the stability for fluid motions.