SOME PROBLEMS AND METHODS IN HYDRODYNAMICS 



L. M. Milne-Thomson, C.B.E. 

 Brown University 



When I received Admiral Bennett's invitation to address you tonight I felt 

 almost as uncomfortable as General Burgoyne must have felt at Saratoga with the 

 thought of the North behind me and hydrodynamicists massing in the South. Indeed 

 I began to reflect on the distance from London to York which as you know, or ought 

 to know, is a long way, and even non-stop trains take several hours to cover it. Once 

 upon a time in one such non-stop train sat an American and an Englishman the only 

 occupants of a compartment. For about an hour each looked out of the window. 

 Then the American said "Do you mind if I talk to you" to which the Englishman 

 replied "What about?" You see my difficulty. An after dinner speech which conveyed 

 any information whatever is outside the range of my experience. An after dinner speech 

 concerning Hydrodynamics must be unique in the history of dining and I therefore feel 

 honored to be chosen as the first to make such history. 



Hydrodynamics as an exact science started with Archimedes. It is true that he 

 treated the particular case of zero velocity, but his work remains today a correct piece 

 of applied mathematics and indeed a giant achievement for his time. 



Scoffers have said that Archimedes' chief claim to fame is that he took a bath 

 and then forgot to dress in spite of his principles. 



It is not my purpose nor indeed am I competent to discuss hydrodynamics from 

 the point of view of the historian, who, as you know, is an amiable gentleman advancing 

 into eternity stern first. 



Rather I should like to say a few words concerning some specific problems and 

 methods available for tackling them. Omission of a name or a topic does not imply 

 undervaluation by me. It merely proves that a choice has to be made. Many other 

 choices would have been possible. 



It is becoming increasingly evident that the most insight giving statement of 

 the equations of motion of continuous media in general and of fluids in particular is 

 by means of tensors. 



There are at least two ways of regarding tensors, namely as quantities attached 

 to a coordinate system, or intrinsically, the latter being to my way of thinking incom- 

 parably superior. 



The intrinsic definition of a tensor T (n) of rank n is recursive. 



A tensor of rank n is a linear operator which operating on an arbitrary vector 

 x, by scalar multiplication, gives rise to a tensor of rank n-1. 



This definition, together with the statement that a tensor of rank zero is a scalar, 

 completely characterizes tensors of all ranks. 



Thus for example a vector a is a tensor of rank 1 since a • x is a scalar, or 

 tensor of rank zero. 



Similarly the dyadic product pq;q combines with x to give pq (q • x) a tensor 

 of rank 1 and so pq;q is a tensor of rank 2, a 2-tensor, indeed the very important 



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