,_ . . , -, Lieber 



limiting case of very small Reynolds numbers are the restrictions implied by 

 the Navier -Stokes equations and the principle of minimum dissipation essentially 

 equivalent; at all finite Reynolds numbers, they are evidently complementary 

 and consequently imply different restrictions on viscous flow fields. This think- 

 ing is consistent with the conclusions Ladyzhenskaya (1963) obtained in her work 

 on the mathematical theory of the Navier -Stokes equations, which indicate that 

 the Navier -Stokes equations are not sufficient to describe the motion of a vis- 

 cous fluid for large Reynolds numbers. 



The principle of minimum dissipation used in conjunction with the Navier - 

 Stokes equations and its extension by a theorem obtained from a variational 

 principle which we formulated in order to give more complete hydrodynamical 

 expression to the principle of maximum uniformity (Lieber and Wan, 1957), im- 

 ply that the structure of actual flow fields is restricted by linear differential 

 relations, which are referred to as a linear substructure, and by a nonlinear 

 compatibility equation that implies certain necessary connections between sym- 

 metry properties of actual flow fields and their time -dependent motion. 



The work of Desai and Lieber reported in the proceedings of this sympo- 

 sium gives further, though indirect, hydrodynamical representation to the prin- 

 ciple of maximum uniformity, through an algorithm which is implicitly endowed 

 with a linear substructure that joins successive steps of an iteration procedure 

 by which the algorithm is defined. The successive steps of this iteration pro- 

 cedure correspond to successive finite steps in the development of a viscous 

 flow field, which are not generally separated by small perturbations but rather 

 by finite changes that become arbitrarily small only when the flows obtained by 

 successive steps virtually converge to a fixed pattern. S. M. Desai (1965) intro- 

 duced the very significant idea of using the potential flow solutions which corre- 

 spond to particular shapes of physical boundaries, as the base flow upon which 

 to initiate the iteration procedure that defines the algorithm cited, and by which 

 analytical representations of viscous incompressible flow fields are obtained on 

 the basis of the complete Navier-Stokes equations and realistic boundary condi- 

 tions. Each step of the iteration process is thus made compatible with the 

 Navier-Stokes equations and then with the law of force equilibrium. 



The potential flow on which the iteration is initiated has a fundamental lin- 

 ear substructure, in the sense that its kinematics is governed by Laplace's 

 equation, and is intrinsically endowed with the maximum uniformity attainable 

 from ideal (slip) boundary conditions. This is because potential flows are a 

 subclass of flows governed by a direct hydrodynamical formulation of the prin- 

 ciple of maximum uniformity in which the integral constructed in formulating 

 this variational principle is a positive, definite measure of force. This force 

 measure of nonuniformity bears therefore a direct connection with the funda- 

 mental scalar measure of force used in the conception and general statement of 

 the principle of maximum uniformity. From these considerations, we envisage 

 the potential flow on which the iteration procedure that defines our algorithm is 

 initiated, as affording a formal representation within the framework of the algo- 

 rithm, of the ideally maximum uniformity state toward which actual flow fields 

 tend in their evolutionary development, in complying with the law governing 

 their development — the principle of maximum uniformity. This, I believe, ex- 

 plains the fact that by only one iteration on the base potential flow, we obtain a 



482 



