Desai and Lieber 



flow around a sphere reminds one of the streamline pattern of the correspond- 

 ing potential flow. 



A potential flow is thus seen to play a fundamental role at the lower as well 

 as the higher ranges of the characteristic parameter, the Reynolds number, of 

 the flow of a real fluid. This being the case, there seems no a priori reason 

 why it should not play a fundamental role all through the range of this parame- 

 ter. The principle of minimum dissipation, if assumed to hold, indicates that it 

 should play such a role. Consequently, we may regard it as the fundamental 

 base flow for all Reynolds numbers. 



MATHEMATICAL CONSIDERATIONS 



In the theory of linear differential operators, the operator and its domain 

 are defined as in the following paragraphs [30]. , . 



First, the linear vector space s of functions on which the differential oper- 

 ator L, say of order n, operates is defined such that (a) the interval of the vari- 

 able, (b) the nature of the functions, and (c) the scalar product are specified. 

 The domain of the operator is the set of all functions u in s which have a piece- 

 wise continuous derivative of the order n, whichsatisfy n independent and lin- 

 ear conditions, and are such that Lu belongs to s. 



The differential equation Lu = f does not have a unique solution unless the 

 conditions to be satisfied by u are given. Different sets of conditions lead to 

 different solutions. Hence for precise notation, a different symbol should be 

 used for the operator each time the conditions are changed. For convenience, 

 however, the same symbol is used for the differential operator under all condi- 

 tions, but the conditions which the solution of Lu = f is to satisfy are specified. 

 Thus the operator is formally the same for all the solutions of Lu = f , but in 

 fact is different for different solutions [30]. 



In solving linear boundary value problems with involved boundary condi- 

 tions it is a common practice to consider the ultimate solution as made up of 

 two or more parts, each part satisfying the governing equations completely, but 

 the boundary conditions only partially so that when added together they satisfy 

 the governing equations and all the boundary conditions due to the linearity of 

 the operator. The number of parts into which the solution is divided is gener- 

 ally finite. For example, if u is the final solution of a second order differential 

 equation Lu = f such that it satisfies a set of conditions B,(u) = ctj, B^iu) = a^ 

 for specified values of the variable, it may be conceived as made up of two parts, 

 u, and Ut, such that 



and 



Due to linearity. 



, Bj(uj) - aj , B2(uj) 



Lu2 = f , BjCu^) - , B^(u^) 



530 



