-..-^ Desai and Lieber 



Consequently, if the boundary conditions are not brought into consideration, this 

 potential flow solution satisfies the Navier-Stokes equations for all Reynolds 

 numbers, i.e., even though the solution is generally referred to as the solution 

 for an inviscid fluid, it is in fact independent of the viscosity of a fluid. 



Let S denote the actual solution which satisfies the Navier-Stokes equa- 

 tions, the continuity equation, and all the boundary conditions, and which, we as- 

 sume, represents the flow field as observed in experiments. Let S^ denote the 

 potential flow solution given by Eqs. (1.17), (1.18), (1.19), and (1.20). Let the 

 flow domain be denoted by M. The solution S, if substituted in the Euler equa- 

 tions, will not, in general, satisfy them everywhere in the domain M. However, 

 it may satisfy them in some subdomain Mj of M to a high degree of approxima- 

 tion. Let Mj denote the remaining subdomain of M so that Mj + Mj = M. The 

 solution Sg satisfies the Navier-Stokes equations as well as the Euler equations 

 everywhere in the domain M and consequently in the subdomain M^, regardless 

 of the viscosity of the fluid. Therefore, in the subdomain Mj the actual solution 

 S must be a close approximation to the potential flow solution Sq, but in the sub- 

 domain M2 it may be significantly different from it. Now A is said to deviate 

 from B under a variable set of conditions C if A changes with C while B does 

 not, and A approaches B when the variable set of conditions C approach one or 

 more fixed sets of conditions D, E, etc. The actual solution s depends on the 

 Reynolds number as a parameter and hence changes with it, whereas the poten- 

 tial solution Sq, being independent of the Reynolds number, rem.ains constant for 

 all Reynolds numbers. Therefore we can identify S with A, Sg with B, Re with 

 the variable set of conditions C, and fixed limiting values of Re with the fixed 

 sets of conditions D, E, etc. If then the subdomain Mj happens to enlarge as 

 the Reynolds number Re approaches any one of the fixed limiting values, this 

 fact may be taken to mean that the actual solution s deviates from the potential 

 flow solution Sq in the whole domain M and that this deviation decreases as the 

 Reynolds number Re approaches any one of the fixed limiting values. The ob- 

 servations at high values of Reynolds number show that the domain in which the 

 real flow can be regarded as a potential flow does in fact increase with increas- 

 ing Reynolds numbers. Similar observations at extremely low values of Reyn- 

 olds numbers, e.g., for Hele-Shaw flows, show that the same is the case with 

 decreasing Reynolds numbers. Consequently, if we assume that s does repre- 

 sent the real flow as observed, the foregoing conclusion becomes compulsive. 



As we have shown, the potential flows satisfy the Navier-Stokes equations 

 in a stricter sense than do the viscous flows, i.e., by leading to the vanishing of 

 the separate parts of the governing equations whatever be the viscosity of the 

 fluid— and this is all the more striking when the individual terms of the Vorticity 

 Transport Equation are observed to vanish separately. The irrotational char- 

 acter of the flow is what leads to this remarkable behavior. This is here re- 

 garded to be of fundamental importance and consequently, for the present work, 

 the third possibility (c) leading to the Euler equations is the meaningful one. In 

 this light, we may think of a real flow as a deviation from the correspotiding 

 potential flow ,'^ made in order to satisfy the no-slip boundary conditions, and 



'■'By corresponding potential flow is meant here the flow which satisfies the N-S 

 equations and all boundary conditions except the no-slip condition. 



522 



