v/. 7-'. Desai and Lieber 



Thus for any obstacle, the characteristic Reynolds number is given by Uh'^/j^L, 

 where L is a characteristic length of the obstacle and h' is the distance sepa- 

 rating the two plates. Since A is proportional to h'^, uj, V2, and Wj all tend 

 to zero as h' — 0. Then, in the limit, the flow reduces to a two-dimensional 

 flow with uj and vj as velocity components, wj being zero. This was the con- 

 clusion of Stokes and the experimental observation of Hele Shaw. Riegels' ex- 

 periments, carried out up to about A = 6, show that for A < 1 there is no dis- 

 cernible deviation from the potential streamline field, but that for higher values 

 of this parameter the deviation is noticeable. 



The pressure field for a creeping flow satisfies the harmonic equation 

 v^p = because the convective terms in the Navier-Stokes equations are com- 

 pletely neglected, but the velocity field does not satisfy the equation. On the 

 other hand, the velocity field for a potential flow always satisfies the harmonic 

 equation, but the pressure field does not and is in fact given by V^p = -p(Bui/3xj 

 Buj/Bxj). When the product term p (duj/Bxj 3uj/3xi) can be neglected, the pres- 

 sure field of the potential flow is harmonic in the limit. Stokes has shown that 

 the velocity field of a creep flow becomes harmonic in the limit. Hence in the 

 limit, the velocity field and the pressure fields of a creep flow are potential. 

 Thus these experiments and theoretical treatments show that there exists a 

 three-dimensional real flow, which apparently satisfies the no-slip conditions, 

 such that the streamline field due to it becomes identical with that due to a two- 

 dimensional potential flow in the same space when the Reynolds number tends to 

 become vanishingly small; and that this potential flow evidently does not satisfy 

 the no-slip conditions. 



There are two noteworthy points in these experiments of Hele-Shaw and 

 Riegels: (a) the three-dimensional flow becomes two-dimensional; and (b) the 

 velocity field attains a potential character, i.e., it becomes irrotational. Of 

 these two, the first may be regarded as peculiar to the particular geometry 

 under consideration, and hence incidental. The second point, however, may be 

 regarded as an expression of a general truth, and hence fundamental, for the 

 following reasons. 



The rate at which energy E is dissipated in a body of fluid is given by the 

 expression 



E = /x j fi2dT+2/xJ [n-(Vxfi)]ds 



(V) (S) 



-2m I [V • (n • grad) V] ds + X f (div V) n ■ V ds , 



(S) (S) 



where 



11 = Curl V = vorticity vector 



V = Velocity vector with components u. 



526 



