Studies on the Motion of Viscous Flows— III 



All fixed surfaces at which the no-slip boundary condition holds constitute 

 the part Sj. Whenever the condition of uniform flow at infinity is valid, all the 

 conditions a = 0, (n • grad) V = 0, and div V = hold because they involve 

 differentiation of the vector V which is constant when the flow is uniform. Hence 

 the enveloping surface at infinity belongs to the class Sj. The external flows 

 for which the bounding surface S is made up of fixed boundaries in the interior 

 and an enveloping boundary at infinity where uniform flow conditions prevail, 

 belong to the class of flows for which the surface integrals 1 2, I3, and 1 4 are 

 zero, no matter whether the fluid is compressible or incompressible. For in- 

 ternal flows where part of the enveloping surface is a fixed boundary and the in- 

 let and outlet conditions are such that (n • grad) V = and div V = 0, but V ^ 

 and i 0, the surface integrals I3 and 1 4 vanish. We then see that for exter- 

 nal flows 



E - i^j n'dr , (1.24e) 



(V) . 



and for internal flows 



f n^dT + 2fij [n-(Vv, n)]ds (i.24f) 



(V) (S3) 



are valid expressions for E. 



If we assert that the principle of minimum dissipation holds, then that 

 would imply that the integrals in Eqs. (1.24e) and (l,24f) achieve their lowest 

 attainable value consistent with conservation principles and the surface condi- 

 tions. We note that these integrals depend on Q, the vorticity field. If n van- 

 ishes or at least becomes insignificantly small in most parts of the flow field, 

 the contribution to the volume integral would be reduced materially. From Eqs. 

 (1.24e) and (1.24f) it is also clear that as n — 0, E — 0. Conversely, E — 

 should imply n — because both the integrands are positive-definite. We ob- 

 serve that though for real fluids E cannot be zero in general, it must tend to 

 zero according to the principle of minimum dissipation. Consequently, the va- 

 lidity of this principle would lead us to infer that q^o- Since fi = implies 

 that the velocity field is irrotational and hence derivable from a potential, we 

 can now state the following as a conclusion: 



Theorem I: For a large class of real flows, the velocity field 

 tends to become irrotational and hence derivable from a poten- 

 tial. 



The Hele-Shaw flows belong to the class of real flows for which the above 

 statement holds. Hence the fact that, as A— '0, the velocity field becomes po- 

 tential can be viewed as the substantiation of the principle of minimum dissipa- 

 tion. That large parts of a flow field become derivable from a potential as the 

 Reynolds number of the flow is increased in experimentally well established and 

 forms the basis of the boundary layer theory. The streamline pattern of a creep 



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