Desai and Lieber 



where 



I J = Volume integral = m n^dr 



CV) 



12 = Surface integral = 2^ J [n • (Vx fl)] ds 



■■■■ ■ (S) 



13 = Surface integral = -2/Li I V • (n • grad)Vds 



(S) 



l^ = Surface integral ~^[ (divV)n-Vds 



(S) 



Let us consider the circumstances under which the sum (I2+I3 + I4) of the 

 surface integrals ij, I3, and i^ vanishes. It would be a very special flow in 

 which the surface conditions are such that the sum vanishes, but none of the in- 

 tegrals 1 2, 1 3, and 1 4 vanishes. Further, the class of flows in which the con- 

 ditions are such that these integrals vanish individually but their integrands are 

 nonzero will also be a special one. The third possibility, when the integrands of 

 1 2, 1 3, and 1 4 vanish identically, is the most important. 



The integrands of I2, I3, and I4 are zero when V = on s. Thus when 

 the fluid is enclosed within fixed boundaries this condition is realized. Alter- 

 natively, the integrands of I2, I3, and I4 vanish when n • (Vxfi) = 0, 

 (n • grad) V = 0, and div V = respectively on s. In general, n will not be 

 perpendicular to the direction of V = Si. Hence n • (V ^ Q) can vanish if 

 V X n = . But since Vi o, this would require n = o. It is important to note 

 that the vanishing of fi on s does not imply that it vanishes everywhere in the 

 fluid. Similarly, div V = on S does not imply that div V = everywhere. 

 However, if the fluid is incompressible, div V = everywhere and, in particu- 

 lar, div V = on the surface s. 



We may visualize the bounding surface S to be divided into the following 

 parts: 



(i) Part Sj such that at least the condition V = o holds. 



(ii) Part s^ such that n = o, (n • grad) V = o, and div V = o, but \ t o. 



(iii) Part S3 such that (n • grad) V = and div V = 0, but V ^ and n t 0. 



(iv) Part S4 such that none of the above conditions hold. 



It may happen that one or more of the above four parts are zero. 



528 



