616 



- 1,10 

 -150 

 -1,70 



FIGURE A.l. Flow pattern for (VjWj). ASTrYi = 0.2. 



W„dndC = «>. 



(A. 12) 



which shows that at n 



does not vanish. 



We must then, if we adopt the procedure of neglecting 

 the pressure gradient, not demand that Vq vanish 

 at infinity, but instead demand 



3ri 



= at 



(A. 14) 



This boundary condition for V. must, for consistency, 

 be demanded of V, i.e., of Vj , V2 , etc., as we 

 proceed to higher and higher approximations. 



In this connection we can also see that it is 

 not possible to add a multiple of W„ to the W^, 

 given by Eq. (A. 3) to make V. vanish at infinity. 

 For, in order to make Vp vanish at infinity, the 

 only possibility is to add to the Wg given by Eq. 

 (A. 3) a multiple of 



r1 



W 



Oc 



f (n) 



(A. 15) 



where f satisfies 



That means 



+ 2nf' 



f = 0. 



But this cannot be true , because integration of 

 (A. 2), by parts if necessary, gives 



and Eq. (A. 11) gives 



-1, 



-31 = 



hpdnd? = -1, 



so that 



I = 



Hence W cannot contain a multiple of W„ , and Eq. 

 (A. 3) is the unique solution. 



Then the equation of continuity gives 



f (n) 



which makes Wq , 

 infinite at 



and therefore Wn if it contains 



^OC 



n 



Wn 



^0 

 Any other dependence of 

 on C than exp(-5^) would, of course, not make 

 Vq vanish at infinity, for the part of Vq that 

 arises from Wg would not be able to cancel out Eq. 

 (A. 13) at |n| = ". 



Hence Wg and Vg are uniquely given by Eqs. (A. 3) 

 and (A. 13). Using them in 



Lhi = Vgh 



0"0r 



+ W„h 



and 



0"0i;' 



(A. 16) 



-fn2 



^0 =^ 



Ce 



(n^ + c^) 



dn, 



(A. 13) 



(L - 3)Wi = 



M + VgWg^ + WgWg^, 



(A. 17) 



we find that 



FIGURE A. 2. Flow pattern for (Vg + XVj , Wq -h XW, ) . 

 aeirl' = 0.2. 



Wi = — 



(A. 18) 



I have computed hj numerically from Eq. (A. 16) 

 and the boundary conditions, and therefore Wj . 

 The velocity component, Vj , is then found from the 

 equation of continuity. The flow pattern corres- 

 ponding to (Vi,Wi) is given in Figure A.l, where 

 the streamlines are shown, with Tj = on the 5 

 axis and A6Tr4'i = 0.2. 



Then the flow field for 



V 



Vg -I- AVi and W = Wg + AWj 



is shown in Figure A. 2, with A = 30, where the 

 AeTry = 0.2. 



It is clear that the "streamlines" do not close 

 to form closed eddies, as in the figures of Yih's 

 paper. Thus the effect of the pressure gradient 

 cannot be neglected in the problem studied by Yih. 



In past studies of jets and plumes, where the 



