450 Lieut.-Col. A. R. Richardson on Stream-line 

 Along the stream-line ^r = ir, i.e. t< —1, 



1_ — gCsinh^V — t m 



dw 



,\ = and q goes from e to 1. 



Hence the equation represents the flow of a fluid through 

 a semi-infinite pipe in cases where the issuing stream is 

 disturbed. 



(ii.) Description of the motion. 



t = ( — 1 + e-* cos yjr)-ie-* sin >jr, . . (17) 



/- • * , • *. — e-^sm'xfr 



\t — ? + "?, SV ~ 2 ' 



dw 



— e c cos /3£ sinh py — ic sin /3£ cosh /3ij __ _ gi'0 



</ is finite everywhere except at co . 



0= -c sin /3% cosh /3<n (18) 



Now 

 j. _ 1 £ -<p sin \Jr 



V2 v/a/(— ^l + ^- (p cos^) 2 + e- 2 ^in 2 ^ + (l — <?-*cos^) 



77 = — - vA/l — Ser* cos ~y + e-*$+ (1 — *-<P cos^r). 

 Hence, as (j> is large and positive, 



i. e. all stream-lines are parallel at the entrance. 



Also q = e- cainh P everywhere over the cross section. 



Owing to the presence of the exponential e~Q this state of 

 affairs holds close up to the mouth of the pipe. 



The stream-lines begin to curl as the mouth of the pipe is 

 approached. 



