108. STRAIGHT CHANNEL VARIED IN WIDTH. 



Consider the flow along a two-dimensional channel of infinite length which has parallel 

 straight sides but whose width undergoes a sudden change at one point from h^ to h„, as illus- 

 trated in Figure 178. Let the fluid approach from the right at uniform speed U. Then its veloc- 

 ity will ultimately become uniform again at the value h, U/h^, since the same volume must 

 pass all cross sections. 



e 







y 





A 



c 





1 



D — 



E X 





— eo 



F 















Figure 178 - Treatment of a straight channel abruptly varied in width. 

 See Section 108. 



The mathematical problem is an extension of that in the last section. The walls taken 

 together in the order ABCDEF as labeled in Figure 178 can be regarded as an infinite polygon 

 with two vertices at infinity, AF and BC. At BC, where a change in direction of n occurs, 

 let t = 0; at D and E, with exterior angles -77/2 and 77/2, let < = 1 and t = a> 1, respectively. 

 Then the Schwarz-Christoffel Equation [31a] becomes 



dz 

 dt 



K t -I 



t \ t 



[108a] 



