To further illuslratc lhc use of lhc Schwarz-Chrisloffel

Chapter 0, Problem 11.17

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To further illuslratc lhc use of lhc Schwarz-Chrisloffel lransfonnation. let us find lhc complex polenlial for lhc flow of a fluid in a channel with an abrupt change in ils bread ch (Fig. 188 ). \Ve lake our unil of lcnglh such that chc brcadlh of chc wide parl of che channel is ;r units: then /m. where 0 < '1 < I. rcprcscms lhc brcadlh of lhc narrow parl. Lcl the real conslanl \10 denote lhc velocity of lhc fluid far from the offset in lhe wide part lhus Jim \I = \"I where lhc complex variable \I reprcsems the velocily vcccor. The rme of flow per unit deplh ch rough lhe channel. or lhc strenglh of lhe source on the le fl and of lhc sink on 410 TI IE SCI IWARZ-CI IRISTOlll~L TRAl'\SIDRMA TIO!\ CllAP. 11 \' I' :ri ll'.1 Im __ , ___ ~t_t ___ _ I X X, X 1 II FIGURE 188 the 1ighl. is then ( 1 ) The cross section of the channel can be considered as the limiting case of the quadrilateral v ... ith the ve11ices u: 1 u 1 2, u_,. and Wi shmvn in Fig. I 88 as the first and last of these vertices arc moved infinitely far to the left and right. respectively. In the limit. the exterior angles become k_,;r = 2 As before. we pnx:eed fonnally, using limiting values whenever it is convenient to do so. If we write .r 1 = 0. XJ = I, .r4 = oo and leave x 2 lo he