56 



1IM)K()I)\ .\ \.\IK:s IN Mill' 1)1 sU.\ 



Sec. (jJ 



^o = +20 plus \fiK = — J t'<|ual.s 4>r = +><», as 

 well as bv 4'o = +'-S plus ^^ = —12 equals 

 ^f. = +1(5. These permit the streamline ^c = +1G 

 to be drawn, rememheriiip that, like the first one 

 descrilMxl, it is tangent to (lie radial linc^o = +1C 

 at the source and the line ^k = —16 at the sink. 

 Actually, for 2-iliml How between a source-sink 

 pair, all the streamlines of the ^c series are drcics 

 which pass through the source and sink centers. 

 These circles have a radius of s cosec 0(thcta), 

 where s is the half-tlistaiice between source and 

 sink, and is the angle between (1) the radial 

 stream function at the sourer for which the circle 

 is drawn and (2) the reference axis. Considering 

 the radial source streamline ypo = +1G, is ir/2 

 or 90 deg and_cosec 6 = \; the radius of the circle 

 for ^c = +16 is therefore s, half the distance 

 from the source to the sink. This particular 

 circle has its center on the reference line and is 

 tangent to the radial strcamhncs if/o — +H> and 

 ^K = —16 at the source-sink axis. When cxteiidetl 

 below the source-sink axis in the diagraraj_thiri 

 circle becomes the stream function ^^ = -flfi • n 

 that side. Similarlj', all other circles extended 

 below the axis become what might be termed 

 supplementary streamlines in that region. For 

 example, the streamline ioTjpc = +4 above the 

 reference line and tpc = +28 below it are parts 

 of the same circle. Numerically, the stream 

 function for these two parts of any complete 

 circle total 32, the same value as for half of either 

 the source or the sink. The centers of the circular 

 streamlines fall on intersections alreadj' given liy 

 the radial-flow lines in the diagram, because if the 

 quadrants are divided into a whole number of 



sectors, there is always a radial stream-function 

 line from the source at right angles to another 

 line from the sink. For example, since the circular 

 streamline \pc = +12 is tangent to rpo = +12 

 at the source, its center lies on the line rpo = +28, 

 where 12 -|- 16 (for one quadrant) = 28. 



On this basis, and with the additional inter- 

 sections on the diagram, the remainder of the 

 circular streamlines are drawn. AVhereas these 

 lines are approximately equidistant from each 

 other at the source and sink centers, they become 

 increasingly farther apart at greater and greater 

 distances from the source and sink, bccau.-je of 

 the slowing down of the liquid in the widening 

 crescent-shaped sectors. However, once drawn by 

 subdividing each quadrant into a given number 

 of sectors, the circular stream pattern nerer 

 changes for any change in source-sink strength or 

 spacing. Once carefully drawn, it can be enlarged 

 or reduced photographicallj^ to suit the source- 

 sink spacing. The numbers on it are altered to 

 .suit the source-sink strengths. 



This completes the first step in the operation. 

 The second step is the combination of this circular 

 source-to-sink flow pattern with the 2-diml 

 uniform-flow pattern parallel to the reference 

 plane. The uppermost sheet of tracing cloth or 

 paper containing the circular-streamline pattern 

 of the \{/c S3'stcm is now transferred and superposed 

 on a sheet which has parallel streamlines of the 

 uniform flow drawn on it. Following this, the 

 i^f/ flow is combined with the ^c flow by adding 

 the ipc and the i^y stream functions algebraically 

 where the strcamhncs cross each other. For 

 example, in the right-hand portion of Fig. 43. D, 



\ Uiiifomi-flow 



■v stream tuifclions^'^'' 



aurtace 



Fi'i. 'l-'t.!) Uankink. Stiikam Form and Simnoit.VDi.vo Stiikam Vuiw Uesi'i.tino Fmni Inskhtion or a '.J-Himi. 

 Soi;k(;k-Sink 1'aiii i\ a U.\ih-oiiM Stiikam 



