Sec. 433 



DELINEATION OF SOURCE-SINK DIAGRAMS 



55 



Fig. 43.C Radial Stream Functions foe a 2-Diml Source-Sink Pair and the Resultant Stream Functions 



Leave a clear space at the center of each sheet 

 where the lines would otherwise be crowded 

 together. One way is to use large circles for the 

 source and sink. Extend the radial lines as far 

 as the drawing area permits or requires. The 

 radial flow from the source is designated as the 

 ^/so- or ^o-system; that from the sink as the 

 ^SK- or \^K-system. Number the radial lines 

 around both source and sink, starting with zero 

 at the reference Hne in the positive x-axis on the 

 right and increasing numerically hoth ways to the 

 negative x-axis. This applies both to the diagram 

 shown and its mirror image. As previously men- 

 tioned, any convenient set of equidifferent 

 numerical values may represent the outflowing 

 and inflowing stream functions, but the sets 

 must be identical. The lines i^o = and ^^ = — 

 both he over the reference trace and point toward 

 the source of the uniform-stream flow and opposite 

 to its velocity vectors. The hnes radiating from 

 the source are marked with positive signs to 

 indicate positive stream functions, while those 

 pointing toward the sink are marked with negative 

 signs. 



With the reference-plane trace as a base, super- 

 pose these separate radial-flow diagrams with 

 their centers at the selected source and sink 

 positions. Fasten the source and sink sheets in 

 place, then over both diagrams lay a third sheet 

 of tracing paper or cloth, or other transparent 

 material, upon which the straight reference line 

 is drawn and the source-sink positions are marked. 

 The hnes visible through the top sheet then appear 

 as the straight-line portions of Fig. 43. C. 



The next step is the construction of the flow 



pattern representing the liquid movement from 

 the source to the sink. For this purpose it is more 

 convenient to start from the axis between the 

 source and sink instead of from the axis stretching 

 to the right from the source. 



Considering four units of liquid emanating 

 from the source in the sector between the hori- 

 zontal reference line and the radial- stream-function 

 hne \po = +28, represented by the four broken- 

 Une vectors, these units must enter the sink 

 between the reference line and the radial line 

 ^K = —4. Where the stream-function hne 

 fo = +28 crosses the hne \pK = —4, at the point 

 Y, the value of the resultant^ stream function 

 i^c = +28 + (-4) = +24. The numerical 

 stream-function values of the fc system are dis- 

 tinguished by single bars over the numerals. 

 However, the four liquid units flowing out of the 

 source and into the sink, corresponding to the 

 fuU-hne curved vectors shown, actually pass 

 between the point Z, where rf/c = +28, and the 

 reference trace. Whereas the streamline \pc = +28 

 is tangent to the radial line ^o = +28 at the 

 source and to ^^ = — 4 at the sink, this streamhne 

 moves downward or inward, away from those 

 hnes, as the distance from the source or sink 

 increases. In other words, instead of passing 

 through Y it passes through Z. 



The intersections of other parts of radial Hnes 

 along a perpendicular to the reference hne, 

 midway between the source and smk, occur at 

 the points m^ked_Y, X, and_W, where the values 

 of ^c are +24, +20, and +16, respectively. Two 

 other points, M and N, on the streamline 

 if/c = +16, are given by the intersection of 



