Sec. 43.S 



DELINEATION OF SOURCE-SINK DIAGRAMS 



63 



Cones of inflow for unity stream function □ 

 between -p^A and fi-^ =0 



Funnels of outfiow for 

 unity stream function 

 between V'q'Z and TiQ'\ 



Fig. 43.K Definition Sketch foe 3-Diml Soubcb-and-Sink Stream Functions for Graphic Construction 

 OF Body Shapes and Flow Patterns 



liquid flows, is a conical "funnel," surrounding 

 the cone. The same quantity rate of liquid is 

 assumed to pass through the annular spherical 

 surface at the base of the "funnel." This funnel is 

 diagrammed at A in Fig. 43. K where, to make it 

 stand out more clearly, the cone around the 

 source-sink axis is omitted. Diametrically opposite 

 one funnel is an identical one, drawn at the left 

 side of the 3-diml source in the figure. With a 

 3-diml stream function ^o = at the source-sink 

 axis, this stream function has a value of unity 

 (1.0) at the outer surface of the cone and the inner 

 surface of the funnel. It has a value of 2 at the 

 outer surface of the funnel, indicated in the 

 figure. Surrounding the inner funnels are a series 

 of other larger funnels, also concentric with the 

 source-sink axis, terminating in zones of area 

 equal to the first, through which unit quantity 

 rates of Hquid flow in and out. 



For a given spherical radius and radial velocity 

 at the source and sink, the quantity rates of 

 liquid flowing through the cones and funnels are 

 directly proportional to the areas of their spherical 

 bases and zones, respectively. For the construction 

 of the 3-diml stream forms to be described, the 

 surface of each source or sink "sphere" is divided 

 into an integral number of zones of equal area, 

 all symmetrical about the axis. 



The area of a spherical segment or zone is 

 expressed by the formula 4s = 2irRb, where R 

 is the radius of the sphere and h is the height of 

 the segment or zone in the direction of the polar 

 axis, normal to the planes dividing the zones. In 

 this case the polar axis corresponds to the source- 

 sink axis, and to the direction of flow. Hence a 

 selected or convenient spherical radius along the 

 direction of flow is divided into a number of 



equal parts corresponding to the selected stream 

 functions for each "hemisphere" of the radial 

 flow, such as k = 10. When perpendiculars are 

 erected on this radius at the spacing h = R/k, 

 the intersections of these perpendiculars with a 

 circle representing the spherical surface give the 

 limits for the bases of the cones and the funnels 

 through which equal quantities of liquid move 

 out from the source and in toward the sink. 



The radial-flow diagram for a 3-diml source (or 

 sink), as projected on a page to represent the 

 traces on any plane passed through the source-sink 

 axis, takes the form indicated in diagram 1 

 of Fig. 43.L, where only one-quarter of the pattern 

 is shown. That for the sink is the same, with 

 negative signs. 



The uniform flow approaching from a distance 

 is assumed to be made up of stream rods and 

 tubes, circular and annular in section and con- 

 centric about the extended source-sink axis, as 

 shown in diagram 2 of Fig. 2.M of Volume I 

 and in Fig. 42.A. The inner stream "rod" of 

 Fig. 2.M has the form of a cylinder of any selected 

 radius, depending upon the value of the 3-diml 

 uniform-stream function assigned. The outer 

 stream "tubes" have radial thicknesses such that 

 the quantity rate of flow through each of them 

 equals that through the center "rod." Their 

 divisional surfaces therefore have equidLfferent 

 stream functions, based upon their common axis 

 as a reference. The prospective inner and outer 

 radii of the rod and the annular stream tubes are 

 determined graphically by laying down a second- 

 order parabola y' = ex, with its vertex on the 

 source-sink axis. Perpendiculars are then erected 

 at uniform intervals along the source-sink axis 

 in the manner indicated at 2 in Fig. 43.L [Taylor, 



