Sec. 43.2 



DELINEATION OF SOURCE SINK DIAGRAMS 



53 



Unifoi'm- -2.0- 



flow 

 stream _|g_ 



function 



Reference avis +oo 



Fig. 43. a Uniform-Flow and Radial-Flow 

 Stream Functions for a 2-Diml Soubce 



starting with at the positive x-axis and going 

 around both ways to 32 at the negative x-axis. 

 Only those hnes above the a;-axis are shown in 

 the figure. 



Off to the right draw the stream comb for the 

 uniform flow and assign suitable equidifferent 

 values, marked from i/'^ = through ^c = ~20 

 in Fig. 43. A. The negative signs for the uniform 

 flow signify liquid moving opposite to the positive 

 direction of the .T-axis. The stream comb is then 

 extended to the left to form the horizontal 

 straight-line portions of Fig. 43. B. In this and 

 succeeding layouts the lower half of the diagram, 

 that is, the mirror image of the upper half, is 

 omitted to save space. 



The horizontal hne extending from — » through 

 the source to + <» , coinciding with the a;-axis, 

 represents the trace of the reference plane for the 

 flow diagram to be constructed. If four radial 

 vectors are drawn from SO, between the lines 

 4'so = ^ and 4/so = +4, they represent sche- 

 matically the four units of the quantity rate of 

 flow from the source in that sector. If four parallel 

 vectors are drawn between the trace of the 

 reference plane, in the x-axis, and the horizontal 

 line ^p = - 4, they represent in the same manner 

 the quantity rate of flow in that part of the 

 stream. These eight vectors are drawn in broken 

 lines in the figure. The two flows, in which the 

 quantity rates are each numerically equal to 4, 

 buck each other along the region AB in Fig. 43. B. 

 At the intersection B of the radial line ^so = +4 

 and the horizontal line ^^ = —4, the resultant 

 stream function value is zero. Assuming for the 

 moment that the reference plane is impenetrable, 

 the uniform flow between the x-axis and ^pu = ~^ 

 is deflected upward. There is estabhshed a line 

 ^s = between A and B, across which no liquid 

 passes. 



The liquid in the uniform stream \}/v , in quan- 

 tity rate equal to —4, turns upward as indicated 



by the full-line vectors beyond B. It then goes 

 through a gap where the algebraic sum of the 

 radial and uniform-stream functions is equal to 

 —4. Such a gap is that portion of the radial hne 

 4'so = +4 represented by the segment BF, lying 

 between the parallel horizontal lines whose 

 stream functions 4/u are —4 and —8, respectively. 

 The original four units of liquid in the parallel 

 flow turn up through this gap and cross the line 

 BF. At the point F, therefore, the stream function 

 ^s of the combined flow is equal to —4. The four 

 units of liquid in the radial flow turn upward 

 inside the point B. 



The second four units of uniform flow, between 

 \pu = —4^ and ypu =^ ~8, now approaching a new 

 barrier at F, turn up between F and J, whereupon 

 the point J lies on the new streamline ^s = —8. 



If the procedure just described is followed for 



Fig. 43.B Combination of Uniform Flow With 

 Radial Flow Feom a 2-Diml Source 



groups of eight liquid units instead of four, a 

 point C is established at the intersection of 

 i^so = +8 and ^u = ~8 where the resultant 

 stream function value \{/s is again zero. There is 

 also found a point G where the stream function 

 of a new series is lAs = —4 and a point K where 

 ^s = —8. 



A whole pattern of intersections is thus quickly 

 determined, at which the stream functions xf/g of 

 the third series are equal to the algebraic sums 

 of those of the second and the first series. For 

 example, at E it is found that i/'p = —16 and 

 4'so = +16, so this point is on the line \ps = 0. 

 At L, xpu = —20 and 4'so = +12, so this is a 

 point on the streamhne 4's = —8. 



The various points so determined are all inter- 

 sections of radial stream-function lines having 

 values greater than the uniform stream-function 

 lines by increments equal to the value of the new 

 stream function 4's ■ This means that the parallel 

 stream flow that lay below 4'u = —4 at infinity 



