12. THREE-DIMENSIONAL SOURCES, SINKS, AND DIPOLES 



Some important types of flow in incompressible fluid of uniform density will now be 

 discussed briefly in preparation for the detailed studies to be made in later chapters. 



Suppose that incompressible fluid is flowing radially outward from a point P with a 

 velocity q that is a function only of the distance r from P. Then, if a sphere of radius r is 

 drawn with its center at P, a volume inr'^q of fluid flows outward from this sphere per second. 

 Since this volume must be the same for all spheres centered at P, p-q must be a constant, and 

 it is possible to write 



?=A [12a] 



where ^ is a constant. The volume of fluid flowing outward per second from P is then 477/4. 

 A velocity potential exists; for, if 



=i, 



A 



T 

 1 = - 



dT 



At P, q is not defined and a singularity is said to occur. It may be imagined that there 

 is a source at P in which fluid is being created at the rate 4n-i4. In Lamb's Hydrodynamics,^ 

 4rr/l is called the strength of the source and is denoted by m\ in Milne-Thomson's Theoretical 

 Hydrodynamics,-^ the symbol m is used for A itself and is called the strength. If A is negative, 

 the flow is inward and a sink may be imagined to exist at P, in which fluid is being annihilated 

 at the rate AiuA. The term "source," when not specifically contrasted with "sink," will be 

 intended in an algebraic sense, covering both sources and sinks. A flow of this type could be 

 produced by a sphere with fixed center whose radius varies with time. 



To find the distribution of pressure in the fluid, substitute in the pressure equation or 

 Equation [9g] q = A/r'- and 



^A^\dA 



- ■ - dt T dt 



Then 



l^ldA_A^ ^^(,) 

 P T dt 2t* 



if for simplicity p is written for p^. At r = «., p = p P(«). Hence if p^ denotes the pressure at 

 infinity (in excess of hydrostatic pressure), assumed uniform all round, 



p^P^_Pf_\p [12c] 



rdt 2r4 

 Other types of flow having a singularity at P can be obtained by differentiating Equation 

 [12b], in accordance with the principle stated in Section 7. Thus, in Cartesian coordinates 

 with origin at P, replacing by (f)^. 



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