15. TWO-DIMENSIONAL OR LINE SOURCES, SINKS, VORTICES, AND DIPOLES 



A two-dimensional source or uniform line source is said to exist on a line when the 

 fluid flows uniformly away at right angles to the line. In the diagram that represents the flow 

 in a particular plane, the line is represented by a point. By considering the flow across a 

 circular cylinder having the line as its axis, it is easily seen that, because of the assumed 

 incompressibility of the fluid, the velocity is 



?=4 [15a] 



0) 



where 4 is a constant, positive or negative, and tTdenotes distance from the line. The 

 volume flowing outward per unit time per unit length of the cylinder is thus 2jtA; either this 

 quantity or A itself may be called the strength of the line source or sink. On the line itself 

 the velocity is not defined. 



The corresponding velocity potential is 



(?!.=- 4 In Sr [15b] 



where In stands for the natural logarithm, since then v = - d4>/dSi'. It is impossible by adding 

 a constant in to prevent it from becoming infinite at infinity; this complication in two 

 dimensions is sometimes annoying. 



The line source can also be built up by distributing infinitesimal three-dimensional 

 point sources uniformly along the line. The constant A then represents twice the point-source 

 strength per unit length of the line, if by source strength is meant the constant A in such 

 formulas as [12a, b]. 



The potential of a two-dimensional line dipole can be obtained by differentiating that 

 of a line source. Since Sr= (a;^ + y^)^ in terms of Cartesian coordinates defined in a plane 

 parallel to the flow, a possible potential for a line source at the origin is 



By differentiating (f>. with respect to x and using the principles stated in Section 7, the 

 following solution of the Laplace equation is obtained, representing a line dipole of strength 



a;2 + y2 ZS 



Here ^ is a constant and is a polar angle measured from the a;-axis, so that cos 6 =x/{x +y )''. 



In using this formula, irrespective of its mode of derivation, d may conveniently be 

 defined as the angle between two planes intersecting along a fixed line, on which the line 

 dipole is situated, and eras a coordinate representing perpendicular distance from this line; 

 one of the planes, from which 6 is measured, is fixed in position, the other rotates about the 

 fixed line. When the motion is studied in a plane parallel to the flow, the intersection of this 

 plane with the fixed plane is a line called the axis of the dipole. 



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