314 MOTION OF THE GAS SPHERE 



An inward radial flow is similarly described by a simple sink. Such 

 sources and sinks are of course purely conceptual devices without physi- 

 cal reality, and are introduced merely for mathematical convenience. 

 If the spherical boundary of the flow has a radius a and velocity da/dt, 

 the rate of flow across the boundary is 47ra2 da/dt, as 4iira'^ is the surface 

 area. For incompressible flow, this must also be the rate at which 

 fluid is emitted by a source at the center and the strength of the source 

 is therefore M = o? da/dt. The velocity potential (p{r) at a point r 

 from the center corresponding to this flow is, from Eq. (8.21), 

 ^(t-) = a^ da/dt- 1/r = M/r. From this equation, the velocity poten- 

 tial is constant for a fixed value of distance r from the source, and the 

 equipotential lines for which (p = constant are thus circles. The flow 

 of fluid is normal to these lines and the streamlines, across which there 

 is no flow, extend radially from the source. 



The equipotentials and streamlines shown in Fig. 8.12a are seen to 

 be analogous to the equipotentials and lines of force about point elec- 

 tric charges or magnetic poles, and the concept of a hydrodynamical 

 source may thus be used to analyze flow problems in the same way as 

 corresponding electrical problems. A further analogy which can often 

 be used to advantage is that of fluid flux with electric or magnetic flux. 

 In problems with axial symmetry, this flux is expressed by the stream 

 function \p introduced by Stokes,^ the value of which at a point not on 

 the axis is defined to be (l/27r) times the rate of flow of fluid through any 

 surface bounded by a circle around the axis of symmetry and passing 

 through the point. Its value is thus determined by the velocity distri- 

 bution, and constant values of the function represent the streamlines 

 (actually surfaces) across which there is no flow. Hence the condition 

 \f/ = constant must be satisfied at geometrical boundaries where the 

 flow is parallel to the surface and the stream function is conveniently 

 used in applying such boundary conditions. 



A second type of source which proves useful is formed by the combi- 

 nation of a source and sink of equal strength. The strength of the 

 combination is taken to be the product of the source strength and dis- 

 tance between the two. If the separation is reduced to zero while keep- 

 ing this product unchanged, the combination is called a dipole source 

 analogous to an electrical dipole, its direction being defined as that of 

 the line from the sink to the source. The velocity potential of such a 

 dipole at any point is easily computed from its definition with the result 



/x cos 6 



<p(r, e) = -— 



where B is the angle which the radius vector r to the point from the 

 dipole makes with its axis. 



8 See, for example, Milne-Thomson (74), Chapter XV. 



