- p{V^-a'^) + V^ [118k] 



where V is the uniform velocity of the fluid at infinity and p^ is the pressure there; p is the 

 density of the fluid. Cases in which a body is moving in steady translation with the fluid 

 at rest at infinity can be reduced as usual to the case of steady motion of the fluid by impart- 

 ing to everything a velocity equal and opposite to that of the body; the pressure and the forces 

 on the body are not thereby affected. 



119. POTENTIAL AND STREAM FUNCTIONS FOR A UNIFORM STREAM, 

 A POINT SOURCE OR A POINT DIPOLE 



The velocity potential for a uniform stream having velocity l) toward negative x can 

 be written 



4>^ Vx [119a] 



for then, by ('118b,c,d), u = - V, v = 0, «,- = 0. The streamlines are straight lines parallel 

 to the a>axis. 



For some purposes it is convenient to regard such a streami as having axial symmetry 

 about some chosen line parallel to the velocity, and to define an axisymmetric stream function 

 with respect to this line. Let the positive direction along the line be taken toward positive x, 

 and through any point P or (a;, y, z) draw a circle of radius w^ about the chosen line QQ' as 

 axis, as in Figure 197. Then, across any surface bounded |)y this circle there flows in unit 

 time a volume nTS^V of fluid. Hence, according to the usual definition as stated in Section 16, 

 the axisymmetric or Stokes stream function at P is nliy^lJ/^Ti or 



ij, ^ - VZ>^ [119b] 



2 



In general, any line parallel to the flow may be chosen as the axis of symmetry, and 

 oT represents the distance of P from this line. If the a;-axis itself is chosen, oJ^ = y'^ + z^. 



If the velocity of the uniform stream is V toward a direction whose direction cosines 

 are ^, m, and n, the velocity potential becomes 



4) = ~ U (Ix -^ my ■¥ m) [119c] 



as is easily verified from [119a] by a rotation of axes. 



For the point source, the velocity potential ^ at a point {x,y,z) is, from Section 12, 



<^ = — [119d] 



299 



