Figure 15 — Illustrating the definition of the 



stream function in axisymmetric 



three-dimensional motion. 



stream function \Jj thus defined, often called 

 the Stol^es stream function, represents the flow 

 between P and the axis taken per radian of 

 rotation about the axis. 



As in two-dimensional motion, however, 

 it is often convenient to relax the definition 

 somewhat by adding an arbitrary constant to tli. 

 The dimensions of i/f are volume per unit time 

 or L /t. In any plane through the axis, the 

 curves, i/y = constant, are again the streamlines. 



As coordinates, take distance x along 

 the axis measured in the positive direction, and 

 the distance oTfrom the axis, and let the 

 corresponding components of the particle 

 velocity be q^ and q^^. (Thus the x- component 

 is denoted by u only when Cartesian coordinates 

 are employed.) Then, if P is displaced a dis- 

 tance dx parallel to tie axis, the flow across 



AP is increased by Inoi'q'^dx, and this equals 1-nd\b. Or, if P is displaced a distance (fST 

 outward from the axis, the flow in increased by - l-nTSq doi = Ind^i. 'ience 



1 d^l, 



1 dxli 

 To d X 



In a similar way it can be shown that 



1 di'f 

 CO an 



[16a, b] 



[16c] 



where q is the magnitude of the velocity and drJi/dn is the space rate of change of ih in tiiat 

 perpendicular direction which is obtained by a clockwise rotation through 90 degrees from the 

 direction of the velocity. 



If a velocity potential exists, from Equations [6p, q] 



dx 



aco 



Thus ^ and i/» are related by the equations 



1 dil/ d<f) 



ST dx 



[16d, e] 



[16f, g] 



d X CO dco dco' 

 It is to be noted that in the axisymmetric case <^ and i/y do not have the same dimensions. 



Since x and oTare really Cartesian coordinates, and (f> does not vary in the third direc- 

 tion, <jS will satisfy the usual Laplace equation in terms of x and STalone. The differential 

 equation for lA is found by substituting from [16f, g] in the identity, d^4>ldxd7^= d^cf^/dco'dx. 



28' 



