EQUATIONS OF MOTION 



63 



Here the axes of co-ordinates have been taken to coincide with th 

 principal axes of stress. If, however, these are transferred, exactly similar 

 equations are obtained. 



In the case where gravity is the only external force acting on the fluid, 

 X = o, Y = o, and Z = </, g acting in the opposite direction to that of 

 z increasing. 



So far, the equations are applicable to motion of any kind, whether 

 steady or unsteady, but in their 

 present form are not obviously z 



useful for the solution of any 

 practical problem. So long as 

 the motion is sinuous and 

 irregular nothing further can 

 be done with them. 



ART. 21. APPLICATION 

 STKEAM LINE MOTION. 



TO 



If, however, the fluid be 

 moving with definite stream 

 line motion, these equations 

 can be considerably simplified. 

 Suppose a particle moving with 

 stream line motion from 0, in 

 the direction OS (Fig. 31), with 

 velocity V 8 , the space OS being 

 S s. The direction cosines of 



this motion in the directions OX. OY, OZ, are -= . -~ 



d s d s 



FIG. 31. 



d z 

 d s' 



Also 



d x 



ds 



u 

 V s 



d y _ v _ 



d~* ~ V~ s ~ 



d z 



ds 



_w___ 



V s 



Again, geometrically we get 



F 8 2 = u* + v 2 + w 2 (18) 



And V s =lu-{-mv-\-nw. (14) 



For stream line motion, and with only gravity acting, the general 

 equations (12) may then be written 



