Flow of an Incompressible Viscous Fluid. 



83 



equal the rate of conversion of energy of motion into heat 

 energy, or vice versa. 



If u, v, w in these equations denote the component velocities 

 &t a given point in the fluid, and if w, v, ~" represent the 

 velocities of the instantaneous centre of gravity of the 

 element of fluid surrounding this point, while u f , v\ w f 

 represent the velocities relative to this centre of gravity, 

 Reynolds has pointed out * that two equations of energy 

 may be obtained, one dealing with the mean motions w, v, w, 

 and the other with the relative motions ic', v', iv', and from 

 a consideration of these equations has shown that the limit 

 of stability of motion is attained when 



-I 



du 



ch 



u u -j— + u v 

 ax ay 



dv 



• an i i 



+ u w 



da 

 dz 



dv 



, ,av av , dv 



-{ + v u -j- -f- W -j~ + v w' -j- 



ax ay dz 



)> dx. dy . dz. 



.. 



, ,dw , dw , dw 



+ iv u —j — \-w v' — — h w w 

 dx 



-I- 



V&+M* 



dz 



+ 



Uliv' d 



\dy 



/dv' 



+ 



dy 



dvy /du dw'\ 

 dx / \ dz dx / 

 iu/\ 2 



dy) 



>dx . dy . dz. (4) 



The left-hand side of this equation represents the conversion 

 of energy of mean motion into energy of relative motion, 

 while the right-hand side represents the conversion of 

 the energy of relative motion into heat. So long as the 

 first of these terms is less than the second, the motion as a 

 whole is steady, while if the second is the smaller, the motion 

 is essentially unstable, and eddies are formed. 



By integrating equation (4) it may be shown f that in the 

 case of flow through a uniform circular tube, the condition 



for stability is — -~<k where k is a definite numerical 

 constant. ^ 



An examination of the equation, however, enables further 

 general conclusions to be drawn as to the effect of any 

 variation from a state of rectilinear motion, on the stability 



* Phil. Trans. 1895. 



t Scientific Papers, " Reynolds," vol. ii. p. 561. Also Phil. Trans, ibid. 



G2 



