﻿Equations of Motion of a Viscous Fluid. 441) 



then the a component of (1) becomes 



= F. - *! & + , ("I f(div. v) + V 2 « - f V*. 



h ^p - - r^ 3 



QOC ll 2 0« /^3 0« 



- 2 ^fyUd + u ^rM-- ■ {9) 



The corresponding equations in v, w, can at once be written 

 down from symmetry. 



Application to cylindrical and polar coordinates. — If we 

 take cylindrical coordinates sr, #, z, we have 



li x = /i 3 = 1 and h 2 = — . 



in 



Most of the terms in (9) vanish and we have * 

 Xu c 2 „ i3/i ri a M 2 3fi 



|_%^ = F - i|g + J" * JL(div.Y) +V 2 , - A + 2 *H 

 SB* ot « p*r30 [_3^-30 ; -a- ™ 2 ~ddJ 



® w & ! Si? . rl 3 ,,. x , v-72 1 



-— — £ . — ^ + „ (div. v) 4- S7 2 w 



$)t " p 3s U3~ ' J" 



From (3) 



t 13/ n . 1 3v ~dw 



div. v = ^ (<mi) +-Sa+^~ 

 and is zero if the fluid is incompressible, while from (8; 



£> _3, c> ^ 3, , 3 

 3X~3* "3*r + ^30 " V 



If r, #, <£ are spherical polar coordinates, 



h=l, // 2 = ~, /'3= .-7. 

 r /-sin fr 



* cy. Basset, ' Hydrodynamics,' vol. ii. p. 244, where the equations are 



given lor an incompressible fluid. 



Phil. Mag. S. 6. Vol. 29. No. 172. April 1915. 2 G 



