8 Colorado College Studies. 



Equations of a Viscous Liquid. 



Ct/Uii(lric(il Co-ordinates. — In treating the equations for 

 cylindrical and spherical co-ordinates, it is found. that the total 



rate of displacement, as, for example, on the face C-K" (Fig. 8) 



along the normal, is not expressed by — , for if the velocity u 



(Ir 



is constant there must be an inflow equivalent to . Conse- 



r 



quently, in the following treatment account is taken of this, 

 and then the element considered as uniform in size, /. e., 

 rdo dz is a constant so long as we are considering the ac- 

 celeration in II along r. We may therefore consider the total 

 rate of displacement in the direction of r at surface vdo dz as 



du II 

 dr r 

 It has been shown in the derivation of the centrifugal force 



2 2 2 2 



that the distance BE= , or a velocity . (See Fig. 10.) 



2?- r 



If rv is a variable the rate of change on surface dr dz=zv. 



Therefore the total rate of displacement in direction of t) at 



surface dr dz is 



du 



rdo 



The rate of displacement along z at surface rdo dr is 



du 



di'. 

 For the direction of 0. 



r — is the linear displacement at surface rdO dr 

 dr 



■2v. 



