HYDEAULICS AND ITS APPLICATIONS 



Differentiating the equations (7) and (8), so as to obtain the values of 



) 



2 -, etc., in the case of a viscous fluid, and inserting the values thus 



obtained in equations (9), the first of these becomes 



y a p 2 a /8 u , a v , a -w\ , ^ 2 u 



~ ?Tx IT ^ die VFI- "*" 97 + si / " ^ a~^ 



, /8 2 U 



-f /x - . 



Va ?/ 2 



; , 



~ -- - + ^-o 



< x . o z a 2 2 



From the equation of continuity we have 



3j.. + i + a s= 



8^^87/8^ - 



(/ /f 



.. 

 Also 



. 

 + 



d x d y d x d 

 So that the equations become 



d w , 

 P TT = / ) ^ 





r 2 



= A(^ + 



')= 



where v 2 denotes the operator ( : 



8 2 





v) l- 



8 2 



(10) 



The terms involving M in these equations are complex, and for purposes 

 of practical application to hydraulic problems are usually neglected. If 

 this is done, the equations of motion for a non-viscous, incompressible 



fluid become 



d u d i 



d v 



dw dp 



Q *f ^ _ ^L 



(U) 



or, writing ^ in terms of - , etc., from equation (2), and dividing 

 a c v x 



throughout by p 



v I d p 8 u . 8 u . d u . d u 



X -%* = u h^^ h w 5 h 5 i 



p d x d x 8 2/ ' 8^'a^ 



18j9_ 8v 8v 9 v .^ 3 t? 



" P Fy " w 8^ "^ * 8~1/ "^ ^ d~z + FT 



1 8 p dw. d IV . dw.dw 



Z - r-^- = 1fcr -- \- V-r -- \- W r -- f-^ 



p d z d x % y d z d t 



(12) 



the Eulerian equations of motion. 



