Figure 226 — Illustrating the equation of 



continuity in terms of the orthogonal 



curvilinear coordinates A, /i, v. 



(^l.fil.i'l) 



Consider first the pair of faces approximately perpendicular to the coordinate direction 

 for A. The face at which A = A^ has sides of length 8s , Ss^ and an area Ss 5s^. Fluid is 

 entering the element across this face at a rate q^ 8s 8s^. The rate at which it is leaving the 

 element across the opposite face, on which A = A^ + SA, can be written 



d 

 IX ^V ^^v + ^^ ^ (^^ ^^y- ^'> 



The difference between this expression and the last, or 



SA — (on Ss Ss^ ), 



is the net rate at which fluid is leaving the element by passing across this pair of faces. 



Treating the other two pairs of faces in a similar way, and adding the three expressions 

 thus found to obtain the total rate of outflow, which must be zero, it is found that 



SA — (o,. Ss Ss ) + Su — (q 8s &s<.) + 8u — (y,, 8s. 8s ) = 0. 

 ^^ '■'/A II u' >^ ^^ ^'^11 V \' Q^ ^"iv A ii' 



Dividing by 8\8ii8u and noting that SA, 5//, 8u are constants. 



aA\s^ Su M 0,1 \8v SA >/ du\^SX 8,1 ^ 



This is the equation of continuity for an incompressible fluid expressed in terms of any 

 orthogonal coordinates. 



343 



