Desai and Lieber 



Conservation of Mass Equation (the Continuity Equation): 



du u 1 Bv 



-7 + 7 + r — ^ ■ (1.3) 



Br r r 'dd 



The set of Eqs. (1.1), (1.2), and (1.3) can be conveniently reduced to a single 

 equation of the fourth order by introducing a stream function, defined as follows: 



= i(?,5,t) , (1.4) 



such that 



(1.5) 

 (1.6) 



We observe that the stream function xp is defined in such a way that the 

 continuity Eq. (L3) is identically satisfied. By differentiating Eq. (1.1) partially 

 with respect to 6 and Eq. (1.2) partially with respect to r and then eliminating 

 between them the common term B^p 3^3,9 jn pressure, we obtain the Vorticity 

 Transport Equation by using Eqs. (1.4), (1.5), and (1.6) to express u v, and 

 their derivatives in terms of \p and its derivatives. 



Vorticity Transport Equation: ~ - /' ^ . 



/~'*°'" Bt r ^e 3r ?3? dO 



where the Laplacian Operator is 



32 1 B 1 32 



- i^vV = , (1.7) 



V 



Br2 r 3? P 3^2 



the Biharmonic Operator is -'' - ^^ •i-'^'"^*-*/- "•' ' 



V^ = V2(V2) 



and, by definition, the Vorticity Field is 



■' • - <. . - - 1 / 3v V 1 Bu \ 1 ~^~ ,^ „v 



2 \B? ? V 20 J 2 



Using the definition of Eq. (1.8) of m, Eq. (1.7) can be written as 



502 



