Bvennen and Whitney 



For Incompressible, planar flow the equations reduce to 

 Continuity ; '^c'^b- YflXi, = F(a,b) (5) 



(or differentiated w. r.t, t): 



UfaYo- U.Yi, + V,Xb - V^X^ = (6) 



Motion : U^X^- U,Xj, + Vj,Y^. V^Y^=:- r{a,b). (7) 



By introducing the vectors Z = X + iY and W = U - iV, (6) and (7) 

 conveniently combine to: 



Z^Wb - Z^Wq = - r(a,b). (8) 



Other types of flow have also been investigated. For example, 

 in the case of a heterogeneous , or non-dispersive stratified liquid in 

 which p is a function of (a,b), Eq. (8) becomes: 



ZoWb- Zi,W„ = - [ r(a,b)],,,^ - ij (X^^ - F)(Pj,X^ - p^X^,) 



+ (Y,, - G)(p^,Y^ - p^Yj,) dt. (9) 



The integral term therefore manufactures vorticity. The methods 

 developed for a homogeneous fluid in Sections 4A to D are naodified 

 in Section 4E to include such effects. 



III. OTHER PARAMETRIC PLANES 



It may be of interest to digress at this point to consider other 

 parametric planes (a,b), which are not necessarily Lagrangian, 

 That is to say the restrictions X|(a,b,t) = U, Y^(a,b,t) = V are 

 abandoned so that U,V are no longer either Eulerian or Lagrangian 

 velocities. Provided J = 8(X, Y)/8(a,b) #0, or oo, the equation for 

 incompressible and irrotationcil planar flow remains 



ZoWb-ZbW^=0- (10) 



To incorporate one of the advantages of the Lagrangian system, it 

 is required that the free surface be fixed and known, say on a line 

 of constant b. Then the kinematic and dynamic free surface conditions 

 are respectively 



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