Lagvangian Solutions of Unsteady Free Surface Flows 



Fig, 2(a). The Cell In the Reference Plane (a,b) 



AQ2 = --g^(Z2-Z4) 



I AQ4 = -AQg 



INCREMENTAL VELOCITY CHANGES 

 WHICH DISPERSE THE CONTINUITY 

 RESIDUAL, Rg (PURE STRETCHING) 



AQ4 = -A02 



INCREMENTAL VELOCITY CHANGES 

 WHICH DISPERSE THE CIRCULATION 

 RESIDUAL, Rj (PURE ROTATION) 



Fig. 2(b). The Cell in the Physical Plane (X,Y) 



such a way that p is some simple analytic function of (a,b). This 

 is particularly desirable because by substituting for p, Pg, p^ in 

 Eq. (9), this can then be integrated over a cell area (as in Section 4B) 

 to produce a convenient additional term, ^^*\j on the L.H.S. of the 

 cell Eq. (18). Since the expression for 9*'*' will depend upon that 

 choice of p(a,b) an example will illustrate this. 



If p is to be constant cilong the free surface, AB, Fig, 1, 

 and eilong the bed, CD, it may be possible to choose Z° such that 

 p is a linear function of b, say p = p^p (1 + yb) where 

 "Y = Pab/Pcd ■ ^ ^^^ b = 1 on AB. Then, 



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