Bvennen and Whitney 



In the case of zero surface tension. 



D. Method of Solution 



It remains to discuss how the equations may be solved to find 

 W at every node. Due to the non-linear terms in (18) and some 



boundary conditions as well as to the fact that a good estlnmate of 

 ^p + i/2 ^^^ i^g made from values at previous time stations, a simple 

 iterative or relaxation scheme was employed. Such a method In- 

 volves visiting each cell In turn and adjusting the W values at Its 

 vertices In such a way that repetition of the process reduces the 

 cell residuals, R, to negligible proportions. But, on arrival at a 

 particular cell, there are an Infinite number of ways In which Its 

 four vertex values can be altered In order to dissipate the single cell 

 residual. However, experience demonstrated that a procedure based 

 on the following changes {^W^ 2 3 and 4) "^^^ superior In convergence 

 and stability to any of the others' tested: 



AW, = - AW3 = a)lR(Z, - Z3)/8A 



(23) 

 AWg = - AW^ = u)lR(Z2 - Z4)/8A 



Here w is an overrelaxation factor and A Is the area of the cell, 

 which Is unchanged with time and given by the expression (15). These 

 Incremental changes have a simple and meaningful physical Inter- 

 pretation. As can be seen from Fig, 2, they are a combination of two 

 changes, one representing pure stretching and the other pure rotation, 

 which dissipate respectively the continuity and circulation components 

 of the residual. 



Having visited each and every cell, the boundary conditions 

 were then Imposed. Where these were given In the forna A.U ' + 

 g^YP*i/2^ C = = Rg, A,B,C being constants and Rg the residual, 

 the following changes were made, the choice being based upon experi- 

 ence: 



AU^*'/' 



AV"*'/' 



B 



J (A^ + B^) 



^ (24) 



The whole process was then repeated to convergence, 

 E, Inhomogeneous Fluid 



In a non-dlsperslve , Inhomogeneous fluid, p(a,b), which Is 

 Independent of time, will be prescribed through the Initial choice^ of 

 Z (a,b). Indeed In many cases It will be convenient to choose Z in 



126 



