In the experiments and simulations of chapters X
and Y, the speed of growth of the evolving solid 
was slower when there was a lower concentration of rejected 
solute ahead of it.  In other words, the structures or 
morphologies, best able to entrain solute, grew most quickly.
This reasoning held for the herringbone structure, 
the dendritic patterns, and even to a limit of absolute stability 
for very high supersaturations.

Applying this idea to arrays of growing dendrites, 
a pair of coupled PDE's is 
developed to describe the envelope containing the evolving 
pattern.  The main motivation of this development comes from 
the observation in the simulations that the growing dendritic 
patterns which are more ``closed off'' serve better to 
contain the rejected solute.  This reduces the solute 
``blinket'' ahead of the growing pattern allowing it a 
greater growth velocity than a pattern (given all of the same 
system parameters) with a more ``open'' structure.  
The faster growing patterns have a smaller diffusion length, 
which allows the individual dendrites to grow more closely to 
one another.  This results in a denser morphology (i.e. more solid).  
With more solid having formed, there is less salt in 
solution, resulting in a lower saturation of the interdendritic 
fluid.  This lower saturation coupled with the higher curvatures 
of the solid results in much faster coarsening.  Since the 
PDE's are being developed to study the envelope of the growing 
array, where the solid is first forming, this coarsening is
again neglected.  
Any kind of fluid flow is also neglected.  Considering the 
strong role it was seen to play in the relative positions 
of ``fast'' dendrites growing in an array, the PDE's of 
this chapter are not expected to yield the correct front 
shape when fluid flow plays a strong role (especially 
given a 2D geometry, as in the experiments, 
which could lead to vortices).  

This system of two PDE's has been developed to model the interface between 
a growing array of dendritic needles and the undercooled 
binary melt into which they are growing.  The growth is assumed 
to be dictated by the diffusion of rejected impurity, in which 
the diffusion through the dendritic array 
differs from the diffusion through the liquid ahead of the
dendritic array.  
The evolution of heat, as well as any macroscopic flow
field in the liquid, are neglected.  The equations are 
primarily for numerical simulation, but can be combined 
to give a single, more analytically 
attractive equation exhibiting behaviour similar to that of a 
rapidly solidifying front.  Although the presented equations 
are nonconservative in the chosen variables, the total amounts 
of both salt and water (or A and B) are indeed conserved.