The evolution equation (10) of Tritscher & Broadbridge can only be used to model a restricted class of anisotropy. Furthermore, the constitutive equations mean that only a material with anisotropy such that the surface diffusion and diffusivity functions must both be either increasing or decreasing together.
We note that equation (5) can be rewritten as
A similarity reduction of equation (12) is possible using the scaling group
giving invariants
This allows the evolution equation (12) to be written as the ordinary differential equation
Katoanga and Lisle (personal communications) attempted a partial
classification of symmetries for equations (5) and (15) . A total
of six classifying equations were found on a 5 parameter group,
third order in
where
These functions give the constitutive relations
The figures at the end of this document provide a useful graphical comparison between the different materials modelled by the constitutive relations. It should be noted that the polar plots of the diffusion and surface tension of the isotropic material gives a circle. |
Derivation of the Evolution Equation New Constitutive Functions |