Notes of a linear stability analysis (for ATG instabilities in thin film assemblies)

It is well known that the planar interface between a non-hydrostratically stressed solid and its melt or vapour is unstable with respect to perturbations. This instability is known as Asaro-Tiller-Grinfeld instability following the independent discovery of the same by Asaro and Tiller (Metallurgical transactions A, 3, 1789-1796, 1972) and Grinfeld (Soviet Physics Doklady, 290, 1358-1363, 1986) for solid-melt interfaces. Similar analysis for solid vapour interfaces was carried out by Srolovitz (Acta Metallurgica, 37, 621-625, 1989). The crucial role played by inhomogeneities in promoting ATG instabilities was shown by Sridhar et al (Acta Metallurgica, 45, 2715-2733, 1997). Some time back, we tried to reproduce some of the linear stability analysis carried out by Sridhar et al, and also redo the analysis assuming a different diffusion mechanism: while Sridhar et al assumed interface diffusion control, we were interested in volume diffusion controlled instabilities. I am attaching the notes of the analysis in case it might be of interest to some of the readers of iMechanica. In addition, I am also attaching the MAPLE(TM) command file used in carrying out some of the analysis, and the C code based on the MAPLE(TM) analysis to calculate the growth rate for different wavevectors (which is named txt so that iMechanica would allow me to upload it).