On the completeness of the Rayleigh-Marangoni and Graetz eigenspaces and the simplicity of their eigenvalues
The study of convection in bounded geometries is often facilitated by consideration of model problems which require little numerical computation. Such problems typically result in a sixth-order ordinary differential system which is non-selfadjoint in the inner product associated with the space of square integrable functions in the sense of Lebesgue. Using theorems of Karlin (1971) and Naimark (1967), we prove the simplicity of the eigenvalues and completeness of the eigenspace. We illustrate some of the simplicity results with numerical calculations. The use of completeness in future problems is explained. We also consider the extended Graetz problem with homogeneous reaction and heterogeneous reaction at the wall. Similar results are shown here. Some of the above-mentioned results can be obtained by other means, but we provide this analysis as an interesting alternative.