A spectral Galerkin approximation of the Orr-Sommerfeld eigenvalue problem in a semi-infinite domain

Summary The convergence of a Galerkin approximation of the Orr-Sommerfeld eigenvalue problem, which is defined in a semi-infinite domain, is studied theoretically. In case the system of trial functions is based on a composite of Jacobi polynomials and an exponential transform of the semi-infinite domain, the error of the Galerkin approximation is estimated in terms of the transformation parametera and the numberN of trial functions. Finite or infinite-order convergence of the spectral Galerkin method is obtained depending on how the transformation parameter is chosen. If the transformation parameter is fixed, then convergence is of finite order only. However, ifa is varied proportional to 1/N with an exponent 0<<1, then the approximate eigenvalue converges faster than any finite power of 1/N asN. Some numerical examles are given.     

Study of the adsorption behavior of the enantiomers of 1-phenyl-1-propanol on a cellulose-based chiral stationary phase

Using single-step frontal analysis, we measured single-component and competitive adsorption isotherm data for the two enantiomers of 1-phenyl-1-propanol (PP). These experimental data were fitted to several competitive bi-Langmuir models (with 8, 6, 5 and 4 parameters) and to the competitive Langmuir model. The latter model accounted well for the behavior of both PP enantiomers on Chiracel OB (cellulose tribenzoate coated on silica gel). The parameters obtained were used in numerical calculations to predict the band profiles of the two single components and of their mixtures under overloaded conditions. The equilibrium-dispersive model provides satisfactory results, with minor differences between the calculated and the experimental profiles. These differences became negligible when a more complex kinetic model was used, with a concentration-dependent rate coefficient.