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.     

Die Fehlordnung des Natriumazides und deren Einfluß auf die thermische Zersetzung, 4. Mitt.: Über die Programmierung von Zersetzungsreaktionen mit Hilfe von Korngrößenverteilungen

It is shown that in the case of thermal decomposition of sodium azide the overall kinetics can be predicted by defined particle size of the decomposed sodium azide crystals. This is always the case if the rate constant is a function of the particle size. Hence this special example can be generalized for similar decomposition reactions. It is necessary that the particles decompose independently which could be proved experimentally with sodium azide. If for this reason we state for true that the pressure/time-function of each particle size add together it is possible to set up a formula for the pressure/time-function of any particle size distribution. With the pressure/time function holding for sodium azide of uniform particle size, the total function for a Gauß distribution can be calculated exactly. Moreover, the trivial case of one single particle size and the case of two different particle sizes are discussed. Furthermore an approximation method for any arbitrary pressure/time-functions and distribution by means of “Schwerpunktdeutung” are discussed which can be carried out graphically as well as numerically. The numerical approximation is illustrated by an example. Pressure/time-functions then loose their characteristic form because of their dependence on the particle size distributions under consideration. In this case, reaction mechanism cannot be derived from pressure/time functions.     

Carbon Participation in the Solvolysis of 6-exo-substituted 2-exo- and 2-endo-Norbornyl p-Toluenesulfonates. Norbornanes part 5

The solvolysis rates and products of the 6-exo-substituted 2-exo- 1a - 1u , and 2-endo-norbornyl p-toluenesulfonates 2a - 2u , have been determined. In general, the rate constants for 1 and 2 (log k) correlate well with the inductive constants σ of the substitutents at C(6); however, their sensitivity to σ is much larger in the 2-exo-series 1 than in the 2-endo-series 2 . This differential transmission of polar effects is the cause of decreasing 2-exo/2-endo rate ratios from 2388 for R = t-C₄H₉ to 0.37 for R = Br, i. e. with increasing electron attraction by the substituent. The high sensitivity of the rate constants for the 2-exo-p-toluenesulfonates 1 to σ indicates an unusually strong inductive interaction between C(6) and the incipient cationic center at C(2). This interaction is ascribed to the participation of the pentacoordinate C(6)-atom, i. e. to 1,3-bridging, a consequence of steric hindrance of nucleophilic solvent participation in norbornanes. Donor substituents enhance 1,3-bridging, lead to faster reactions and to the formation of 2-exo substitution products. Conversely, acceptor substituents reduce 1,3-bridging, decrease rates and facilitate the formation of 2-endo substitution products. Graded 1,3-bridging is discussed in the light of Winstein's nonclassical ion concept.