Series solutions for flow in stratified aquifers with natural geometry

2010 ◽  
Vol 33 (1) ◽  
pp. 48-54 ◽  
Author(s):  
Sanders Wong ◽  
James R. Craig
2020 ◽  
Vol 24 ◽  
Author(s):  
Hubert Hettegger ◽  
Andreas Hofinger ◽  
Thomas Rosenau

: The regioselectivity of the reaction of 2,5-dihydroxy-[1,4]-benzoquinone (DHBQ) with diamines could not be explained satisfactorily so far. In general, the reaction products can be derived from the tautomeric ortho-quinoid structure of a hypothetical 4,5-dihydroxy-[1,2]-benzoquinone. However, both aromatic and aliphatic 1,2-diamines form in some cases phenazines, formally by diimine formation on the quinoid carbonyl groups, and in other cases the corresponding 1,2- diamino-[1,2]-benzoquinones, by nucleophilic substitution of the OH groups, the regioselectivity apparently not following any discernible pattern. The reactivity was now explained by an adapted theory of strain-induced bond localization (SIBL). Here, the preservation of the "natural" geometry of the two quinoid C–C double bonds (C3=C4 and C5=C6) as well as the N–N distance of the co-reacting diamine are crucial. A decrease of the annulation angle sum (N–C4–C5 + C4–C5–N) is tolerated well and the 4,5-diamino-ortho-quinones, having relatively short N–N spacings are formed. An increase in the angular sum is energetically unfavorable, so that diamines with a larger N–N distance afford the corresponding ortho-quinone imines. Thus, for the reaction of DHBQ with diamines, exact predictions of the regioselectivity, and the resulting product structure, can be made on the basis of simple computations of bond spacings and product geometries.


Author(s):  
Walter Ott

Descartes’s treatment of perception in the Optics, though published before the Meditations, contains a distinct account of sensory experience. The end of the chapter suggests some reasons for this oddity, but that the two accounts are distinct is difficult to deny. Descartes in the present work topples the brain image from its throne. In its place, we have two mechanisms, one purely causal, the other inferential. Where the proper sensibles are concerned, the ordination of nature suffices to explain why a given sensation is triggered on the occasion of a given brain motion. The same is true with regard to the common sensibles. But on top of this purely causal story, Descartes re-introduces his doctrine of natural geometry.


2010 ◽  
Vol 88 (3) ◽  
pp. 217-227 ◽  
Author(s):  
Mark Ingratta ◽  
Manoj Mathew ◽  
Jean Duhamel

A series of polystyrenes randomly labeled with 1-pyrenebutanol were prepared by copolymerizing styrene and 1-pyrenebutylacrylate yielding the CoBuE–PS series. Solutions of CoBuE–PS were prepared in nine organic solvents having viscosities ranging from 0.36 to 5.5 mPa·s and the fluorescence spectra and pyrene monomer and excimer fluorescence decays were acquired. Analysis of the fluorescence spectra yielded the IE/IM ratio, whereas analysis of the fluorescence decays with the fluorescence blob model (FBM) yielded the parameters N blobo , <kblob × Nblob> , and k blobo . These parameters were compared to those obtained with two other series of pyrene-labeled polystyrenes, which had been studied earlier, namely CoA–PS and CoE–PS where pyrene was attached to the polymer backbone via a methylamide and benzyl methylether linker, respectively. Although the parameters IE/IM, N blobo , <kblob × Nblob>, and k blobo took different values according to the specific nature of the linker connecting pyrene to the polystyrene backbone, they exhibited trends that were quite similar for all the pyrene-labeled polystyrene constructs. The excellent agreement between the parameters retrieved for the three different types of pyrene-labeled polystyrenes suggests that the FBM accounts satisfyingly for differences in the nature of the label used, while still retrieving information pertinent to the polymer of interest.


1951 ◽  
Vol 18 (3) ◽  
pp. 285-292
Author(s):  
H. J. Plass

Abstract Timoshenko has applied plate theory to each leg of an angle-section bar to determine the critical compressive load needed to cause sinusoidal torsional buckling. In this paper his idea is used to calculate the critical bending load needed to cause sinusoidal torsional buckling of an angle bar. The bending is assumed to be applied so that the extreme fibers of the angle are in compression, the vertex in tension. Approximate results are first obtained by means of the Rayleigh-Ritz method. The approximate deflection functions from which the energy terms are computed are based upon certain infinite-series solutions. After having obtained approximate results, exact values are obtained, using the approximate values as a guide to limit the amount of calculation. The results of this calculation are shown in Fig. 5, where they are compared with those predicted by bar theory. Differences between the two theories become more noticeable as the bar becomes short compared to its flange width. It is found that the critical bending load becomes larger very rapidly as the ratio of length to width of the flanges decreases. Bar theory predicts no such increase. The reason for this difference is explained.


Author(s):  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Tian-Tian Zhang

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.


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