On the bidirectional vortex and other similarity solutions in spherical coordinates

The bidirectional vortex refers to the bipolar, coaxial swirling motion that can be triggered, for example, in cyclone separators and some liquid rocket engines with tangential aft-end injectors. In this study, we present an exact solution to describe the corresponding bulk motion in spherical coordinates. To do so, we examine both linear and nonlinear solutions of the momentum and vorticity transport equations in spherical coordinates. The assumption will be that of steady, incompressible, inviscid, rotational, and axisymmetric flow. We further relate the vorticity to some power of the stream function. At the outset, three possible types of similarity solutions are shown to fulfill the momentum equation. While the first type is incapable of satisfying the conditions for the bidirectional vortex, it can be used to accommodate other physical settings such as Hill’s vortex. This case is illustrated in the context of inviscid flow over a sphere. The second leads to a closed-form analytical expression that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. The third type is more general and provides multiple solutions. The spherical bidirectional vortex is derived using separation of variables and the method of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions that rarely appear in spherical geometry. It is hoped that these special forms will permit extending the current approach to other complex fluid motions that are easier to capture using spherical coordinates.     

Practical and quality-control aspects of multi-element analysis with quadrupole ICP–MS with special attention to urine and whole blood

Two screening methods were developed for rapid analysis of a great number of urine and blood samples within the framework of an exposure check of the population after a firework explosion. A total of 56 elements was measured including major elements. Sample preparation consisted of simple dilution. Extensive quality controls were applied including element addition and the use of certified reference materials. Relevant results at levels similar to those found in the literature were obtained for Co, Ni, Cu, Zn, Sr, Cd, Sn, Sb, Ba, Tl, and Pb in urine and for the same elements except Ni, Sn, Sb, and Ba in blood. However, quadrupole ICP–MS has limitations, mainly related to spectral interferences, for the analysis of urine and blood, and these cause higher detection limits. The general aspects discussed in the paper give it wider applicability than just for analysis of blood and urine—it can for example be used in environmental analysis.     

Insight into Oxide‐Bridged Heterobimetallic Al/Zr Olefin Polymerization Catalysts

Reaction of (TBBP)AlMe ? THF with [Cp*₂Zr(Me)OH] gave [(TBBP)Al(THF)?O?Zr(Me)Cp*₂] (TBBP=3,3’,5,5’‐tetra‐tBu‐2,2'‐biphenolato). Reaction of [DIPPnacnacAl(Me)?O?Zr(Me)Cp₂] with [PhMe₂NH]⁺[B(C₆F₅)₄]^? gave a cationic Al/Zr complex that could be structurally characterized as its THF adduct [(DIPPnacnac)Al(Me)?O?Zr(THF)Cp₂]⁺[B(C₆F₅)₄]^? (DIPPnacnac=HC[(Me)C=N(2,6‐iPr₂?C₆H₃)]₂). The first complex polymerizes ethene in the presence of an alkylaluminum scavenger but in the absence of methylalumoxane (MAO). The adduct cation is inactive under these conditions. Theoretical calculations show very high energy barriers (ΔG=40–47 kcal mol^?1) for ethene insertion with a bridged AlOZr catalyst. This is due to an unfavorable six‐membered‐ring transition state, in which the methyl group bridges the metal and ethene with an obtuse metal‐Me‐C angle that prevents synchronized bond‐breaking and making. A more‐likely pathway is dissociation of the Al‐O‐Zr complex into an aluminate and the active polymerization catalyst [Cp*₂ZrMe]⁺.     

Well‐Defined Molecular Magnesium Hydride Clusters: Relationship between Size and Hydrogen‐Elimination Temperature

A new tetranuclear magnesium hydride cluster, [{ NN ‐(MgH)₂}₂], which was based on a N? N‐coupled bis‐β‐diketiminate ligand ( NN ^2?), was obtained from the reaction of [{ NN ‐(MgnBu)₂}₂] with PhSiH₃. Its crystal structure reveals an almost‐tetrahedral arrangement of Mg atoms and two different sets of hydride ions, which give rise to a coupling in the NMR spectrum (J=8.5 Hz). To shed light on the relationship between the cluster size and H₂ release, the thermal decomposition of [{ NN ‐(MgH)₂}₂] and two closely related systems that were based on similar ligands, that is, an octanuclear magnesium hydride cluster and a dimeric magnesium hydride species, have been investigated in detail. A lowering of the H₂‐desorption temperature with decreasing cluster size is observed, in line with previously reported theoretical predictions on (MgH₂)_n model systems. Deuterium‐labeling studies further demonstrate that the released H₂ solely originates from the oxidative coupling of two hydride ligands and not from other hydrogen sources, such as the β‐diketiminate ligands. Analysis of the DFT‐computed electron density in [{ NN ‐(MgH)₂}₂] reveals a counterintuitive interaction between two formally closed‐shell H^? ligands that are separated by 3.106 Å. This weak interaction could play an important role in H₂ desorption. Although the molecular product after H₂ release could not be characterized experimentally, DFT calculations on the proposed decomposition product, that is, the low‐valence tetranuclear Mg(I) cluster [( NN ‐Mg₂)₂], predict a structure with two almost‐parallel, localized Mg? Mg bonds. As in a previously reported β‐diketiminate Mg^I dimer, the Mg? Mg bond is not characterized by a bond critical point, but instead displays a local maximum of electron density midway between the atoms, that is, a non‐nuclear attractor (NNA). Interestingly, both of the NNAs in [( NN ‐Mg₂)₂] are connected through a bond path that suggests that there is bonding between all four Mg^I atoms.     

Insights into LiAlH4 Catalyzed Imine Hydrogenation

Commercial LiAlH₄ can be used in catalytic quantities in the hydrogenation of imines to amines with H₂. Combined experimental and theoretical investigations give deeper insight in the mechanism and identifies the most likely catalytic cycle. Activity is lost when Li in LiAlH₄ is exchanged for Na or K. Exchanging Al for B or Ga also led to dramatically reduced activities. This indicates a heterobimetallic mechanism in which cooperation between Li and Al is crucial. Potential intermediates on the catalytic pathway have been isolated from reactions of MAlH₄ (M=Li, Na, K) and different imines. Depending on the imine, double, triple or quadruple imine insertion has been observed. Prolonged reaction of LiAlH₄ with PhC(H)=NtBu led to a side-reaction and gave the double insertion product LiAlH₂[N]₂ ([N]=N(tBu)CH₂Ph) which at higher temperature reacts further by ortho-metallation of the Ph ring. A DFT study led to a number of conclusions. The most likely catalyst for hydrogenation of PhC(H)=NtBu with LiAlH₄ is LiAlH₂[N]₂. Insertion of a third imine via a heterobimetallic transition state has a barrier of +23.2 kcal mol⁻¹ (ΔH). The rate-determining step is hydrogenolysis of LiAlH[N]₃ with H₂ with a barrier of +29.2 kcal mol⁻¹. In agreement with experiment, replacing Li for Na (or K) and Al for B (or Ga) led to higher calculated barriers. Also, the AlH₄⁻ anion showed very high barriers. Calculations support the experimentally observed effects of the imine substituents at C and N: the lowest barriers are calculated for imines with aryl-substituents at C and alkyl-substituents at N.     

Localizations of Transfors

Let be n-dimensional teisi, i.e., higher-dimensional Gray-categorical structures. The following questions can be asked. Does a left q-transfor , i.e., a functor 2 _q , induce a right q-transfor , i.e., a functor More generally, does a functor induce a functor For k-arrows c and whose (k – 1)-sources and targets agree, does a q-transfor induce a q-transfor , for appropriate k-arrows For k-arrows c and whose (k – 1)-sources and targets agree, does a q-transfor induce a (q + k + 1)-transfor , for appropriate k-arrows I give answers to these questions in the cases where n-dimensional teisi and their tensor product have been defined, i.e., for n 3, and for n up to 5 in some cases that do not need all data and axioms of n-dimensional teisi.I apply the above to compositions in teisi, in particular to braidings and syllepses. One of the results is that a braiding on a monoidal 2-category induces a pseudo-natural transformation , where is the reverse of ? –, which is almost, but not quite, equal to – ?. However, in higher dimensions need not be reversible, so a braiding on a higher-dimensional tas can not be seen as a transfor A B B A.