The Effect of Mean Flows on Enhanced Diffusivity in Transport by Incompressible Periodic Velocity Fields

Avellaneda and one of the authors ([1], [3]) have recently established that an upper bound for the enhanced diffusivity in the large scale, long time advection-diffusion with periodic steady incompressible velocity fields has the form where Pe is the Peclet number and is the reciprocal of the Prandtl number. In this paper, flow fields with maximal and minimal enhanced diffusion are studied. Maximal enhanced diffusion requires that in some directions the enhanced diffusion tensor also has the lower bound . For minimal enhanced diffusion, the effect of the velocity field is to boost the enhanced diffusivity by a negligible amount that is bounded by a fixed constant times the bare diffusivity regardless of Peclet number. Stieltjes measure formulas are used to develop a simple, necessary, and sufficient condition for maximal enhanced diffusion and also to characterize minimal enhanced diffusion. It is established here that constant mean flows can have a dramatic effect on maximal and minimal enhanced diffusion. In particular, for flows in two space dimensions, an explicit criterion is developed that guarantees the surprising fact that mean flows with rational ratios typically generate maximal enhanced diffusion through interaction with an arbitrary steady periodic incompressible flow with zero mean. In contrast, a simple criterion for flows without stagnation points is developed here that guarantees that the effect of mean flows with irrational ratios on advection-diffusion in two dimensions creates minimal enhanced diffusion. The theory for the phenomena mentioned above is elementary yet mathematically rigorous. Examples are emphasized throughout this work including a discussion of enhanced diffusivity for a class of flows recently introduced by Childress and Soward [8]. The theory developed here is also supplemented by a series of numerical experiments that both verify the theoretical predictions and display interesting crossover phenomena at rather large but finite Peclet numbers.     

Key tricyclic synthetic intermediates for the preparation of the sesquiterpenes α- and β-cedrene

A completely novel and direct route towards the synthesis of the natural sesquiterpenes α-cedrene and β-cedrene delivered the compounds (3β,3aβ,7β)-(±)-6,6-ethyl­ene­dioxy-3,8,8-tri­methyl-2,3,3a,4,5,6,7,8-octa­hydro-3a,7-methano­azulen-2-one, C₁₆H₂₂O₃, and (3β,3aβ,7β,8aα)-(±)-6,6-ethyl­ene­dioxy-3,8,8-tri­methyl-1,2,3,3a,4,5,6,7,8,8a-deca­hydro-3a,7-methano­azulen-2-one, C₁₆H₂₄O₃, at key stages of the preparative programme. Structural elucidation showed the latter compound to have added an H atom to the same face of the cyclo­pentenone ring as that occupied by the methyl substituent, and also allowed correct isomer identification for further reaction.     

2-Nitro-N,N-bis(2-nitro­phenyl­thio)­benzenesulfon­amide

The central N atom of C₁₈H₁₂N₄O₈S is essentially planar, lying 0.071 (2) Å out of the plane defined by the three S atoms. N—S distances are 1.712 (2) and 1.721 (2) Å to S—Ar, and 1.681 (2) Å to SO₂—Ar. The nitro group on the phenyl ring carrying the SO₂ group lies out of plane, with C—C—N—O– torsion angle 69.8 (3)°, while the other two nitro groups are near coplanarity, with torsion angle magnitudes 10.4 (3) and 14.0 (3)°.     

Complex Allylation by the Direct Cross‐Coupling of Imines with Unactivated Allylic Alcohols

Regioselective, stereoselective : The convergent coupling of allylic alcohols with imines to deliver stereodefined homoallylic amines is described (see scheme). The process proceeds with net allylic transposition without the intermediacy of allylic organometallic reagents. Two stereodefined centers and a geometrically defined di‐ or trisubstituted alkene are forged with high selectivity.

     

Lifting Bailey pairs to WP-Bailey pairs

A pair of sequences such that and

     

Temperature-dependent Raman scattering of silicon nanowires

Silicon nanowires with narrowly distributed diameters of 20-30 nm have been fabricated by chemical vapor deposition on an anodized aluminum oxide (AAO) substrate. The first-order and second-order Raman scatterings of the silicon nanowires have been studied in a temperature range from 123 to 633 K. Both of the first-order and second-order Raman peaks were found to shift and broaden with increasing temperature. The experimental results were analyzed by combining the phonon confinement effect, anharmonic phonon processes and lattice stress effect. It was found that the intensities of the first-order and second-order Raman bands have different dependences on temperature. The value of relative intensities I(2TA)int/I(2TO)int for silicon nanowires was found to be larger than that of bulk silicon, and increase with rising measurement temperature. We ascribe this phenomenon to the participation of phonons with a large wave vector value of Raman scattering caused by both the phonon confinement effect and the temperature effect.     

Efficient access to azaindoles and indoles

[Structure: see text] An expedient, catalytic method for the synthesis of diverse azaindoles and indoles, starting from readily available and inexpensive starting materials, is described. Conditions were developed for effective reductive alkylation of electron-deficient o-chloroarylamines, substrates previously viewed as poor partners in this reaction. The derived N-alkylated o-chloroarylamines were elaborated to N-alkylazaindoles and N-alkylindoles via a novel one-pot process comprising copper-free Sonogashira alkynylation and a base-mediated indolization reaction.