Development of LBGK and incompressible LBGK‐based lattice Boltzmann flux solvers for simulation of incompressible flows

This paper presents lattice Boltzmann Bhatnagar–Gross–Krook (LBGK) model and incompressible LBGK model‐based lattice Boltzmann flux solvers (LBFS) for simulation of incompressible flows. LBFS applies the finite volume method to directly discretize the governing differential equations recovered by lattice Boltzmann equations. The fluxes of LBFS at each cell interface are evaluated by local reconstruction of lattice Boltzmann solution. Because LBFS is applied locally at each cell interface independently, it removes the major drawbacks of conventional lattice Boltzmann method such as lattice uniformity, coupling between mesh spacing, and time interval. With LBGK and incompressible LBGK models, LBFS are examined by simulating decaying vortex flow, polar cavity flow, plane Poiseuille flow, Womersley flow, and double shear flows. The obtained numerical results show that both the LBGK and incompressible LBGK‐based LBFS have the second order of accuracy and high computational efficiency on nonuniform grids. Furthermore, LBFS with both LBGK models are also stable for the double shear flows at a high Reynolds number of 10⁵. However, for the pressure‐driven plane Poiseuille flow, when the pressure gradient is increased, the relative error associated with LBGK model grows faster than that associated with incompressible LBGK model. It seems that the incompressible LBGK‐based LBFS is more suitable for simulating incompressible flows with large pressure gradients. Copyright © 2014 John Wiley & Sons, Ltd.     

Inversion twinning in a second polymorph of the hydrochloride salt of the recreational drug ethylone

A second polymorph of the hydrochloride salt of the recreational drug ethylone, C₁₂H₁₆NO₃⁺·Cl⁻, is reported [systematic name: (±)‐2‐ethylammonio‐1‐(3,4‐methylenedioxyphenyl)propane‐1‐one chloride]. This polymorph, denoted form (A), appears in crystallizations performed above 308 K. The originally reported form (B) [Wood et al. (2015). Acta Cryst. C 71 , 32–38] crystallizes preferentially at room temperature. The conformations of the cations in the two forms differ by a 180° rotation about the C—C bond linking the side chain to the aromatic ring. Hydrogen bonding links the cations and anions in both forms into similar extended chains in which any one chain contains only a single enantiomer of the chiral cation, but the packing of the ions is different. In form (A), the aromatic rings of adjacent chains interleave, but pack equally well if neighbouring chains contain the same or opposite enantiomorph of the cation. The consequence of this is then near perfect inversion twinning in the structure. In form (B), neighbouring chains are always inverted, leading to a centrosymmetric space group. The question as to why the polymorphs crystallize at slightly different temperatures has been examined by density functional theory (DFT) and lattice energy calculations and a consideration of packing compactness. The free energy (ΔG) of the crystal lattice for polymorph (A) lies some 52 kJ mol⁻¹ above that of polymorph (B).     

Koszul filtrations and finite lattices

In this note, we characterize when a finite lattice is distributive in terms of the existences of some particular classes of Koszul filtrations.     

Stability,periodicity, and symmetries of certain second‐order fractional difference equation with quadratic terms via KAM theory

By using the Kolmogorov–Arnold–Moser theory, we investigate the stability of the equilibrium solution of the difference equation where A ,B ,D > 0,u _?1,u ₀>0. We also use the symmetries to find effectively the periodic solutions with feasible periods. Copyright © 2016 John Wiley & Sons, Ltd.     

Analytic invariant curves for an iterative equation related to dissipative standard map

In this paper, we study existence of invariant curves of an iterative equation which is from dissipative standard map. By constructing an invertible analytic solution g (x ) of an auxiliary equation of the form invertible analytic solutions of the form g (λ g  ^? 1(x )) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < |λ | < 1, we focus on those λ on the unit circle S ¹, that is, |λ | = 1. We discuss not only those λ at resonance, that is, at a root of the unity, but also those λ near resonance under the Brjuno condition. Copyright © 2016 John Wiley & Sons, Ltd.