Limiting analytical solutions for complex saturated and unsaturated transport problems

Non-linear diffusion and velocity-dependent dispersion problems are under consideration. The necessary and sufficient conditions allowing the comparison of solutions to the two dimensional convection-dispersion equations with different coefficients are obtained. These conditions provide a framework within which solutions to the complex non-linear problems mentioned above can be estimated by solutions to the problems possessing analytical solvability.Nomenclature c(x, y, t) concentration of solute in solution,ML ^–3 - C(h)=d/dh moisture capacity function - D,D _ij hydrodynamic dispersion coefficient, a second order tensor,L ² T ^–1 - D _L longitudinal hydrodynamic dispersion coefficient,L ² T ^–1 - D _m molecular diffusion coefficient,L ² T ^–1 - D _T transverse hydrodynamic coefficient,L ² T ^–1 - G flow domain for the unsaturated flow problem - G _z , G _w flow domain and complex potential domain, respectively, for the hydrodynamic dispersion problem - h piezometric head,L - I _n given mass flux normal to the boundary,MLT ^–1 - k hydraulic conductivity,LT ^–1 - K(h) unsaturated hydraulic conductivity,LT ^–1 - L continuously differentiable function with respect to all arguments - m porosity - n(x,t) outer normal vector to the boundary - t time,T - V(x, y, t) seepage velocity vector withV=V,LT ^–1 - x Cartesian coordinate system - x horizontal coordinate,L - y vertical coordinate (elevation),L - (x),(x,t) given functions in initial and boundary conditions (3), (4) - ₁(,) angle between vectors ₁c andV - boundary of the flow domain - _L , _T longitudinal and transverse dispersivities, respectively,L - water mass density,ML ^–3 - v _i components of a unit vector in the direction of the outward normal to the boundary - =–kh velocity potential - =/m - stream function defined such thatw=+i is the complex potential - =/m     

Stability of vertical miscible displacements with developing density and viscosity gradients

Viscous fingering and gravity tonguing are the consequences of an unstable miscible displacement. Chang and Slattery (1986) performed a linear stability analysis for a miscible displacement considering only the effect of viscosity. Here the effect of gravity is included as well for either a step change or a graduated change in concentration at the injection face during a downward, vertical displacement. If both the mobility ratio and the density ratio are favorable (the viscosity of the displacing fluid is greater than the viscosity of the displaced fluid and, for a downward vertical displacement, the density of the displacing fluid is less than the density of the displaced fluid), the displacement will be stable. If either the mobility ratio or the density ratio is unfavorable, instabilities can form at the injection boundary as the result of infinitesimal perturbations. But if the concentration is changed sufficiently slowly with time at the entrance to the system, the displacement can be stabilized, even if both the mobility ratio and the density ratio are unfavorable. A displacement is more likely to be stable as the aspect ratio (ratio of thickness to width, which is assumed to be less than one) is increased. Commonly the laboratory tests supporting a field trial use nearly the same fluids, porous media, and displacement rates as the field trial they are intended to support. For the laboratory test, the aspect ratio may be the order of one; for the field trial, it may be two orders of magnitude smaller. This means that a laboratory test could indicate that a displacement was stable, while an unstable displacement may be observed in the field.     

On the asymptotic behaviour of solution for the generalized double dispersion equation

In this article, we investigate the Cauchy problem for the generalized double dispersion equation in n-dimensional space. We establish the decay estimates of solution to the corresponding linear equation. Under smallness condition on the initial data, we prove the global existence and asymptotic behaviour of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.     

A study on separation and extraction of four main alkaloids in Macleaya cordata (Willd) R. Br. with strip dispersion hybrid liquid membrane

A technique based on strip dispersion hybrid liquid membrane was developed for the separation and extraction of four main alkaloids from fruits of Macleaya cordata (Willd) R. Br. A microporous polypropylene membrane impregnated with an organic membrane solution comprised the heart of the strip dispersion hybrid liquid membrane system. The membrane solution was made by dissolving a cationic carrier, di‐(2‐ethylhexyl) phosphoric acid in an inexpensive, less toxic membrane solvent, kerosene. The transport of alkaloids from an aqueous feed solution through the membrane to a strip dispersion phase was driven by the concentration gradient of H⁺ and facilitated by di‐(2‐ethylhexyl) phosphoric acid. The effects of the extraction time and reuse times of the membrane, the strip solution composition, the carrier concentration, the volume ratio of the aqueous strip solution to the organic membrane solution, and the flow rates of the feed solution and the strip dispersion phase on the transport of alkaloids were investigated. Under the optimal conditions, the permeability coefficients obtained for the four main alkaloids allocryptopine, protopine, sanguinarine, and chelerythrine were 1.66, 1.99, 2.98, and 3.06×10^?4 cm/s, and the transport efficiencies were as high as 68, 77, 83, and 85%, respectively.     

Can Raman spectroscopy detect cumulenic structures of linear carbon chains?

The molecular and vibrational structures of cumulenic carbon chains are investigated by density functional theory calculations and compared with that of hydrogen‐capped polyynes. The small value of bond length alternation (BLA) along the CC bonds sequence obtained by geometry optimization of uncapped C_n chains and vinyl‐capped carbon chains confirms their cumulenic structure. It is demonstrated that for finite length chains the structural parameters are determined by end effects as far as the Peierls distortion, expected for very long molecules, does not occur. The Raman spectra of such molecules are calculated to verify the possibility of identifying markers of cumulenic chains by means of vibrational spectroscopy. As expected, the longitudinal mode consisting of the BLA oscillation, which is responsible for the strongest Raman transition of polyynes, becomes very weak for cumulenes; this behaviour is rationalized in terms of local polarizability derivatives. However, other longitudinal modes can be observed in the Raman spectra of C_n chains. The wavenumber behaviour and the optical activity of these modes are interpreted on the basis of the phonon dispersion branch of an ideally infinite cumulenic polymer. Raman intensities computed for chains of different lengths allow to conclude that cumulenic molecules could be detected and identified by means of Raman spectroscopy. Copyright © 2009 John Wiley & Sons, Ltd.     

The effect of phase change materials on the frontal polymerization of a triacrylate

The production of smoke and fumes is a major obstacle to the practical use of thermal frontal polymerization. The front temperature and the amount of smoking can be reduced by adding inert fillers, such as clay and silica, to the reactive mixture. Here we investigate the possibility of incorporating inert materials that melt (so-called phase change materials) to the mixture. By performing both experiments and mathematical modeling, we demonstrate that, in addition to the standard parameters of frontal polymerization, the front temperature and velocity depend on the melting point and heat of fusion of the phase change material. We use the method of matched asymptotic expansions to develop an explicit expression for the velocity of the reaction front. The expression demonstrates that the behavior of the front is determined by the difference between the reaction temperature and the melting temperature, with the front being slower and cooler if melting occurs farther ahead of the reaction front. The theoretical trends are hard to confirm directly because different characteristics of the phase change material cannot be varied separately.     

Structure and properties of star‐shaped solution‐polymerized styrene‐butadiene rubber and its co‐coagulated rubber filled with silica/carbon black‐I: morphological structure and mechanical properties

The morphological structure and mechanical properties of the star‐shaped solution‐polymerized styrene‐butadiene rubber (SSBR) and organically modified nanosilica powder/star‐shaped SSBR co‐coagulated rubber (N‐SSBR) both filled with silica/carbon black (CB) were studied. The results showed that, compared with SSBR, silica powder could be mixed into N‐SSBR much more rapidly, and N‐SSBR/SiO₂ nanocomposite had better filler‐dispersion and processability. N‐SSBR/SiO₂/CB vulcanizates displayed higher glass‐transition temperature and lower peak value of internal friction loss than SSBR/SiO₂/CB vulcanizates. In the N‐SSBR/SiO₂/CB vulcanizates, filler was dispersed in nano‐scale resulting in good mechanical properties. Composites filled with silica/CB doped filler exhibited more excellent mechanical properties than those filled with a single filler because of the better filler‐dispersion and stronger interfacial interaction with macromolecular chains. N‐SSBR/SiO₂/CB vulcanizates exhibited preferable performance in abrasion resistance and higher bound rubber content as the blending ratio of silica to CB was 20:30. Copyright © 2008 John Wiley & Sons, Ltd.     

Analysis of polybrominated diphenyl ethers in sewage sludge by matrix solid‐phase dispersion and isotope dilution GC–MS

A method for the determination of 14 polybrominated diphenyl ethers (PBDEs) in sludge from wastewater treatment plants is presented. PBDEs were extracted by matrix solid‐phase dispersion assisted by sonication and determined by isotope dilution gas chromatography with electron impact mass spectrometric detection in the selected ion monitoring mode, using labelled ¹³C‐PBDEs as internal standards. The limits of detection and quantification for the tri‐ to hepta‐BDEs were in the range of 0.05 to 0.5 ng/g dry weight and 0.15 to 1.8 ng/g dry weight, respectively, and 1.6 ng/g dry weight and 5.6 ng/g dry weight for deca‐BDE‐209. The proposed analytical method was applied to determine PBDE levels in sewage sludge samples collected from 19 water treatment plants located in the province of Madrid (Spain). In all of the examined samples, BDE‐100 and BDE‐154 were the main compounds found with a mean concentration of 3.9 and 2.0 ng/g, respectively. PBDEs were detected in all of the samples, and their total concentrations not considering BDE‐209 were between 3.9 and 23.0 ng/g dry weight. The dominant PBDE congener in sewage sludge was BDE‐209, which constituted 38.7 to 97.3% of the total, and showed concentration levels ranging from 8.1 to 717.2 ng/g dry weight.     

Solutions of the generalized KdV equation with time-dependent damping and dispersion

Solitary wave solutions are obtained for the generalized Korteweg-de-Vries (gKdV) equation with time-dependent damping and dispersion by using the tanh-coth method, the exp-function method and the modified sine-cosine method. These methods are useful and efficient and a variety of solitary wave solutions are obtained that possess variable coefficients.