A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

A numerical and experimental investigation of the interactions between a non‐uniform planar array of incompressible free jets

The interaction between multiple incompressible air jets has been studied numerically and experimentally. The numerical predictions have been first validated using experimental data for a single jet configuration. The spreading features of five unequal jets in the configuration of one larger central jet surrounded by four smaller equi‐distant jets, have been studied, for different lateral spacing ratios of 1.5, 2.0 and 2.5 and a central jet Reynolds number of 1.24×105 (corresponding to a Mach number of 0.16). Flow of five equal jets has also been simulated, for the sake of comparison. The jet interactions commence at an axial distance of about 3–4 diameters and complete by an axial distance of about 10 diameters for the lowest spacing ratio of 1.5. For larger spacing ratios, the length required for the start and completion of jet interaction increase. Peripheral jets bend more towards the central jet and merge at a smaller distance, when their sizes are smaller than that of the central jet. The entrainment ratio for multiple jets is higher than that for a single jet. Excellent agreement is observed between the experimental data and theoretical predictions for both mean flow field and turbulent quantities, at regions away from the jet inlet. The potential core length and initial jet development, however, are not predicted very accurately due to differences in the assumed and actual velocity profiles at the jet inlet. Copyright © 2004 John Wiley & Sons, Ltd.     

Robust handling of non‐linear constraints for GA optimization of aerodynamic shapes

A new approach to the robust handling of non‐linear constraints for GAs (genetic algorithms) optimization is proposed. A specific feature of the approach consists of the change in the conventional search strategy by employing search paths which pass through both feasible and infeasible points (contrary to the traditional approach where only feasible points may be included in a path). The method (driven by full Navier–Stokes computations) was applied to the problem of multiobjective optimization of aerodynamic shapes subject to various geometrical and aerodynamic constraints. The results demonstrated that the method retains high robustness of conventional GAs while keeping CFD computational volume to an acceptable level, which allowed the algorithm to be used in a demanding engineering environment. Copyright © 2004 John Wiley & Sons, Ltd.     

Synthesis and characterization of oxidized cellulose

Samples of oxidized cellulose (OC) with various carboxyl contents and degrees of crystallinity were obtained by the oxidation of native and mercerized cellulose with a solution of nitrogen(IV) oxide in CCl₄. A detailed characterization of these OC samples was performed. The effect of oxidation conditions (concentration of N₂O₄ in the solution and oxidation time) and starting cellulose material on OC characteristics (carboxyl, carbonyl and nitrogen content, degree of crystallinity and polymerization, surface area and swelling, and acidic properties) was investigated. Reactivity in the oxidation process was higher in mercerized cellulose than in native cellulose. The action of dilute solutions (10–15%) of N₂O₄ did not affect the degree of crystallinity of cellulose samples. Under these conditions, the oxidation took place mainly in amorphous regions and on the surface of crystallites. Oxidation in a concentrated (40%) N₂O₄ solution led to the destruction of crystallites, which increased the surface area and swelling of cellulose in water. The surface area and the swelling of OC samples increased with a decrease in the index of crystallinity. The acidic properties of OC were shown to increase with an increase of swelling in water. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 4785–4791, 2004     

Functionalization of polyethylene/silane copolymers in posttreatment reactions

7‐Octenyldimethylphenylsilane was copolymerized with ethylene via Et(Ind)₂ZrCl₂ methylaluminoxane catalyst system without loss of catalyst activity or decrease in molar mass. The comonomer contents in the polymer samples were at a level of 0.15–1.0 mol % and the reactive phenylsilane groups were posttreated to different alcoxy‐ and halosilane groups, for example, Si? F, Si? Cl, Si? OCH₃, and Si? OCH₂CH₃. The posttreatment reactions had no major effect on the molar masses or on the thermal properties (measured with differential scanning calorimetry) of the copolymers. The reaction pathways were nearly independent of the comonomer contents and the reactions reached 70–100% conversions. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 1461–1467, 2004     

Atom transfer radical polymerization of methyl methacrylate by copper(I)‐pyridine‐2‐carboximidate catalysts

Pyridine‐2‐carboximidates [methyl ( 1a ), ethyl ( 1b ), isopropyl ( 1c ), cyclopentyl ( 1d ), cyclohexyl ( 1e ), n‐octyl ( 1f ), and benzyl ( 1g )] were prepared from the reaction of 2‐cyanopyridine with the corresponding alcohols. Cyclopentyl‐substituted 1d was found to be a highly effective ligand for copper‐catalyzed atom transfer radical polymerization (ATRP) of methyl methacrylate (MMA). For example, the observed rate constant for a CuBr/ 1d catalytic system was found to be nearly twice as high as the cyclohexyl‐substituted CuBr/ 1e catalytic system [k_obs = (1.19 vs 0.56) × 10^?4 s^?1). The effects of the solvents, temperature, catalyst/initiator, and solvent/monomer ratio on the ATRP of MMA were studied systematically for the CuBr/ 1d catalytic system. The optimum condition for the ATRP of MMA was found to be a 1:2:1:400 [CuBr]_o/[ 1d ]_o/[ethyl 2‐bromoisobutyrate]_o/[MMA]_o ratio at 60 °C in veratrole solution, which yielded well‐defined poly(MMA) with a narrow molecular weight distribution of 1.14. The catalytically active copper complex 2d was isolated from the reaction of CuBr with 1d . Narrow molecular weight distributions as low as 1.06 were achieved for the CuBr/ 1d catalytic system by employing 10% of the deactivator CuBr₂. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2747–2755, 2004